Calabi–Yau 3-folds of Picard number 2 with hypersurface Cox rings

Mauz, Christian

doi:10.1007/s40879-022-00553-5

Calabi–Yau 3-folds of Picard number 2 with hypersurface Cox rings

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Published: 07 July 2022

Volume 8, pages 533–573, (2022)
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Calabi–Yau 3-folds of Picard number 2 with hypersurface Cox rings

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Christian Mauz ORCID: orcid.org/0000-0002-8211-9472¹

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Abstract

We classify all smooth Calabi–Yau threefolds of Picard number two that have a general hypersurface Cox ring.

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1 Introduction

This article contributes to the explicit classification of Calabi–Yau threefolds. Recall that a Calabi–Yau variety is a normal projective complex variety X with trivial canonical class ${\mathscr {K}}_X$, at most canonical singularities and $h^i(X, {\mathscr {O}}_X) = 0$ for . Calabi–Yau varieties form a vast and actively studied area of research, also aiming for classification results such as [12, 15, 27, 33, 34] or more recently [14, 16, 17, 38].

The present paper takes up the classification approach based on positively graded rings [9,10,11, 37] yet in the multigraded setting. We study Calabi–Yau threefolds X in terms of their Cox ring. Recall that the Cox ring of a normal projective variety X with finitely generated divisor class group is the graded algebra

We consider the case that our Calabi–Yau threefold X comes with a hypersurface Cox ring, that means that we have a K-graded presentation

$$\begin{aligned} {\mathscr {R}}(X) = R_g = {{\mathbb {C}}}[T_1, \dotsc , T_r] / \langle g \rangle \end{aligned}$$

with a homogeneous polynomial g of degree $\mu \in K$ such that $T_1, \dotsc , T_r$ form a minimal system of K-prime generators for $R_g$. In particular ${\mathscr {R}}(X) = R_g$ is a finitely generated ${{\mathbb {C}}}$-algebra, hence X is a Mori dream space in the sense of [23]. Note that a smooth Calabi–Yau variety of dimension at most three is a Mori dream space if and only if its cone of effective divisors is rational polyhedral [30]. More generally, Mori dream spaces of Calabi–Yau type are completely characterized via the singularities of their total coordinate space $\mathrm{Spec}\,{\mathscr {R}}(X)$ [22].

Following [20], we say that $R_g$ resp. g is spread if each monomial of degree $\mu $ is a convex combination over those monomials showing up in g with non-zero coefficient. Besides, we call $R_g$ general (smooth, Calabi–Yau) if g admits an open neighbourhood U in the finite dimensional vector space of all $\mu $-homogeneous polynomials such that every $h \in U$ yields a hypersurface Cox ring $R_{h}$ of a normal (smooth, Calabi–Yau) variety $X_h$ with divisor class group K. In [20] general hypersurface Cox rings were applied to the classification of smooth Fano fourfolds of Picard number two.

In dimension two Calabi–Yau varieties are K3 surfaces. Their Cox rings have been studied in [3, 4, 35], in particular describing several classes of K3 surfaces with a hypersurface Cox ring. Numbers 1, 2, 6 and 12 from Oguiso’s classification of smooth Calabi–Yau threefolds that arise as a general complete intersection in some weighted projective space [33] comprise all smooth Calabi–Yau threefolds X with having a general hypersuface Cox ring; see also [24]. Besides, Przyjalkowski and Shramov have established explicit bounds for smooth Calabi–Yau weighted complete intersections in any dimension [36].

Our main result concerns smooth Calabi–Yau threefolds of Picard number two with a hypersurface Cox ring $R_g$. Any projective variety X with class group K and Cox ring $R_g$ is encoded by $R_g$ and an ample class $u \in K$ in the sense that X occurs as the GIT quotient of the set of u-semistable points of $\mathrm{Spec}\, R_g$ by the quasitorus $\mathrm{Spec}\, {{\mathbb {C}}}[K]$. In this setting, we write and refer to the Cox ring generator degrees $w_1, \ldots , w_r \in K$, the relation degree $\mu \in K$ and an ample class $u \in K$ as specifying data of the variety X. Note , hence a hypersurface Cox ring $R_g$ of a threefold with Picard number two has six generators $w_1, \dotsc , w_6$ .

Theorem 1.1

The following table lists specifying data, $w_1, \dotsc , w_6$, $\mu $ and u in for all smooth Calabi–Yau threefolds X of Picard number two that have a spread hypersurface Cox ring.

No.	$[w_1, \dotsc , w_6]$	$ \mu $	u	No.	$[w_1, \dotsc , w_6]$	$ \mu $	u
1	$\left[ \begin{array}{cccccc} 1 &{} 1 &{} 1 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 1 &{} 1 &{} 1 \end{array}\right] $	$\left[ \begin{array}{c} 3 \\ 3 \end{array}\right] $	$\left[ \begin{array}{c} 1 \\ 1 \end{array}\right] $	15	$\left[ \begin{array}{cccccc} 1 &{} 1 &{} 1 &{} 1 &{} 0 &{} 0 \\ -1 &{} 0 &{} 0 &{} 1 &{} 1 &{} 1 \end{array}\right] $	$\left[ \begin{array}{c} 4 \\ 2 \end{array}\right] $	$ \left[ \begin{array}{c} 2 \\ 1 \end{array}\right] $
2	$\left[ \begin{array}{cccccc} 1 &{} 1 &{} 1 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 1 &{} 1 &{} 1 \\ \overline{0} &{} \overline{1} &{} \overline{2} &{} \overline{0} &{} \overline{1} &{} \overline{2} \end{array}\right] $	$\left[ \begin{array}{c} 3 \\ 3 \\ \overline{0} \end{array}\right] $	$\left[ \begin{array}{c} 1 \\ 1 \\ \overline{0} \end{array}\right] $	16	$\left[ \begin{array}{cccccc} 1 &{} 1 &{} 1 &{} 1 &{} 0 &{} 0 \\ -1 &{} 0 &{} 0 &{} 1 &{} 1 &{} 1 \end{array}\right] $	$\left[ \begin{array}{c} 4 \\ 2 \end{array}\right] $	$ \left[ \begin{array}{c} 1 \\ 2 \end{array}\right] $
3	$ \left[ \begin{array}{cccccc} 1 &{} 1 &{} 1 &{} 1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 1 &{} 1 &{} 1 &{} 1 \end{array}\right] $	$\left[ \begin{array}{c} 4 \\ 4 \end{array}\right] $	$ \left[ \begin{array}{c} 2 \\ 1 \end{array}\right] $	17	$\left[ \begin{array}{cccccc} 1 &{} 1 &{} 1 &{} 1 &{} 0 &{} 0 \\ -2 &{} 0 &{} 0 &{} 1 &{} 1 &{} 1 \end{array}\right] $	$\left[ \begin{array}{c} 4 \\ 1 \end{array}\right] $	$\left[ \begin{array}{c} 2 \\ 1 \end{array}\right] $
4	$\left[ \begin{array}{cccccc} 1 &{} 1 &{} 1 &{} 3 &{} 0 &{} 0 \\ 0 &{} 0 &{} 1 &{} 3 &{} 1 &{} 1 \end{array}\right] $	$\left[ \begin{array}{c} 6 \\ 6 \end{array}\right] $	$ \left[ \begin{array}{c} 2 \\ 1 \end{array}\right] $	18	$\left[ \begin{array}{cccccc} 1 &{} 1 &{} 1 &{} 1 &{} 0 &{} 0 \\ -2 &{} 0 &{} 0 &{} 1 &{} 1 &{} 1 \end{array}\right] $	$\left[ \begin{array}{c} 4 \\ 1 \end{array}\right] $	$ \left[ \begin{array}{c} 1 \\ 2 \end{array}\right] $
5	$\left[ \begin{array}{cccccc} 1 &{} 1 &{} 1 &{} 0 &{} 0 &{} -1 \\ 0 &{} 0 &{} 0 &{} 1 &{} 1 &{} 1 \end{array}\right] $	$\left[ \begin{array}{c} 2 \\ 3 \end{array}\right] $	$ \left[ \begin{array}{c} 1 \\ 1 \end{array}\right] $	19	$\left[ \begin{array}{cccccc} 1 &{} 1 &{} 1 &{} 2 &{} 1 &{} 0 \\ 0 &{} 0 &{} 0 &{} 1 &{} 1 &{} 1 \end{array}\right] $	$\left[ \begin{array}{c} 6 \\ 3 \end{array}\right] $	$ \left[ \begin{array}{c} 3 \\ 1 \end{array}\right] $
6	$\left[ \begin{array}{cccccc} 1 &{} 1 &{} 1 &{} 0 &{} 0 &{} -2 \\ 0 &{} 0 &{} 0 &{} 1 &{} 1 &{} 1 \end{array}\right] $	$\left[ \begin{array}{c} 1 \\ 3 \end{array}\right] $	$ \left[ \begin{array}{c} 1 \\ 1 \end{array}\right] $	20	$\left[ \begin{array}{cccccc} 1 &{} 1 &{} 1 &{} 4 &{} 1 &{} 0 \\ 0 &{} 0 &{} 0 &{} 2 &{} 1 &{} 1 \end{array}\right] $	$\left[ \begin{array}{c} 8 \\ 4 \end{array}\right] $	$ \left[ \begin{array}{c} 3 \\ 1 \end{array}\right] $
7	$\left[ \begin{array}{cccccc} 1 &{} 1 &{} 1 &{} 0 &{} 0 &{} -3 \\ 0 &{} 0 &{} 0 &{} 1 &{} 1 &{} 1 \\ \overline{0} &{} \overline{1} &{} \overline{2} &{} \overline{1} &{} \overline{2} &{} \overline{0} \end{array}\right] $	$\left[ \begin{array}{c} 0 \\ 3 \\ \overline{0} \end{array}\right] $	$ \left[ \begin{array}{c} 1 \\ 1 \\ \overline{0} \end{array}\right] $	21	$\left[ \begin{array}{cccccc} 1 &{} 1 &{} 1 &{} 5 &{} 2 &{} 0 \\ 0 &{} 0 &{} 0 &{} 2 &{} 1 &{} 1 \end{array}\right] $	$\left[ \begin{array}{c} 10 \\ 4 \end{array}\right] $	$ \left[ \begin{array}{c} 3 \\ 1 \end{array}\right] $
8	$\left[ \begin{array}{cccccc} 1 &{} 1 &{} 1 &{} 0 &{} 0 &{} -3 \\ 0 &{} 0 &{} 0 &{} 2 &{} 3 &{} 1 \end{array}\right] $	$\left[ \begin{array}{c} 0 \\ 6 \end{array}\right] $	$ \left[ \begin{array}{c} 1 \\ 1 \end{array}\right] $	22	$\left[ \begin{array}{cccccc} 1 &{} 1 &{} 2 &{} 5 &{} 1 &{} 0 \\ 0 &{} 0 &{} 0 &{} 2 &{} 1 &{} 1 \end{array}\right] $	$\left[ \begin{array}{c} 10 \\ 4 \end{array}\right] $	$ \left[ \begin{array}{c} 3 \\ 1 \end{array}\right] $
9	$\left[ \begin{array}{cccccc} 1 &{} 1 &{} 1 &{} 1 &{} 0 &{} 0 \\ -2 &{} 0 &{} 0 &{} 0 &{} 1 &{} 1 \end{array}\right] $	$\left[ \begin{array}{c} 4 \\ 0 \end{array}\right] $	$ \left[ \begin{array}{c} 1 \\ 1 \end{array}\right] $	23	$\left[ \begin{array}{cccccc} 1 &{} 1 &{} 2 &{} 7 &{} 3 &{} 0 \\ 0 &{} 0 &{} 0 &{} 2 &{} 1 &{} 1 \end{array}\right] $	$\left[ \begin{array}{c} 14 \\ 4 \end{array}\right] $	$ \left[ \begin{array}{c} 4 \\ 1 \end{array}\right] $
10	$\left[ \begin{array}{cccccc} 1 &{} 1 &{} 1 &{} 3 &{} 0 &{} 0 \\ -2 &{} 0 &{} 0 &{} 0 &{} 1 &{} 1 \end{array}\right] $	$\left[ \begin{array}{c} 6 \\ 0 \end{array}\right] $	$ \left[ \begin{array}{c} 1 \\ 1 \end{array}\right] $	24	$\left[ \begin{array}{cccccc} 1 &{} 1 &{} 1 &{} 2 &{} 1 &{} 0 \\ -2 &{} 0 &{} 0 &{} 0 &{} 1 &{} 1 \end{array}\right] $	$\left[ \begin{array}{c} 6 \\ 0 \end{array}\right] $	$\left[ \begin{array}{c} 2 \\ 1 \end{array}\right] $
11	$\left[ \begin{array}{cccccc} 1 &{} 1 &{} 1 &{} 1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 1 &{} 1 &{} 1 \end{array}\right] $	$\left[ \begin{array}{c} 4 \\ 3 \end{array}\right] $	$ \left[ \begin{array}{c} 2 \\ 1 \end{array}\right] $	25	$\left[ \begin{array}{cccccc} 1 &{} 1 &{} 1 &{} 1 &{} 3 &{} 0 \\ -2 &{} 0 &{} 0 &{} 0 &{} 1 &{} 1 \end{array}\right] $	$\left[ \begin{array}{c} 7 \\ 0 \end{array}\right] $	$\left[ \begin{array}{c} 4 \\ 1 \end{array}\right] $
12	$\left[ \begin{array}{cccccc} 1 &{} 1 &{} 1 &{} 1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 1 &{} 1 &{} 1 \end{array}\right] $	$\left[ \begin{array}{c} 4 \\ 3 \end{array}\right] $	$ \left[ \begin{array}{c} 1 \\ 2 \end{array}\right] $	26	$\left[ \begin{array}{cccccc} 2 &{} 1 &{} 1 &{} 1 &{} 3 &{} 0 \\ -2 &{} 0 &{} 0 &{} 0 &{} 1 &{} 1 \end{array}\right] $	$\left[ \begin{array}{c} 8 \\ 0 \end{array}\right] $	$\left[ \begin{array}{c} 4 \\ 1 \end{array}\right] $
13	$\left[ \begin{array}{cccccc} 1 &{} 1 &{} 1 &{} 3 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 2 &{} 1 &{} 1 \end{array}\right] $	$\left[ \begin{array}{c} 6 \\ 4 \end{array}\right] $	$ \left[ \begin{array}{c} 1 \\ 1 \end{array}\right] $	27	$\left[ \begin{array}{cccccc} 1 &{} 1 &{} 2 &{} 5 &{} 1 &{} 0 \\ 0 &{} 0 &{} 1 &{} 3 &{} 1 &{} 1 \end{array}\right] $	$\left[ \begin{array}{c} 10 \\ 6 \end{array}\right] $	$ \left[ \begin{array}{c} 3 \\ 1 \end{array}\right] $
14	$\left[ \begin{array}{cccccc} 1 &{} 1 &{} 1 &{} 3 &{} 0 &{} 0 \\ -1 &{} 0 &{} 0 &{} 1 &{} 1 &{} 1 \end{array}\right] $	$\left[ \begin{array}{c} 6 \\ 2 \end{array}\right] $	$ \left[ \begin{array}{c} 1 \\ 1 \end{array}\right] $	28	$\left[ \begin{array}{cccccc} 1 &{} 1 &{} 2 &{} 5 &{} 1 &{} 0 \\ 0 &{} 0 &{} 1 &{} 3 &{} 1 &{} 1 \end{array}\right] $	$\left[ \begin{array}{c} 10 \\ 6 \end{array}\right] $	$ \left[ \begin{array}{c} 3 \\ 2 \end{array}\right] $
				29	$\left[ \begin{array}{cccccc} 1 &{} 1 &{} 1 &{} 4 &{} 1 &{} 0 \\ -1 &{} 0 &{} 0 &{} 1 &{} 1 &{} 1 \end{array}\right] $	$\left[ \begin{array}{c} 8 \\ 2 \end{array}\right] $	$ \left[ \begin{array}{c} 5 \\ 1 \end{array}\right] $
				30	$\left[ \begin{array}{cccccc} 1 &{} 2 &{} 1 &{} 1 &{} 1 &{} 0 \\ -1 &{} -1 &{} 0 &{} 0 &{} 1 &{} 1 \end{array}\right] $	$\left[ \begin{array}{c} 6 \\ 0 \end{array}\right] $	$ \left[ \begin{array}{c} 2 \\ 1 \end{array}\right] $

Moreover, each of the items 1 to 30 even defines a general smooth Calabi–Yau hypersurface Cox ring and thus provides the specifying data for a whole family of smooth Calabi–Yau threefolds. Any two smooth Calabi–Yau threefolds of Picard number two with specifying data from distinct items of the table are not isomorphic to each other.

Note that the varieties from Theorem 1.1 constitute a finite number of families. For recent general results on boundedness of Calabi–Yau threefolds we refer to [13] as well as [39] for the case of Picard number two.

Hypersurfaces in toric Fano varieties form a rich source of examples for Calabi–Yau varieties, e.g. [1, 6, 7]. Theorem 1.1 comprises several varieties of this type.

Remark 1.2

Any Mori dream space X can be embedded into a projective toric variety by choosing a graded presentation of its Cox ring ${\mathscr {R}}(X)$; see [5, Section 3.2.5] for details. The following table shows for which varieties X from Theorem 1.1 the presentation ${\mathscr {R}}(X) = R_g$ gives rise to an embedding into a (possibly singular) toric Fano variety. Observe that in our situation this simply means .

1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
✓	✓	✗	✗	✓	✓	✗	✗	✗	✗	✓	✗	✓	✓	✓
16	17	18	19	20	21	22	23	24	25	26	27	28	29	30
✗	✓	✗	✓	✓	✓	✓	✓	✗	✗	✗	✗	✓	✓	✗

2 Background on Mori dream spaces

A Mori dream space is an irreducible normal projective variety X with finitely generated divisor class group and finitely generated Cox ring ${\mathscr {R}}(X)$. As observed by Hu and Keel [23], Mori dream spaces are characterized by their eponymous feature, which is optimal behavior with respect to the Mori program [31, 32]. In this section we gather basic facts on the combinatorial description of Mori dream spaces from [5]. The ground field ${{\mathbb {K}}}$ is algebraically closed and of characteristic zero.

Let us first recall some terminology concerning graded algebras. Consider a finitely generated abelian group K and an integral ${{\mathbb {K}}}$-algebra $R = \bigoplus _{w \in K} R_w$ with a K-grading. A non-zero non-unit $f \in R$ is K-prime if it is homogeneous and $f \,{|}\, gh$ with homogeneous elements $g, h \in R$ implies $f \,{|}\, g$ or $f \,{|}\, h$. We say that R is K-factorial or that the K-grading on R is factorial if any homogeneous non-zero non-unit is a product of K-prime elements. Fix a system $f_1, \dotsc , f_r \in R$ of pairwise non-associated K-prime generators for R. The effective cone and the moving cone of R in the rational vector space associated with K are

This definition does not depend on the choice of $f_1, \dotsc , f_r$. The K-grading on R is called pointed if $R_0 = {{\mathbb {K}}}$ holds and contains no line; it is called almost free if any $r-1$ of generate K as a group.

Moreover, by an abstract Cox ring we mean a K-graded algebra R such that

R is normal, integral and finitely generated,
R has only constant homogeneous units, the K-grading is almost free, pointed and factorial, and
the moving cone is of full dimension in $K_{{\mathbb {Q}}}$.

The Cox ring of a Mori dream space always satisfies the conditions of an abstract Cox ring. Vice versa, we can produce Mori dream spaces from abstract Cox rings using the following construction [5, Construction 3.2.1.3].

Construction 2.1

Let R be an abstract Cox ring and consider the action of the quasitorus $H = \mathrm{Spec}\, {{\mathbb {K}}}[K]$ on the affine variety ${\overline{X}} = \mathrm{Spec}\, R$. For every GIT-cone $\lambda \in \Lambda (R)$ with , we set

Then $X = X(\lambda )$ is normal, projective and of dimension . The divisor class group and the Cox ring of X are given as

Moreover, the cones of effective, movable, semiample and ample divisor classes of X are given in as

See [5, Theorem 3.2.1.4] for the description of the Cox ring and [5, Proposition 3.3.2.9] for the description of the cones of effective, movable, semiample and ample divisors. Moreover, [5, Theorem 3.2.1.9] guarantees that indeed all Mori dream spaces arise from Construction 2.1

Table 1 Binomials used to ensure primeness of $T_1, \dotsc , T_6 \in R_g$ in the proof of Theorem 1.1

Full size table

Choosing homogeneous generators for an abstract Cox ring gives rise to a closed embedding into a projective toric variety [5, Construction 3.2.5.7].

Construction 2.2

In the situation of Construction 2.1, consider a graded presentation

$$\begin{aligned} R = {\mathbb {K}}[T_1, \dotsc , T_r] / {\mathfrak {a}} \end{aligned}$$

where $T_1, \dotsc , T_r$ define pairwise non-associated K-primes in R and ${\mathfrak {a}} \subseteq S = {\mathbb {K}}[T_1, \dotsc , T_r]$ is a homogeneous ideal. The GIT-fan $\Lambda (S)$ w.r.t. the diagonal H-action on refines the GIT-fan $\Lambda (R)$. Let $\tau \in \Lambda (S)$ with . Running Construction 2.1 for S and $\tau $ yields a projective toric variety Z fitting in the following diagram:

The embedding $\imath :X \rightarrow Z$ is neat, i.e. it is a closed embedding, the torus invariant prime divisors on Z restrict to pairwise different prime divisors on X and the induced pullback of divisor class groups is an isomorphism.

Varieties arising from abstract Cox rings admit the following description for their local behavior [5, Corollary 3.3.1.9] and [5, Corollary 3.3.1.12].

Proposition 2.3

In the situation of Construction 2.2 the following hold:

(i)
X is ${{\mathbb {Q}}}$-factorial if and only if $\lambda $ is of full dimension.
(i)
X is smooth if and only if ${\overline{X}}{}^\mathrm {ss}$ is smooth and $X \subseteq Z^{\mathrm{reg}}$ holds.

Furthermore, for hypersurface Cox rings we have an explicit formula for the anticanonical class [5, Proposition 3.3.3.2].

Proposition 2.4

Consider the situation of Construction 2.2. If ${\mathfrak {a}} = \langle g \rangle $ holds, then the anticanonical class of X is given in as

We call an irreducible normal variety X weakly Calabi–Yau if its canonical class ${\mathscr {K}}_X$ vanishes. For varieties with hypersuface Cox ring this notion only depends on the generator degrees and the relation degree. Moreover, it turns out that smooth weakly Calabi–Yau hypersurfaces are Calabi–Yau varieties in the strong sense.

Remark 2.5

In the situation of Construction 2.2 assume ${\mathfrak {a}} = \langle g \rangle $.

(i)
From Proposition 2.4 we deduce that X is weakly Calabi–Yau if and only if $\mu = w_1 + \cdots + w_r$ holds. In particular, $\mu $ lies in the relative interior of whenever X is weakly Calabi–Yau.
(ii)
If X is weakly Calabi–Yau, then Proposition 2.4 shows that X is an anticanonical hypersurface of a projective toric variety Z as in Construction 2.2. If, in addition, X is smooth, then Proposition 2.3 allows us to apply [2, Proposition 6.1]. From this we infer $h^i(X, {\mathscr {O}}_X) = 0$ for all , hence X is Calabi–Yau.

Let us briefly recall the notion of flops [25, 26] in our situation. A Weil divisor D on a variety X is said to be relatively ample w.r.t. a morphism $\varphi :X \rightarrow Y$ of varieties, or just $\varphi $-ample, if there is an open affine covering $Y = \bigcup V_i$ such that D restricts to an ample divisor on each $\varphi ^{-1}(V_i)$. Moreover, a proper birational morphism $\varphi :X \rightarrow Y$ of normal varieties is called extremal, if X is ${{\mathbb {Q}}}$-factorial and for each two Cartier divisors $D_1, D_2$ on X there are $a_1, a_2 \in {{\mathbb {Z}}}$, not both zero, such that $a_1 D_1 - a_2 D_2$ is linearly equivalent to the pullback $\varphi ^* C$ of some Cartier divisor C on Y. A birational map $\psi :X^- \dashrightarrow X^+$ of ${{\mathbb {Q}}}$-factorial weakly Calabi–Yau varieties is a flop if it fits into a commutative diagram

where and are small proper birational morphims, $\varphi ^-$ is extremal, and there is a Weil divisor D on $X^-$ such that $-D$ is $\varphi ^-$-ample and the proper transform of D on $X^+$ is $\varphi ^+$-ample.

Later we will use that weakly Calabi–Yau Mori dream spaces of Picard number two that share a common Cox ring are connected by flops; for convenience we give a direct proof here.

Proposition 2.6

Let R be an abstract Cox ring with grading group K of rank two and $\lambda , \eta \in \Lambda (R)$ full-dimensional cones with . Consider the varieties $X(\lambda )$ and $X(\eta )$ arising from Construction 2.1. If the canonical class of $X(\lambda )$ is trivial, then there is a sequence of flops

$$\begin{aligned} X(\lambda ) \dashrightarrow X_1 \dashrightarrow \cdots \dashrightarrow X_k \dashrightarrow X(\eta ). \end{aligned}$$

We study the toric setting first. Consider $S = {{\mathbb {K}}}[T_1, \dotsc , T_r]$ with a linear, pointed, almost free grading of an abelian group K of rank two and the associated action of the quasitorus $H = \mathrm{Spec}\, {{\mathbb {K}}}[K]$ on . Let us recall some facts about toric varieties arising from GIT-cones as treated e.g. in [5, Chapters 2–3]. The degree homomorphism $Q:{{\mathbb {Z}}}^r \rightarrow K$, gives rise to a pair of mutually dual exact sequences:

For a given GIT-cone $\tau \in \Lambda (S)$ with , the associated toric variety is described by the fan $\Sigma (\tau )$, which is obtained from a collection of faces of the positive orthant as follows:

In particular all such fans share the same one-skeleton consisting of the pairwise different rays generated by $v_1, \dotsc , v_r$ where . Moreover, we denote $Z_{\gamma _0}$ for the affine toric variety associated with the lattice cone . The covering of Z by affine toric charts then formulates as

Lemma 2.7

Let $\tau _1, \tau _2 \in \Lambda (S)$ with . Then for any , we have

$$\begin{aligned} P(\gamma _2^*) \subseteq P(\gamma _1^*) \quad \Longleftrightarrow \quad \gamma _1 \subseteq \gamma _2. \end{aligned}$$

Proof

The implication “$\Leftarrow $” is clear. We show “$\Rightarrow $”. Note that the cones $P(\gamma _1^*) \in \Sigma (\tau _1)$ and $P(\gamma _2^*) \in \Sigma (\tau _2)$ both live in lattice fans having precisely $v_1, \dotsc , v_r$ as primitive ray generators. Thus, for $j = 1, 2$ and any $v_i$ we have

$$\begin{aligned} v_i \in P(\gamma _j^*) \; \Longleftrightarrow \; {{\mathbb {Q}}}_{\geqslant 0}\, v_i \text { is an extremal ray of } P(\gamma _j^*) \; \Longleftrightarrow \; e_i \in \gamma _j^*. \end{aligned}$$

From this we infer that $P(\gamma _2^*) \subseteq P(\gamma _1^*)$ implies $\gamma _2^* \subseteq \gamma _1^*$. This in turn means $\gamma _1 \subseteq \gamma _2$.$\square $

Let be full-dimensional GIT-cones with intersecting in a common ray .

Consider the projective toric varieties associated with and denote , and for the describing fans. Moreover, the inclusions of the respective semistable points induce proper birational toric morphims , described by the refinements of fans $\Sigma ^- \!\preceq \Sigma ^0$ and respectively. This yields a small birational map as shown in the diagram

Lemma 2.8

Let $-D$ be an ample divisor on , then D regarded as a divisor on $Z^+$ is $\varphi ^+$-ample.

Proof

By suitably applying an automorphism of K and relabeling $w_1, \dotsc , w_r \in K$ we achieve counter-clockwise ordering, i.e.

and for all , . Moreover, we name the indices of the weights that approximate $\tau ^0$ from the outside

The geometric constellation of $w_1, \dotsc , w_r$ in ${{\mathbb {Q}}}^2$ directly yields that the set of minimal cones of is

where . The corresponding cones $P(\gamma _0)^*$ are precisely the maximal cones of , in particular the associated toric charts $Z_{\gamma _0}$ form an open affine covering of . We show that D is ample on each open subset $(\varphi ^+)^{-1}(Z_{\gamma _0})$ of .

First, note that $\varphi ^+$ is an isomorphism over the affine toric charts of $Z^0$ associated with the common minimal cones of and , namely all $Z_{\gamma _{i,j}}$ where $i \leqslant i^+$ and . In particular, each preimage $(\varphi ^+)^{-1}(Z_{\gamma _{i,j}})$ is affine. Since $Z^+$ is ${{\mathbb {Q}}}$-factorial by Proposition 2.3, the divisor D is ${{\mathbb {Q}}}$-Cartier thus restricts to an ample divisor on any open affine subvariety of .

It remains to consider the charts of $Z^0$ defined by the faces of the form $\gamma _j$. Let us fix some $i^-< j < i^+$. The minimal cones with $\gamma _j \subseteq \gamma _0$ are precisely those of the form $\gamma _{j,i}$ where $i \geqslant j$. As the toric morphism $\varphi ^+$ is described by the refinement , Lemma 2.7 yields

Note that $U \subseteq Z^+$ is an open toric subset and the maximal cones of the associated subfan $\Sigma '$ of $\Sigma ^+$ are precisely the cones $P(\gamma _{j,i}^*)$ where . This shows that the rays of $\Sigma '$ are the rays of $\Sigma ^+$ minus $\varrho _j$. Thus, the divisor class group of U is given by with the projection corresponding to the restriction of divisor classes

Taking ${{\,\mathrm{rank}\,}}K = 2$ into account, we may choose suitable coordinates leading to an isomorphism such that for any the restriction $\imath ^*(w)$ to shares the same sign with . Graphically this means that the sign of is positive if w lies above the ray $\tau ^0$ and negative if w lies below .

Since we know the maximal cones of , we may compute the ample cone of U as

Note that lies below $\tau $, thus the class of $-D$ (regarded on $Z^+)$ restricted to U is negative, hence . In other words, D is ample on U. Altogether, we conclude that D is $\varphi ^+$-ample.$\square $

Proof of Proposition 2.6

First, we deal with the case that $\lambda $ and $\eta $ intersect in a common ray . Consider a K-graded presentation

$$\begin{aligned} R = {\mathbb {K}}[T_1, \dotsc , T_r] / {\mathfrak {a}} \end{aligned}$$

where $T_1, \dotsc , T_r$ define pairwise non-associated K-primes in R and ${\mathfrak {a}} \subseteq S = {\mathbb {K}}[T_1, \dotsc , T_r]$ is a homogeneous ideal. The GIT-fan $\Lambda (S)$ w.r.t. the H-action on S refines the GIT-fan $\Lambda (R)$. We may choose such that

The toric morphisms $\varphi _Z^-$, $\varphi ^+_Z$ arising from the face relations of GIT-cones are compatible with the toric morphisms arising from $\varrho \preceq \lambda , \eta $ as shown in the following diagram where the vertical arrows are neat embeddings as in Construction 2.2

We claim that the resulting birational map $\psi :X(\lambda ) \dashrightarrow X(\eta )$ is a flop. First, observe that $X^-$ is ${{\mathbb {Q}}}$ factorial by Proposition 2.3 (i) and that $\varphi ^-, \varphi ^+$ are small birational morphisms; see [5, Remark 3.3.3.4].

We show that $\varphi ^-$ is extremal. Let $D_1, D_2$ be Cartier divisors on X. If $D_1, D_2$ lie on a common ray in , we find $a_1, a_2 \in {{\mathbb {Z}}}$, not both zero, such that $a_1 D_1 - a_2 D_2$ is principal, thus linearly equivalent to the pullback of any principal divisor on Y. Otherwise, the subgroup spanned by is of rank two and thus of finite index in . Hence, for any Cartier divisor C on Y, we find $b \in {{\mathbb {Z}}}_{> 0}$ such that $b [(\varphi ^-)^*(C)] \in G$, i.e., $a_1 [D_1] - a_2[D_2] = b [(\varphi ^-)^*(C)]$ for some $a_1, a_2 \in {{\mathbb {Z}}}$. In other words, $a_1 D_1 - a_2 D_2$ is linearly equivalent to the pullback of bC. If C is not principal, then $(\varphi ^-)^* C$ is not principal either, thus at least one of $a_1, a_2$ is non-zero.

Let $D_Z$ be a torus invariant divisor on $Z(\tau ^-)$ such that $-D_Z$ is ample for $Z(\tau ^-)$. Since $X(\lambda ) \subseteq Z(\tau ^-)$ is neatly embedded, we may restrict $D_Z$ to a divisor $D_X$ on $X(\lambda )$. Note that $-D_X$ is ample since $-D_Z$ is so. In particular $-D_X$ is $\varphi ^-$-ample. Lemma 2.8 yields that $D_Z$ is $\varphi _Z^+$-ample. Let $U \subseteq Z(\varrho )$ be an affine open subset such that $D_Z$ is ample on

The further restriction of $D_Z$ from V to $V \cap X(\eta )$ is still ample. In other words, $D_X$ restricted to $(\varphi ^+)^{-1}(X(\tau ) \cap U)$ is ample. We conclude that $D_X$ is $\varphi ^+$-ample.

Altogether $\psi :X(\lambda ) \dashrightarrow X(\eta )$ is a flop.

In the general case we find full-dimensional GIT-cones $\lambda = \eta _1, \dotsc , \eta _k = \eta $ where holds for all i and each intersection $\eta _i \cap \eta _{i+1}$ is a ray of $\Lambda (R)$. According to the preceeding discussion, we may successively construct the desired sequence of flops.$\square $

3 Combinatorial constraints on smooth hypersurface Cox rings

The proof of Theorem 1.1 basically uses the combinatorial framework for the classification of smooth Mori dream spaces of Picard number two with hypersurface Cox ring established in [20, Section 5]; see also [29]. Let us recall the notation from there and slightly extend it to address the torsion subgroup of the grading group explicitly. We also present the accompanying toolkit. Moreover we add some new tools for dealing with torsion.

We work over an algebraically closed field ${{\mathbb {K}}}$ of characteristic zero.

Setting 3.1

Consider where $\Gamma $ is some finite abelian group of order t, a K-graded algebra R and $X = X(\lambda )$, where $\lambda \in \Lambda (R)$ with , as in Construction 2.1. Assume that we have an irredundant K-graded presentation

$$\begin{aligned} R = R_g = {{\mathbb {K}}}[T_1, \ldots , T_r] / {\langle g \rangle } \end{aligned}$$

such that the $T_i$ define pairwise nonassociated K-primes in R. Write , for the degrees in K. According to the presentation we denote

$$\begin{aligned} w_i = (u_i, \zeta _i), \quad \mu = (\alpha , \theta ), \qquad u_i, \alpha \in {{\mathbb {Z}}}^2,\;\; \zeta _i, \theta \in \Gamma . \end{aligned}$$

Similarly the degree matrix $Q = [w_1, \dotsc , w_r]$ is divided into a free part $Q^0$ and a torsion part $Q^{\mathrm {tor}}$, i.e., we set

$$\begin{aligned} Q^0 = \begin{bmatrix} u_1&\cdots&u_r \end{bmatrix}, \quad Q^{\mathrm {tor}} = \begin{bmatrix} \zeta _1&\cdots&\zeta _r \end{bmatrix}. \end{aligned}$$

Regarded as elements of $K_{{\mathbb {Q}}}$ we identify $w_i$ with $u_i$ and $\mu $ with $\alpha $. Suitably numbering $w_1, \ldots , w_r$, we ensure counter-clockwise ordering, that means that we always have

Note that each ray of $\Lambda (R)$ is of the form , but not vice versa. We assume X to be ${{\mathbb {Q}}}$-factorial. According to Proposition 2.3 this means . Then the effective cone of X is uniquely decomposed into three convex sets,

where $\lambda ^-$ and $\lambda ^+$ are convex polyhedral cones not intersecting and consists of the origin.

Remark 3.2

Setting 3.1 is respected by orientation preserving automorphisms of K. If we apply an orientation reversing automorphism of K, then we regain Setting 3.1 by reversing the numeration of $w_1, \ldots , w_r$. Moreover, we may interchange the numeration of $T_i$ and $T_j$ if $w_i$ and $w_j$ share a common ray without affecting Setting 3.1. We call these operations admissible coordinate changes. Note that any automorphism of ${{\mathbb {Z}}}^2$ naturally extends to an automorphism of acting as the identity on $\Gamma $.

We state an adapted version of [20, Proposition 2.4] locating the relation degree.

Proposition 3.3

In the situation of Setting 3.1 we have .

A further important observation [29, Corollary 2.5.10] is that the GIT-fan structure of $R_g$ can be read of from the geometric constellation of $w_1, \dotsc , w_r$ and $\mu $.

Proposition 3.4

Situation as in Setting 3.1. Assume that $X(\lambda )$ is locally factorial and R is a spread hypersurface Cox ring. Then the full-dimensional cones of $\Lambda (R)$ are precisely the cones where $\varrho _i \ne \varrho _j$ and one of the following conditions is satisfied:

(i):: $\mu \in \varrho _i$ holds, $\varrho _i$ contains at least two generator degrees and $\eta ^\circ $ contains no generator degree,
(ii):: $\mu \in \varrho _j$ holds, $\varrho _j$ contains at least two generator degrees and $\eta ^\circ $ contains no generator degree,
(iii):: $\mu \in \eta ^\circ $ holds and there is at most one , which lays on the ray through $\mu $,
(iv):: $\mu \notin \eta $ holds and $\eta ^\circ $ contains no generator degrees.

Recall that a point $x \in X$ of a variety X is factorial if its stalk ${\mathscr {O}}_{X,x}$ admits unique factorization. We call X locally factorial if every point $x \in X$ is factorial. In particular smooth varieties are locally factorial.

The following lemmas are crucial in gaining constraints on specifying data of hypersurface Cox rings. For proofs see [20, Lemmas 5.6, 5.7] and [29, Lemma 2.5.4].

Lemma 3.5

Situation as in Setting 3.1. Let i, j with . If $X = X(\lambda )$ is locally factorial, then either $w_i,w_j$ generate K as a group, or g has precisely one monomial of the form , where $l_i+l_j > 0$.

Lemma 3.6

Let $X = X(\lambda )$ be as in Setting 3.1 and let $1 \leqslant i< j < k \leqslant r$. If X is locally factorial, then $w_i, w_j, w_k$ generate K as a group provided that one of the following holds:

(i):: , $w_k \in \lambda ^+$ and g has no monomial of the form $T_k^{l_k}$,
(ii):: , $w_j, w_k \in \lambda ^+$ and g has no monomial of the form $T_i^{l_i}$,
(iii):: , , .

Moreover, if (iii) holds, then g has a monomial of the form $T_j^{l_j}$ where $l_j$ is divisible by the order of the factor group $K / \langle w_i, w_k \rangle $. In particular $l_j$ is a multiple of .

Lemma 3.7

Assume $u, w_1, w_2$ generate the abelian group . If $w_i = a_i w$ holds with a primitive $w \in {\mathbb {Z}}^2$ and $a_i \in {\mathbb {Z}}$, then (u, w) is a basis for ${\mathbb {Z}}^2$ and u is primitive.

Now we present some structural observations which prove useful at different places inside the proof of Theorem 1.1 when we deal with specific configurations of generator and relation degrees.

Lemma 3.8

In Setting 3.1, assume that $X = X(\lambda )$ is locally factorial and $R_g$ a spread hypersurface Cox ring. If $w_i$ lies on the ray through $\mu $, then g has a monomial of the form $T_i^{l_i}$ where $l_i \geqslant 2$.

Lemma 3.9

In Lemma 3.1 assume that and hold. Let $\Omega $ denote the set of two-dimensional cones $\eta \in \Lambda (R)$ with .

(i)
If $X(\eta )$ is locally factorial for some $\eta \in \Omega $, then is a regular cone and every $u_i$ on the boundary of is primitive.
(ii)
If $X(\eta )$ is locally factorial for all $\eta \in \Omega $, then, for any , we have $u_i = u_1 + u_r$ or g has a monomial of the form $T_i^{l_i}$.

Lemma 3.10

Situation as in Setting 3.1. If we have $w_2 = w_3$ and $\mu \in \varrho _2$, then $w_4 \in \varrho _2$ holds.

Proof

Suppose $w_4 \notin \varrho _2$. Then every monomial of g not being divisible by $T_1$ is of the form $T_2^{l_2} T_3^{l_3}$ where $l_2 + l_3 > 0$. Since g is prime, thus not divisible by $T_1$, at least one such monomial occurs with non-zero coefficient in g. From $w_2 = w_3$ we deduce that is a classical homogeneous polynomial in $T_2$, $T_3$, thus admits a presentation $g_1 = \ell _1 \cdots \ell _m$ where $\ell _1, \dotsc , \ell _m$ are linear forms in $T_2$ and $T_3$. Here $w_2 = w_3$ ensures that $\ell _1, \dotsc , \ell _m$ are homogeneous w.r.t. the K-grading. Observe $m > 1$ as the presentation of R is irredundant. We conclude that $g_1$ is not K-prime, hence $T_1 \in R$ is not K-prime either. A contradiction.$\square $

We have to bear in mind that the divisor class group of a smooth Calabi–Yau threefold X is not necessarily torsion-free. The following lemmas show that in the case of a hypersurface Cox ring the order of the torsion subgroup is bounded in terms of monomials of the relation degree. A first important constraint is that the torsion subgroup of K is cyclic.

Lemma 3.11

Situation as in Setting 3.1. If $X = X(\lambda )$ is locally factorial and $\mu \notin \lambda $, then $K \cong {{\mathbb {Z}}}^2$ holds.

Proof

We have for some generator degrees $w_i, w_j$ lying on the boundary of $\lambda $. Due to $\mu \notin \lambda $, there is no monomial $T_i^{l_i} T_j^{l_j}$ of degree $\mu $. Lemma 3.5 yields that K is generated by $w_i, w_j$. Since ${{\,\mathrm{rank}\,}}(K) = 2$, this implies .$\square $

Lemma 3.12

Situation as in Setting 3.1. If X is locally factorial, then holds.

Proof

Both $\lambda ^+$ and $\lambda ^-$ contain at least two Cox ring generator degrees. This allows us to choose $w_i, w_j, w_k$ such that Lemma 3.6 applies. This ensures that K is generated by three elements. By Setting 3.1 we have , thus K is as claimed.$\square $

Lemma 3.13

Situation as in Setting 3.1. Let $1 \leqslant i, j \leqslant n$ with . If $X = X(\lambda )$ is locally factorial and $\mu \in \lambda $ holds, then there is a monomial $T_i^{l_i} T_j^{l_j}$ of degree $\mu $ where $l_i + l_j > 0$.

Proof

Since g is $\mu $-homogeneous, we are done when g has a monomial of the form $T_i^{l_i} T_j^{l_j}$ with $l_i + l_j > 0$.

We assume that g has no monomial of the form . Then $\varrho _i$ and $\varrho _j$ both are GIT-rays, thus none of $w_i, w_j$ lies in . This forces . Then Lemma 3.5 tells us that $w_i, w_j$ generate K as a group. Using we deduce that $\mu $ is a positive integral combination over $w_i, w_j$, i.e., there exists a monomial as desired.$\square $

Lemma 3.14

Situation as in Setting 3.1. Let $1 \leqslant i, j, k \leqslant r$ such that $w_i, w_j, w_k$ generate K as a group, and . If X is locally factorial, then $t \,{|}\, l_k$ holds for any monomial $T_i^{l_i} T_k^{l_k}$ of degree $\mu $.

Proof

Because of , we find an integral -matrix S with $Su_i = e_1$ and $S u_j = e_2$. Then again, the group automorphism

is an admissible orientation preserving coordinate change such that $\zeta _i = \zeta _j = 0$. Moreover we may assume $\mu \in \lambda $; otherwise Lemma 3.11 yields $t = 1$ and there is nothing left to show. This allows us to use Lemma 3.13. From this we infer that $\mu = (\alpha , \theta )$ is an integral positive combination over $w_i, w_j$, thus $\theta = 0$. Since $w_i, w_j, w_k$ generate K as a group, $\zeta _k$ is a generator for $\Gamma $. Using $\zeta _i = 0$ we obtain $l_k \zeta _k = \theta = 0$ whenever $T_i^{l_i} T_k^{l_k}$ is of degree $\mu $. This implies $t \,{|}\, l_k$.$\square $

Lemma 3.15

Situation as in Setting 3.1. Assume that $X = X(\lambda )$ is locally factorial. If and $\alpha = l_k u_k$ holds, then $t \,{|}\, l_k$.

Proof

Lemma 3.6 yields that $w_1, w_k, w_r$ generate K as a group. Besides $T_k^{l_k}$ is of degree $\mu $ by Lemma 3.8. Now Lemma 3.14 tells us $t \,{|}\, l_k$.$\square $

Lemma 3.16

Let for $1 \leqslant i \leqslant 3$. If $u_1 = u_2$ holds and $w_1, w_2, w_3$ span as a group, then $\zeta _1 - \zeta _2$ is a generator for ${{\mathbb {Z}}}/ t {{\mathbb {Z}}}$.

Lemma 3.17

Situation as in Setting 3.1. If X is locally factorial, , and $u_i = u_j$ holds for some $1< i< j < r$, then $\zeta _1 - \zeta _2$ is a generator for $\Gamma $. In particular K is torsion-free or $t \ne 2, 4$ holds.

Proof

First note that $w_i, w_j$ share a common ray in $K_{{\mathbb {Q}}}$, thus do not lie in the relative interior of the GIT-cone $\lambda $; see Proposition 3.4. So we have $w_i, w_j \in \lambda ^-$ or . By applying an orientation reversing coordinate change if necessary we achieve $w_i, w_j \in \lambda ^-$.

We have ; see Lemma 3.12. Using enables us to apply a suitable admissible coordinate change such that $\zeta _1 = \zeta _r = \overline{0}$. Remark 2.5 ensures that g has no monomial of the form . Hence Lemma 3.6 yields that both triples $w_1, w_i, w_r$ and $w_1, w_j, w_r$ generate K as a group. In particular $\zeta _i,\zeta _j$ both are generators for ${{\mathbb {Z}}}/ t {{\mathbb {Z}}}$. Moreover Lemma 3.6 tells us that $w_i, w_j, w_r$ form a generating set for K. Lemma 3.16 yields that $\zeta _i - \zeta _j$ is a generator for ${{\mathbb {Z}}}/ t {{\mathbb {Z}}}$. The proof is finished by the fact that the difference of two generators for ${{\mathbb {Z}}}/ 2 {{\mathbb {Z}}}$ resp. ${{\mathbb {Z}}}/ 4 {{\mathbb {Z}}}$ is never a generator for the respective group.$\square $

4 Proof of Theorem 1.1: collecting candidates

The first and major task in the proof of Theorem 1.1 is to show that we find specifying data for any given smooth Calabi–Yau threefold X with spread hypersurface Cox ring among the items displayed in Theorem 1.1. This is done by a case-by-case analysis of the geometric constellation of the Cox ring generator degrees.

Now the ground field is ${{\mathbb {K}}}= {{\mathbb {C}}}$. The sole reason for this is the reference involved in the proof of the following proposition.

Proposition 4.1

Consider the situation of Setting 3.1. If $X(\lambda )$ is a smooth weakly Calabi–Yau threefold, then any variety $X(\eta )$ arising from a full-dimensional GIT-cone $\eta $ satisfying is smooth.

Proof

Lemma 2.6 provides us with a sequence of flops

$$\begin{aligned} X(\lambda ) = X_1 \dashrightarrow \cdots \dashrightarrow X_k = X(\eta ). \end{aligned}$$

According to [26, Theorem 6.15], see also [25], flops of threefolds preserve smoothness. So we successively obtain smoothness for all varieties in the above sequence, especially for $X(\eta )$.$\square $

Given a positive integer n, a sum of the form $n_1 + \cdots + n_k = n$ where $n_1, \dotsc , n_k \in {{\mathbb {Z}}}_{\geqslant 1}$ is called an integer partition of n. If one wants to emphasize the order of the summands, one calls such a sum an integer composition of n. For instance, $1 + 1 + 2 = 4$ and $1 + 2 + 1 = 4$ are two different integer compositions of 4 but they are equal as integer partitions.

Remark 4.2

In Setting 3.1 the geometric constellation of $w_1, \dotsc , w_r$ is described by an integer composition of r in the following sense: First, we take into account that some of the rays may coincide and label the actual rays properly. Let $1 \leqslant j_1< \cdots < j_s \leqslant r$ such that holds for $j_k \ne j_l$ and each $\varrho _i$ equals some $\varrho _{j_k}$. Set . We denote $N_k$ for the number of Cox ring generator degrees $w_i$ lying on $\sigma _k$. Then the distribution of the degrees $w_i$ on the rays $\sigma _k$ is encoded by the composition

$$\begin{aligned} N_1 + \cdots + N_s = r. \end{aligned}$$

For example, when $r = 4$ holds, the integer compositions $1 + 1 + 2 = 4$ and $1 + 2 + 1 = 4$ correspond to the constellations of $w_1, \dotsc , w_4$ illustrated below.

Proposition 4.3

Situation as in Proposition 3.1. If X is a weakly Calabi–Yau threefold, then $r = 6$ holds and the constellation of $w_1, \dotsc , w_6$ corresponds to one of the following integer partitions $N_1 + \dotsc + N_s = 6$ in the sense of Remark 4.2.

$$\begin{aligned} \begin{array}{cccccccc} \hline &{} s &{} N_1 &{} N_2 &{} N_3 &{} N_4 &{} N_5 &{} N_6 \\ \hline \mathrm{I} &{} 2 &{} 3 &{} 3 &{} \text {---} &{} \text {---} &{} \text {---} &{} \text {---} \\ \mathrm{II} &{} 3 &{} 2 &{} 2 &{} 2 &{} \text {---} &{} \text {---} &{} \text {---} \\ \mathrm{III} &{} 3 &{} 1 &{} 2 &{} 3 &{} \text {---} &{} \text {---} &{} \text {---} \\ \mathrm{IV} &{} 4 &{} 1 &{} 1 &{} 2 &{} 2 &{} \text {---} &{} \text {---} \\ \mathrm{V} &{} 4 &{} 1 &{} 1 &{} 1 &{} 3 &{} \text {---} &{} \text {---} \\ \mathrm{VI} &{} 5 &{} 1 &{} 1 &{} 1 &{} 1 &{} 2 &{} \text {---} \\ \mathrm{VII} &{} 6 &{} 1 &{} 1 &{} 1 &{} 1 &{} 1 &{} 1 \\ \hline \end{array} \end{aligned}$$

Proof

Observe . The subsequent table shows all integer partitions $N_1 + \cdots + N_s = 6$.

$$\begin{aligned} \begin{array}{cccccccc} \hline &{} s &{} N_1 &{} N_2 &{} N_3 &{} N_4 &{} N_5 &{} N_6 \\ \hline &{} 1 &{} 6 &{} \text {---} &{} \text {---} &{} \text {---} &{} \text {---} &{} \text {---} \\ &{} 2 &{} 1 &{} 5 &{} \text {---} &{} \text {---} &{} \text {---} &{} \text {---} \\ &{} 2 &{} 2 &{} 4 &{} \text {---} &{} \text {---} &{} \text {---} &{} \text {---} \\ \text {I} &{} 2 &{} 3 &{} 3 &{} \text {---} &{} \text {---} &{} \text {---} &{} \text {---} \\ \text {II} &{} 3 &{} 2 &{} 2 &{} 2 &{} \text {---} &{} \text {---} &{} \text {---} \\ \text {III} &{} 3 &{} 1 &{} 2 &{} 3 &{} \text {---} &{} \text {---} &{} \text {---} \\ &{} 3 &{} 1 &{} 1 &{} 4 &{} \text {---} &{} \text {---} &{} \text {---} \\ \text {IV} &{} 4 &{} 1 &{} 1 &{} 2 &{} 2 &{} \text {---} &{} \text {---} \\ \text {V} &{} 4 &{} 1 &{} 1 &{} 1 &{} 3 &{} \text {---} &{} \text {---} \\ \text {VI} &{} 5 &{} 1 &{} 1 &{} 1 &{} 1 &{} 2 &{} \text {---} \\ \text {VII} &{} 6 &{} 1 &{} 1 &{} 1 &{} 1 &{} 1 &{} 1 \\ \hline \end{array} \end{aligned}$$

Our task is to show that in the situation of Setting 3.1 those partitions without roman label do not admit a composition corresponding to the constellation of $w_1, \dotsc , w_6$ in $K_{{\mathbb {Q}}}$.

Observe that in the cases $s = 1$ and $s = 2$ where $N_1 = 1$, $N_2 = 5$ the moving cone of R must be one-dimensional; a contradiction. From Proposition 3.3 we deduce that any constellation given by $N_1 + N_2 = 2 + 4 = 6$ forces $\mu $ to live in the boundary of . This contradicts Remark 2.5. Furthermore, the partition $N_1 + N_2 + N_3 = 1 + 1 + 4$ comprises precisely two compositions, that is to say

$$\begin{aligned} N_1 + N_2 + N_3 = 1 + 4 + 1 \quad \text {and} \quad N_1 + N_2 + N_3 = 1 + 1 +4. \end{aligned}$$

The first of them implies that is one-dimensional; a contradiction. Considering the latter, Proposition 3.3 shows that $\mu $ lies on the boundary of ; a contradiction to Remark 2.5.$\square $

Throughout the proof of Theorem 1.1 we will often encounter inequations of the following type.

Remark 4.4

The following table describes the solutions of the inequality:

$$\begin{aligned} x_1 \cdots x_n \leqslant x_1 + \cdots + x_n, \quad x_1, \dotsc , x_n \in {{\mathbb {Z}}}_{\geqslant 1}, \end{aligned}$$

for $n = 3,4,5$ where $x_1, \dotsc , x_n$ are in ascending order. Here, $*$ stands for an arbitrary positive integer.

We work in Setting 3.1 for the proof of Theorem 1.1. According to Remark 2.5 (i) it suffices to determine the degree matrix $Q = [w_1, \dotsc , w_6]$ in order to figure out candidates for specifying data of X since the relation degree $\mu $ is given by

$$\begin{aligned} \mu = w_1 + \cdots + w_6. \end{aligned}$$

When Q and $\mu $ are fixed, we cover all possibilities (up to isomorphism) by picking an interior point u of each full-dimensional GIT-chamber $\lambda $ with .

Our proof of Theorem 1.1 will be split into Parts I, ..., VII discussing the constellations of $w_1, \dotsc , w_6$ in the sense of Remark 4.2 given by the accordingly labeled integer partition of six from Proposition 4.3. In the present article we elaborate Parts I–IV and VII. The remaining parts are treated with similar arguments and can be found in [29].

4.1 Part I

We consider $3 + 3 = 6$, i.e. the generator degrees $w_i$ are evenly distributed on two rays $\sigma _1,\sigma _2$. So $w_1, \dotsc , w_6$ lie all in the boundary of .

Lemma 3.9 (i) tells us that each $w_i$ is primitive and is regular. In particular $u_1 = u_2 = u_3$ and $u_4 = u_5 = u_6$. A suitable admissible coordinate change leads to

$$\begin{aligned} Q^0 = \begin{bmatrix} 1 &{} 1 &{} 1 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 1 &{} 1 &{} 1 \end{bmatrix}. \end{aligned}$$

If K is torsion-free, this leads to specifying data as in Number 1 from Theorem 1.1.

We assume that K admits torsion. Remark 2.5 (i) implies $\alpha = u_1 + \cdots + u_6 = (3,3)$. Lemma 3.13 guarantees that $T_2^3 T_4^3$ is of degree $\mu $. Lemma 3.6 tells us that $w_1, w_2, w_4$ generate K as a group and we have . Thus we may apply Lemma 3.14. From this we infer $t \,{|}\, 3$, hence $t = 3$, i.e. ; see also Lemma 3.12. Furthermore, Lemma 3.6 yields that K is generated by each of the triples

$$\begin{aligned} (w_1, w_2, w_4), \quad (w_1, w_3, w_4), \quad (w_2, w_3, w_4). \end{aligned}$$

Since $u_1 = u_2 = u_3$, we conclude that $\zeta _1, \zeta _2, \zeta _3$ are pairwise different. Otherwise two of $w_1, w_2, w_3$ coincide, hence K is generated by two elements; a contradiction. In the same manner we obtain that $\zeta _4, \zeta _5, \zeta _6$ are pairwise different. After suitably reordering $T_1, \dotsc , T_6$ we arrive at specifying data as in Number 2 from Theorem 1.1.

4.2 Part II

We discuss the degree constellation determined by $2 + 2 + 2 = 6$. Here the generator degrees $w_i$ are evenly distributed on three rays $\sigma _1, \sigma _2, \sigma _3$.

We have $\mu \in \sigma _2$ by Proposition 3.3. Proposition 3.4 provides us with two GIT-cones

According to Proposition 4.1 the associated varieties $X(\eta _1)$, $X(\eta _2)$ both are smooth. Lemma 3.9 (i) yields $u_1 = u_2$, $u_5 = u_6$ and . After applying a suitable admissible coordinate change the degree matrix is of the form

$$\begin{aligned} Q^0 = \begin{bmatrix} 1 &{} 1 &{} a_3 &{} a_4 &{} 0 &{} 0 \\ 0 &{} 0 &{} b_3 &{} b_4 &{} 1 &{} 1 \end{bmatrix}, \quad a_3, a_4 \in {{\mathbb {Z}}}_{\geqslant 1}. \end{aligned}$$

We may assume $a_3 \leqslant a_4$. Let $v = (v_1, v_2) \in {{\mathbb {Z}}}^2$ be the primitive vector lying on $\sigma _2$. Applying Lemma 3.6 to $X(\eta _2)$ and the triple $w_3, w_4, w_5$ shows . In addition, we obtain $v_1 = 1$ from Lemma 3.7. Lemma 3.6 again, this time applied to $X(\eta _1)$ and $w_1, w_2, w_3$, gives $v_2 = 1$. From $v_1 = v_2$ we deduce $a_3 = b_3$ and $a_4 = b_4$. Lemma 3.8 ensures that $\mu _1$ is divisible by both $a_3$ and $a_4$, thus $a_3 a_4 \,{|}\, \mu _1$. Remark 2.5 (i) says $\mu = w_1 + \cdots + w_6$. We conclude

$$\begin{aligned} a_3 a_4 \mid \mu _1 = a_3 + a_4 + 2. \end{aligned}$$

First we deduce $a_4 \,{|}\, a_3 + 2$. Moreover we obtain $a_3 \leqslant 4$ due to $a_3 \leqslant a_4$. Altogether the integers $a_3,a_4$ are bounded, so we just have to examine the possible configurations.

$a_3 = 1$: From $a_4 \,{|}\, a_3 + 2 = 3$ we infer $a_4 = 1, 3$. Now we show that K is torsion-free. For $a_4 = 1$ we have
$$\begin{aligned} Q^0 = \begin{bmatrix} 1 &{} 1 &{} 1 &{} 1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 1 &{} 1 &{} 1 &{} 1 \end{bmatrix}, \quad \alpha = (4, 4). \end{aligned}$$
Observe $\mu ^0 = 4 u_3$. Lemma 3.11 tells us $t \,{|}\, 4$, thus K is torsion-free according to Lemma 3.17. Similarly, for $a_4 = 3$ we have
$$\begin{aligned} Q^0 = \begin{bmatrix} 1 &{} 1 &{} 1 &{} 3 &{} 0 &{} 0 \\ 0 &{} 0 &{} 1 &{} 3 &{} 1 &{} 1 \end{bmatrix}, \quad \alpha = (6, 6). \end{aligned}$$
Observe $\alpha = 2 u_4$. Lemma 3.11 tells us $t \,{|}\, 2$, thus K is torsion-free according to Lemma 3.17. We arrive at specifying data as in Numbers 3 and 4 from Theorem 1.1. Observe $X(\eta _1) \cong X(\eta _2)$ in both cases due to the symmetry of the geometric constellation of $w_1, \dotsc , w_6, \mu $. Thus it suffices to list an ample class for $X(\eta _1)$ only.
$a_3 = 2$: From $a_4 \,{|}\, a_3 + 2 = 4$ and $a_3 \leqslant a_4$ we infer $a_4 = 2,4$. This contradicts .
$a_3 = 3$: From $a_4 \,{|}\, a_3 + 2 = 5$ and $a_3 \leqslant a_4$ we infer $a_4 = 5$. This leads to $\mu _1 = a_3 + a_4 + 2 = 10$. A contradiction to $a_3 \,{|}\, \mu _1$.
$a_3 = 4$: From $a_4 \,{|}\, a_3 + 2 = 6$ and $a_3 \leqslant a_4$ we infer $a_4 = 6$. This contradicts .

4.3 Part III

In this part we consider the arrangements of $w_1, \dotsc , w_6$ associated with the integer partition $1 + 2 + 3 = 6$. Here we have precisely three rays $\sigma _1, \sigma _2, \sigma _3$ each of which contains a different number of Cox ring generator degrees. A suitable admissible coordinate change turns the setting into one of the following:

Case III-i. Here we have . Let $v \in {{\mathbb {Z}}}^2$ be a primitive vector on $\sigma _2$. Proposition 3.3 and Remark 2.5 (i) tell us $\mu \in \lambda ^\circ \cup \sigma _2$. This allows us to apply Lemma 3.6 to $w_i, w_4, w_5$ for $i = 1,2,3$. From this we infer for $i = 1,2,3$. In particular $u_1, u_2, u_3$ are primitive, thus $u_1 = u_2 = u_3$. Applying Lemma 3.6 to the triple $w_1, w_2, w_6$ shows . A suitable admissible coordinate change amounts to $v = (0,1)$ and

$$\begin{aligned} Q^0 = \begin{bmatrix} 1 &{} 1 &{} 1 &{} 0 &{} 0 &{} -a_6 \\ 0 &{} 0 &{} 0 &{} b_4 &{} b_5 &{} 1 \end{bmatrix}, \quad a_6, b_4, b_5 \in {{\mathbb {Z}}}_{\geqslant 1}. \end{aligned}$$

We may assume $b_4 \leqslant b_5$. To proceed we have to take the position of $\mu $ into account.

Assume . Then we may apply Lemmas 3.6 and 3.7 to the two triples $w_1, w_2, w_4$ and $w_1, w_2, w_5$. We obtain that $u_4$ and $u_5$ both are primitive, hence

$$\begin{aligned} u_4 = u_5 = v = (0, 1). \end{aligned}$$

From Remark 2.5 (i) we infer $\alpha = (3-a_6, 3)$. Since $\mu $ lives in the relative interior of $\lambda $, which is the positive orthant, we end up with $a_6 = 1, 2$. We show that K is torsion free in both cases.

$a_6 = 1$. The free parts of the specifying data are given as
$$\begin{aligned} Q^0 = \begin{bmatrix} 1 &{} 1 &{} 1 &{} 0 &{} 0 &{} -1 \\ 0 &{} 0 &{} 0 &{} 1 &{} 1 &{} 1 \end{bmatrix}, \quad \alpha = (2, 3). \end{aligned}$$
Lemma 3.13 ensures that $T_1^2 T_4^3$ is of degree $\mu $. Moreover, Lemma 3.6 shows that both triples $w_1, w_4, w_5$ and $w_1, w_2, w_4$ generate K as a group. Applying Lemma 3.14 to $w_1, w_4, w_5$ and $T_1^2 T_4^3$ yields $t \,{|}\, 3$. Again Lemma 3.14, this time applied to $w_1, w_2, w_4$ and $T_1^2 T_3^3$ shows $t \mid 2$. Altogether $t = 1$, thus K is torsion-free.
$a_6 = 2$. The free parts of the specifying data are given as
$$\begin{aligned} Q^0 = \begin{bmatrix} 1 &{} 1 &{} 1 &{} 0 &{} 0 &{} -2 \\ 0 &{} 0 &{} 0 &{} 1 &{} 1 &{} 1 \end{bmatrix}, \quad \alpha = (1, 3). \end{aligned}$$
Lemma 3.13 ensures that $T_1^1 T_4^3$ of degree $\mu $. Moreover Lemma 3.6 shows that $w_1, w_2, w_4$ generate K as a group. Applying Lemma 3.14 to $w_1, w_2, w_4$ and $T_1 T_4^3$ shows $t = 1$ i.e. K is torsion-free.

Eventually this leads to specifying data as in Numbers 5 and 6 from Theorem 1.1.

Assume $\mu \in \sigma _2$. Recall that $v = (0,1)$ spans the ray $\sigma _2$. So here we have $\alpha _1 = 0$. From Remark 2.5 (i) we obtain $a_6 = 3$ and $\alpha _2 = b_4 + b_5 + 1$. Lemma 3.8 yields $b_4, b_5 \,{|}\, \alpha _2$. Applying Lemma 3.6 to $w_1, w_4, w_5$ shows . We conclude

$$\begin{aligned} b_4 b_5 \mid \alpha _2 = b_4 + b_5 + 1. \end{aligned}$$

This implies $b_5 \,{|}\, b_4 + 1$. Moreover we deduce $b_4 \leqslant 3$. We discuss the resulting cases:

$b_4 = 1$: From $b_5 \mid b_4 + 1 = 2$ we deduce $b_5 = 1,2$. For $b_5 = 1$ we have
$$\begin{aligned} Q^0 = \begin{bmatrix} 1 &{} 1 &{} 1 &{} 0 &{} 0 &{} -3 \\ 0 &{} 0 &{} 0 &{} 1 &{} 1 &{} 1 \end{bmatrix}, \quad \alpha = (0, 3). \end{aligned}$$
Suppose that K is torsion-free. Then $w_4 = w_5$ holds. Reversing the order of the variables by applying a suitable admissible coordinate change enables us to use Lemma 3.10. This forces two of the rays $\sigma _i$ to coincide; a contradiction. So K has torsion. From $\alpha = 3 u_4$ and Lemma 3.15 we obtain $t \,{|}\, 3$, hence $t = 3$. So we have . As seen earlier, using enables us to apply a suitable admissible coordinate change such that $\zeta _1 = \zeta _6 = 0$. Now Lemma 3.6 shows that both triples $w_1, w_2, w_6$ and $w_1, w_3, w_6$ generate K as a group. From this we infer that $\zeta _2$ and $\zeta _3$ both are generators for . Lemma 3.6 yields that $w_2, w_3, w_6$ form a generating set for K as well. This forces $\zeta _2 \ne \zeta _3$. Otherwise $w_2 = w_3$ holds, thus K is spanned by two elements; a contradiction. Similarly, Lemma 3.6 applied to $w_1, w_4, w_6$ and $w_1, w_5, w_6$ yields that $\zeta _4$ and $\zeta _5$ both are generators for ${{\mathbb {Z}}}/ 3 {{\mathbb {Z}}}$. Moreover applying Lemma 3.6 to $w_1, w_4, w_5$ ensures $\zeta _4 \ne \zeta _5$. After suitably reordering $T_2$, $T_3$ and $T_4, T_5$ we arrive at Number 7 from Theorem 1.1.

We turn to $b_5 = 2$. Here the free parts of degree matrix and relation degree are given by
$$\begin{aligned} Q^0 = \begin{bmatrix} 1 &{} 1 &{} 1 &{} 0 &{} 0 &{} -3 \\ 0 &{} 0 &{} 0 &{} 1 &{} 2 &{} 1 \end{bmatrix}, \quad \alpha = (0, 4). \end{aligned}$$
Note $\alpha = 2 u_5$. From Lemma 3.15 we infer $t \,{|}\, 2$, hence K is torsion-free according to Lemma 3.17. Moreover, every $\mu $-homogeneous polynomial not depending on $T_6$ is a linear combination over the monomials $T_4^4,T_4^2 T_5,T_5^2$, thus reducible. This implies that $T_6 \in R$ is not prime. A contradiction.
$b_4 = 2$: From $b_5 \mid b_4 + 1 = 3$ and $b_4 \leqslant b_5$ follows $b_5 = 3$. This leads to
$$\begin{aligned} Q^0 = \begin{bmatrix} 1 &{} 1 &{} 1 &{} 0 &{} 0 &{} -3 \\ 0 &{} 0 &{} 0 &{} 2 &{} 3 &{} 1 \end{bmatrix}, \quad \alpha = (0, 6). \end{aligned}$$
Observe $\alpha = 3 u_2 = 2 u_3$. Lemma 3.15 yields $t \,{|}\, 2$ and $t \,{|}\, 3$, hence $t = 1$. So K is torsion-free. We end up with Number 8 from Theorem 1.1.
$b_4 = 3$: From $b_5 \mid b_4 + 1 = 4$ and $b_4 \leqslant b_5$ we infer $b_5 = 4$. This implies $\alpha _2 = 8$; a contradiction to $b_4 \,{|}\, \alpha _2$.

Case III-ii. Here, we have . Proposition 3.3 says $\mu \in \sigma _2$. Let $v \in {{\mathbb {Z}}}^2$ be a primitive vector on $\sigma _2$. Applying Lemma 3.6 to $w_2, w_3, w_5$ as well as $w_2, w_3, w_6$ shows and . In particular $u_5, u_6$ are primitive and lie on the same ray hence coincide. Again by Lemma 3.6, now applied to $w_1, w_5, w_6$, we obtain . A suitable admissible coordinate change amounts to $v = (1,0)$ and $u_5 = (0,1)$. As a result the free part $Q^0$ of the degree matrix Q is of the form

$$\begin{aligned} Q^0 = \begin{bmatrix} 1 &{} a_2 &{} a_3 &{} a_4 &{} 0 &{} 0 \\ -2 &{} 0 &{} 0 &{} 0 &{} 1 &{} 1 \end{bmatrix}, \quad a_2, a_3, a_4 \in {{\mathbb {Z}}}_{\geqslant 1}. \end{aligned}$$

Note that the second coordinate of $u_1$ is determined by $\alpha _2 = 0$ and Remark 2.5 (i). Furthermore, we may assume $a_2 \leqslant a_3 \leqslant a_4$. Lemma 3.8 shows that $\alpha _1$ is divisible by each of $a_2, a_3, a_4$. From applying Lemma 3.6 to all triples $w_i, w_j, w_6$ where $2 \leqslant i < j \leqslant 4$ we infer that $a_2, a_3, a_4$ are pairwise coprime. This leads to

$$\begin{aligned} a_2 a_3 a_4 \mid a_2 + a_3 + a_4 + 1. \end{aligned}$$

According to Remark 4.4 we have $a_2 = 1$ and one of the following two configurations:

$$\begin{aligned} a_3 = 1, \quad a_3 = 2\quad \text {and}\quad a_4 = 3. \end{aligned}$$

Note that $a_3 = 2$ and $a_4 = 3$ amounts to $\alpha _1 = 7$; a contradiction to $a_3 \,{|}\, \alpha _1$. So we have $a_3 = 1$. Then $a_4 \,{|}\, \alpha _1 = 3 + a_4$ holds. We conclude $a_4 \,{|}\, 3$, i.e. $a_4 = 1, 3$. We show that K is torsion-free in both cases:

$a_4 = 1$. Here we have
$$\begin{aligned} Q^0 = \begin{bmatrix} 1 &{} 1 &{} 1 &{} 1 &{} 0 &{} 0 \\ -2 &{} 0 &{} 0 &{} 0 &{} 1 &{} 1 \end{bmatrix}, \quad \alpha = (4, 0). \end{aligned}$$
Note $\alpha = 4 u_2$, thus $t \,{|}\, 4$ by Lemma 3.15. Now Lemma 3.17 ensures that K is torsion-free.
$a_4 = 2$. Here we have
$$\begin{aligned} Q^0 = \begin{bmatrix} 1 &{} 1 &{} 1 &{} 3 &{} 0 &{} 0 \\ -2 &{} 0 &{} 0 &{} 0 &{} 1 &{} 1 \end{bmatrix}, \quad \alpha = (6, 0). \end{aligned}$$
Note $\alpha = 2 u_4$, thus $t \,{|}\, 2$ by Lemma 3.15. Now Lemma 3.17 ensures that K is torsion-free.

We have arrived at Numbers 9 and 10 from Theorem 1.1.

Case III-iii. From Lemma 3.9 (i) we obtain

A suitable admissible coordinate change brings the degree matrix into the following form:

$$\begin{aligned} Q^0 = \begin{bmatrix} 1 &{} 1 &{} 1 &{} a_4 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} b_4 &{} 1 &{} 1 \end{bmatrix}, \quad a_4, b_4 \in {{\mathbb {Z}}}_{\geqslant 1}. \end{aligned}$$

Moreover, Proposition 3.3 tells us or $\mu \in \varrho _4$. Let us first assume . According to Proposition 3.4 we have GIT-cones

both of them giving rise to a smooth variety $X(\eta _i)$; see Proposition 4.1. We obtain that K is torsion-free by applying Lemma 3.11 to $X(\eta _2)$. Applying Lemma 3.9 (ii) gives $u_4 = u_1 + u_6 = (1,1)$. We have arrived at Numbers 11 and 12 from Theorem 1.1.

The next step is to consider $\mu \in \varrho _4$. Lemma 3.8 provides us with some $k \in {{\mathbb {Z}}}_{\geqslant 2}$ such that $\mu = k w_4$ holds. Using Remark 2.5 (i) gives $k b_4 = \alpha _2 = b_4 + 2$. We conclude $b_4 \,{|}\, 2$. This leads to one of the following two configurations

$$\begin{aligned} k = 3 \quad \text {and}\quad b_4 = 1, \qquad k = 2 \quad \text {and} \quad b_4 = 2. \end{aligned}$$

Suppose $k = 3$. Using Remark 2.5 (i) again shows $3 a_4 = 3 + a_4$, equivalently $2 a_4 = 3$. A contradiction. We must have $k = 2$ and $b_4 = 2$. Here Remark 2.5 (i) implies $2 a_4 = 3 + a_4$, thus $a_4 = 3$. We have

$$\begin{aligned} Q^0 = \begin{bmatrix} 1 &{} 1 &{} 1 &{} 3 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 2 &{} 1 &{} 1 \end{bmatrix}, \quad \alpha = (6, 4). \end{aligned}$$

From $\alpha = 2 u_4$ we infer $t \,{|}\, 2$ by Lemma 3.15. Thus Lemma 3.17 yields that K is torsion-free. This amounts to Number 13 from Theorem 1.1.

4.4 Part IV

This part deals with the case of $w_1, \dotsc , w_6$ being disposed on four rays according to the integer partition $1 + 1 + 2 + 2 = 6$. A suitable admissible coordinate change leads to one of the subsequent constellations:

Case IV-i. Here, Proposition 3.3 tells us $\mu \in \sigma _3$. As a result, Proposition 3.4 provides us with two GIT-cones

Proposition 4.1 ensures that the associated varieties $X(\eta _1)$ and $X(\eta _2)$ both are smooth. Let $v \in {{\mathbb {Z}}}^2$ denote the primitive lattice vector lying on $\sigma _3$. Consider $X(\eta _2)$. Applying Lemmas 3.6 and 3.7 to the triples $w_3, w_4, w_5$ and $w_3, w_4, w_6$ yields $u_5 = u_6$ and . Thus we may apply a suitable admissible coordinate change such that $v = (1,0)$ and $u_5 = u_6 = (0,1)$. We apply Lemma 3.6 again, this time to $w_1, w_5, w_6$ and $w_2, w_5, w_6$. This shows that the first coordinate of both $u_1$ and $u_2$ equals one. Now, consider $X(\eta _1)$. We apply Lemma 3.6 to $w_1, w_3, w_4$, hence obtain $u_1 = (1, -1)$. Analogously, we obtain $u_2 = (1, -1)$, thus $u_1 = u_2$. This contradicts $\sigma _1 \ne \sigma _2$.

Case IV-ii. Proposition 3.3 says . First, we assume $\mu \in \varrho _4 = \sigma _3$. Then Proposition 3.4 ensures . Let $v \in {{\mathbb {Z}}}^2$ be the primitive lattice vector on $\sigma _2$. Applying Lemmas 3.6 and 3.7 to all four triples

$$\begin{aligned} (w_2, w_3, w_5), \quad (w_2, w_3, w_6), \quad (w_2, w_5, w_6), \quad (w_3, w_5, w_6) \end{aligned}$$

shows that $u_2, u_3, u_5, u_6$ are primitive, thus $u_2 = u_3$ and $u_5 = u_6$. Additionally we obtain . Lemma 3.6 again, this time applied to $w_1, w_5, w_6$, tells us . A suitable admissible coordinate change eventually amounts to

$$\begin{aligned} Q^0 = \begin{bmatrix} 1 &{} 1 &{} 1 &{} a_4 &{} 0 &{} 0 \\ -b_1 &{} 0 &{} 0 &{} b_4 &{} 1 &{} 1 \end{bmatrix}, \quad a_4, b_1, b_4 \in {{\mathbb {Z}}}_{\geqslant 1}. \end{aligned}$$

From Remark 2.5 (i) we infer $\alpha = (a_4 + 3,\, b_4 - b_1 + 2)$. Lemma 3.8 provides us with some $k \in {{\mathbb {Z}}}_{\geqslant 2}$ such that $\mu = k w_4$. In particular $a_4 \,{|}\, \alpha _1 = a_4 + 3$. This implies $a_4 = 1, 3$. Suppose $a_4 = 1$. Then $k = 4$ holds. This leads to $4b_4 = \alpha _2 = b_4 - b_1 + 2$, hence $3 b_4 = 2 - b_1$. A contradiction to $b_1, b_4 \geqslant 1$. We are left with $a_4 = 3$ and $k = 2$. Inserting into $\alpha = k u_3$ gives $2b_4 = b_4 - b_1 + 2$, thus $b_4 = 2 - b_1$. This forces $b_1 = 1$ and $b_4 = 1$ due to $b_1, b_4 \geqslant 1$. Moreover, $k = 2$ implies $t \,{|}\, 2$ by Lemma 3.15. Thus K is torsion-free according to Lemma 3.17. We have arrived at Number 14 from Theorem 1.1.

We turn to the case $\mu \notin \sigma _3$. Here Proposition 3.4 provides us with two GIT-cones

According to Proposition 4.1, the according varieties $X(\eta _1)$ and $X(\eta _2)$ both are smooth. Consider $X(\eta _2)$. Lemma 3.5 applied to $w_4, w_5$ and $w_4, w_6$ yields as well as . Besides, Lemmas 3.6 and 3.7 applied to $w_1, w_5, w_6$ give us . Now consider $X(\eta _1)$. Let $v \in {{\mathbb {Z}}}^2$ be the primitive vector contained in $\sigma _2$. Applying Lemmas 3.6 and 3.7 to $w_2, w_3, w_4$ and $w_2, w_3, w_5$ shows and . Now we apply an admissible coordinate change such that $v = (1,0)$ and $u_5 = (0,1)$ holds. Taking the determinantal equations from above into account amounts to the following degree matrix:

$$\begin{aligned} Q^0 = \begin{bmatrix} 1 &{} a_2 &{} a_3 &{} 1 &{} 0 &{} 0 \\ -b_1 &{} 0 &{} 0 &{} 1 &{} 1 &{} 1 \end{bmatrix}, \quad a_2, a_3, b_1 \in {{\mathbb {Z}}}_{\geqslant 1}. \end{aligned}$$

We may assume $a_2 \leqslant a_3$. From Remark 2.5 (i) follows $\alpha _2 = 3 - b_1$. Proposition 3.3 guarantees that $\mu $ lives in the positive orthant, hence $b_1 \leqslant 3$. Furthermore, Lemma 3.5 applied w.r.t. $X(\eta _1)$ and the pairs $w_2, w_5$ and $w_3, w_5$ shows $a_2, a_3 \,{|}\, \alpha _1$. Applying Lemma 3.6 to $w_2, w_3, w_5$ shows . Consequently $a_2 a_3 \,{|}\, \alpha _1 = a_2 + a_3 + 2$ holds. We end up with $a_2 = 1$ and $a_3 = 1, 3$.

Let us discuss the case $a_3 = 1$. Here specifying data looks as follows:

$$\begin{aligned} Q^0 = \begin{bmatrix} 1 &{} 1 &{} 1 &{} 1 &{} 0 &{} 0 \\ -b_1 &{} 0 &{} 0 &{} 1 &{} 1 &{} 1 \end{bmatrix}, \quad \alpha = (4,\, 3 - b_1), \quad b_1 \in \{1, 2, 3\}. \end{aligned}$$

Suppose $b_1 = 3$. This implies $\alpha = (4, 0) = 4 u_2$. Lemmas 3.15 and 3.17 yield that K is torsion-free. So $w_2 = w_3$ holds. Note that $\alpha _2 = 0$ means $\mu \in \varrho _2$. In this situation Lemma 3.10 says $w_4 \in \varrho _2$. A contradiction to $\sigma _2 \ne \sigma _3$. So we have $b_1 = 1, 2$. Observe $\mu \in \eta _1^\circ $. Applying Lemma 3.11 to $X(\eta _2)$ guarantees that K is torsion-free. We end up with Numbers 15 to 18 from Theorem 1.1.

We turn to $a_3 = 3$. Here we have $\alpha _1 = 6$. According to Lemma 3.5 applied to $X(\eta _1)$ and $w_3, w_4$, there must be some monomial $T_3^{l_3} T_4^{l_4}$ of degree $\mu $ because of . As the second coordinate of $u_3$ vanishes, $l_4 = \alpha _2 = 3 - b_1$ holds. Inserting this into the equation $\alpha _1 = l_3 a_3 + l_4 a_4$ yields $3l_3 + 3 - b_1 = \alpha _1 = 6$. This forces $b_1$ to be divisible by 3, hence $b_1 = 3$. We arrive at the following data

$$\begin{aligned} Q^0 = \begin{bmatrix} 1 &{} 1 &{} 3 &{} 1 &{} 0 &{} 0 \\ -3 &{} 0 &{} 0 &{} 1 &{} 1 &{} 1 \end{bmatrix}, \quad \alpha = (6, 0). \end{aligned}$$

Observe that this grading does not admit any monomial of the form $T_1^{l_1} T_4^{l_4}$ of degree $\mu $. Thus by Lemma 3.5 applied to $X(\eta _1)$ and $w_1, w_4$. A contradiction.

Case IV-iii. Proposition 3.4 ensures . Let be the primitive ray generators of $\sigma _2$, $\sigma _3$. We may apply Lemmas 3.6 and 3.7 to at least one of the triples $w_2, w_3, w_4$ and $w_2, w_4, w_5$. From this we infer . A suitable admissible coordinate change leads to $v = (1,0)$ and . Applying Lemma 3.6 to $w_2, w_3, w_6$ yields $w_6 = (-a_6, 1)$ for some $a_6 \in {{\mathbb {Z}}}_{\geqslant 1}$. Similarly, one obtains $w_1 = (1, -b_1)$ with $b_1 \in {{\mathbb {Z}}}_{\geqslant 1}$. Counter-clockwise orientation yields . We conclude $b_1 = a_6 = 1$, hence $w_1 = - w_6$. This contradicts being pointed.

Case IV-iv. We have by Proposition 3.3. Suppose . Proposition 4.1 allows us to apply Lemma 3.9 (ii). From this we infer $u_3 = u_4$, thus $\sigma _2 = \sigma _3$; a contradiction. So we have $\mu \in \sigma _2 \cup \sigma _3$. Taking the symmetry in the geometric constellation of $w_1, \dotsc , w_6$ into account, a suitable admissible coordinate change amounts to $\mu \in \sigma _2$. Lemma 3.9 yields $u_1 = u_2$, $u_5 = u_6$ and $u_4 = u_1 + u_6$. Furthermore we obtain . After applying another suitable admissible coordinate change the degree matrix is of the following form:

$$\begin{aligned} Q^0 = \begin{bmatrix} 1 &{} 1 &{} a_3 &{} 1 &{} 0 &{} 0 \\ 0 &{} 0 &{} b_3 &{} 1 &{} 1 &{} 1 \end{bmatrix}, \quad a_3, b_3 \in {{\mathbb {Z}}}_{\geqslant 1}. \end{aligned}$$

From X being Calabi–Yau we infer $\alpha = (a_3 + 3, b_3 + 3)$; see Remark 2.5. Besides, Lemma 3.8 provides us with some $k \in {{\mathbb {Z}}}_{\geqslant 2}$ such that $\mu = k w_3$ holds. Altogether we obtain , hence $a_3 = b_3$. This contradicts $\sigma _2 \ne \sigma _3$.

4.5 Part VII

We work out the constellation where the Cox ring generator degrees $w_1, \dotsc , w_6$ lie on pairwise different rays, i.e. we have $\sigma _i = \varrho _i$ for all $i = 1, \dotsc , 6$. Proposition 3.3 says . After a applying a suitable admissible coordinate change we have either or $\mu \in \varrho _3$.

Case VII-a. Here, we assume . According to Proposition 3.4 the cones

are GIT-cones leading to smooth varieties $X(\eta _i)$; see also Proposition 4.1. Let us consider $X(\eta _1)$. Lemma 3.5 applied to $w_1, w_3$ and $w_2, w_3$ yields and . Thus a suitable admissible coordinate change leads to

$$\begin{aligned} Q^0 = \begin{bmatrix} 1 &{} 1 &{} 0 &{} - a_4 &{} -a_5 &{} -a_6 \\ -b_1 &{} 0 &{} 1 &{} b_4 &{} b_5 &{} b_6 \end{bmatrix}, \quad a_i, b_i \in {{\mathbb {Z}}}_{\geqslant 1}. \end{aligned}$$

Consider $X(\eta _3)$. Applying Lemma 3.5 to $w_4, w_5$ and $w_4, w_6$ gives and . Since $a_4 \ne 0$, this is equivalent to

$$\begin{aligned} b_5 = \dfrac{a_5 b_4 - 1}{a_4}, \quad b_6 = \dfrac{a_6 b_4 - 1}{a_4}. \end{aligned}$$

(1)

Now consider $X(\eta _2)$. Applying Lemma 3.5 to the pair $w_3, w_i$ for $i = 4, 5, 6$ shows that $\alpha _1$ is divisible by each of $a_4, a_5, a_6$. Moreover, Lemma 3.6 applied to $w_2, w_i, w_j$ where $4 \leqslant i < j \leqslant 6$ ensures that $a_4, a_5, a_6$ are pairwise coprime. Together with Remark 2.5 (i) we obtain

$$\begin{aligned} a_4 a_5 a_6 \mid \alpha _1 = a_4 + a_5 + a_6 - 2. \end{aligned}$$

One quickly checks that this forces two of $a_4, a_5, a_6$ to equal one. Suppose $a_5 = a_6 = 1$. Then (1) implies $b_5 = b_6$, thus $u_5 = u_6$. A contradiction. So we must have $a_4 = 1$, in particular

$$\begin{aligned} b_5 = a_5b_4 - 1, \quad b_6 = a_6 b_4 - 1. \end{aligned}$$

(2)

Furthermore, Lemma 3.5 applied to $w_2, w_j$ gives $b_j \,{|}\, \alpha _2$ for $j = 4, 5, 6$. In addition, applying Lemma 3.6 to all triples $w_2, w_i, w_j$ where $4 \leqslant i < j \leqslant 6$ shows that $b_4, b_5, b_6$ are pairwise coprime. Once again by Remark 2.5 (i) we obtain

$$\begin{aligned} b_4 b_5 b_6 \mid \alpha _2 = b_4 + b_5 + b_6 + 1 - b_1. \end{aligned}$$

(3)

Note that the right-hand side is positive due to the position of $\mu $. From this we deduce $b_4 b_5 b_6 \leqslant b_4 + b_5 + b_6$. According to Remark 4.4, this inequality implies that either two of $b_4, b_5, b_6$ equal one or $\{b_4, b_5, b_6\} = \{1,2,3\}$.

We exclude the first option. Here we have $b_5 \ne b_6$ by (2), thus $b_4 = 1$. However, we also have $a_i = 1$ for some $i \in \{5,6\}$. Then again (2) implies $b_i = a_i - 1 = 0$. A contradiction. So we have $\{b_4, b_5, b_6\} = \{1,2,3\}$.

Inserting into (3) amounts to $b_1 = 1$. Currently the degree matrix has the form

$$\begin{aligned} Q^0 = \begin{bmatrix} 1 &{} 1 &{} 0 &{} -1 &{} -a_5 &{} -a_6 \\ -1 &{} 0 &{} 1 &{} b_4 &{} b_5 &{} b_6 \end{bmatrix}. \end{aligned}$$

Recall that $a_5 = 1$ or $a_6 = 1$ holds. So we have $b_4 > b_5$ or $b_4 > b_6$ due to the counter-clockwise orientation of $w_4, w_5, w_6$. From this we infer $b_4 \ne 1$, hence $b_i = 1$ for some $i \in \{5, 6\}$. We are left with the cases $b_4 = 2,3$. With $b_4 = 3$, inserting into (2) gives $3 a_i - 1 = b_i = 1$. A contradiction to $a_i \in {{\mathbb {Z}}}_{\geqslant 1}$. With $b_4 = 2$ we deduce $a_i = 1$ from (2). In particular $w_i = (-1, 1) = - w_1$ holds. A contradiction to being pointed.

Case VII-b. Here, we assume $\mu \in \varrho _3$. Proposition 3.4 provides us with two GIT-cones

Both of them give rise to a smooth variety $X(\eta _i)$; see Proposition 4.1. Consider $X(\eta _2)$. Applying Lemma 3.5 to both pairs $w_4, w_5$ and $w_4, w_6$ yields and . A suitable admissible coordinate change leads to

$$\begin{aligned} Q^0 = \begin{bmatrix} a_1 &{} a_2 &{} a_3 &{} 1 &{} a_5 &{} 0 \\ -b_1 &{} -b_2 &{} -b_3 &{} 0 &{} 1 &{} 1 \end{bmatrix}, \quad a_i, b_i \in {{\mathbb {Z}}}_{\geqslant 1}. \end{aligned}$$

Lemma 3.8 provides us with some $k \in {{\mathbb {Z}}}_{\geqslant 2}$ such that $\mu = k w_3$ holds. In particular, we have $\alpha _1 = k a_3$. Now consider $X(\eta _1)$. Lemma 3.6 applied to the triples $w_1, w_3, w_5$ and $w_2, w_3, w_5$ shows that $a_1 + b_1a_5$ and $a_2 + b_2 a_5$ both divide k. Moreover, applying Lemma 3.6 to $w_1, w_2, w_5$ yields . Together with Remark 2.5 (i) we obtain

We expand the left-hand side and give a rough estimation:

Since is true for every integer n, this inequality shows that equality holds in the above divisibility condition. From this we infer

$$\begin{aligned} a_2 (a_1 a_3 - 1) + a_1(a_3 a_5 b_2 - 1) + a_3(a_2 a_5 b_1 - 1) + a_5 (a_3 a_5 b_1 b_2 - 1) = 1. \end{aligned}$$

Observe that every summand on the left-hand side is non-negative, hence precisely one of them equals one while the other vanish. Since $a_1, a_2, a_3, a_5$ are non-zero, the factor in the parenthesis vanishes whenever the whole summand vanishes. There are two summands where $b_1$ shows up in the second factor. At least one of those parenthesis must vanish, hence $b_1 = 1$. Repeating this argument yields $b_2 = 1$ as well as $a_3 = 1$. Similarly, we obtain $a_1 = 1$ or $a_2 = 1$. Altogether we have $u_3 = (1, -b_3)$ and $u_i = (1, -1)$ where $i \in \{1,2\}$. This implies ; a contradiction to our assumption that $w_1, \dotsc , w_6$ are in counter-clockwise order.

5 Constructing general hypersurface Cox rings

This section is devoted to the construction of general hypersurface Cox rings with prescribed specifying data. First, we describe the toolbox for producing general hypersurface Cox rings from [20, Section 4] in outline. Then we extend it with new explicit factoriality criterions for general hypersurface rings; see Corollaries 5.12 and 5.13.

Construction 5.1

Consider a linear, pointed, almost free K-grading on the polynomial ring and the quasitorus action , where

We write , for the degree map. Assume that is of full dimension and fix $\tau \in \Lambda (S)$ with . Set

Then Z is a projective toric variety with divisor class group and Cox ring ${\mathscr {R}}(Z) = S$. Moreover, fix $0 \ne \mu \in K$, and for $g \in S_\mu $ set

Then the factor algebra $R_g$ inherits a K-grading from S and the quotient $X_g \subseteq Z$ is a closed subvariety. Moreover, we have

$$\begin{aligned} X_g \subseteq Z_g \subseteq Z \end{aligned}$$

where $Z_g \subseteq Z$ is the minimal ambient toric variety of $X_g$, that means the (unique) minimal open toric subvariety containing $X_g$.

The spread $\mu $-homogeneous polynomials form an open subset $U_\mu \subseteq S_\mu $. Moreover, all polynomials $g \in U_\mu $ share the same minimal ambient toric variety $Z_g$. We call , where $g \in U_\mu $, the $\mu $-minimal ambient toric variety. The following propositions enable us to verify smoothness of $Z_\mu $ and the general $X_g$ in a purely combinatiorial manner.

Proposition 5.2

In the situation of Construction 5.1 the following are equivalent:

(i)
The $\mu $-minimal ambient toric variety $Z_\mu $ is smooth.
(ii)
For each $\gamma _0 \preceq \gamma $ with and $|Q^{-1}(\mu ) \cap \gamma _0| \ne 1$ the group K is generated by $Q(\gamma _0 \cap {{\mathbb {Z}}}^r)$.

Proposition 5.3

In the setting of Construction 5.1, assume and that $Z_{\mu } \subseteq Z$ is smooth. If $\mu \in \tau $ holds, then $\mu $ is base point free. Moreover, then there is a non-empty open subset of polynomials $g \in S_\mu $ such that $X_g$ is smooth.

Remark 5.4

In the situation of Construction 5.1 asume that $R_g$ is normal, factorially graded and $T_1, \ldots , T_r$ define pairwise non-associated K-primes in $R_g$. Then $R_g$ is an abstract Cox ring and we find a GIT-cone $\lambda \in \Lambda (R_g)$ with and ${\hat{X}}_g = {\overline{X}}{}^\mathrm {ss}(\lambda )$. This brings us into the situation of Constructions 2.1 and 2.2, so we have

Moreover, for any $g \in U_\mu $ the variables $T_1, \ldots , T_r$ form a minimal system of generators for all $R_g$ if and only if we have $\mu \ne w_i$ for $i = 1, \ldots , r$.

Constructing a general hypersurface Cox ring with prescribed specifying data essentially means to find a suitable open subset $U \subseteq S_\mu $ such that $R_g$, where $g \in U$, satisfies the conditions from the above remark. In the subsequent text we present several criterions to check these conditions.

Proposition 5.5

Consider the setting of Construction 5.1. For $1 \leqslant i \leqslant r$ denote by $U_i \subseteq S_\mu $ the set of all $g \in S_\mu $ such that g is prime in S and $T_i$ is prime in $R_g$. Then $U_i \subseteq S_\mu $ is open. Moreover, $U_i$ is non-empty if and only if there is a $\mu $-homogeneous prime polynomial not depending on $T_i$.

By a Dolgachev polytope we mean a convex polytope $\Delta \subseteq {{\mathbb {Q}}}_{\geqslant 0}^r$ of dimension at least four such that each coordinate hyperplane of ${{\mathbb {Q}}}^r$ intersects $\Delta $ non-trivially and the dual cone of is regular for each one-dimensional face $\Delta _0 \preceq \Delta $.

Proposition 5.6

In the situation of Construction 5.1, there is a non-empty open subset of polynomials $g \in S_\mu $ such that the ring $R_g$ is factorial provided that one of the following conditions is fulfilled:

(i)
K is of rank at most $r-4$ and torsion free, there is a $g \in S_\mu $ such that $T_1, \ldots , T_r$ define primes in $R_g$, we have $\mu \in \tau ^\circ $ and $\mu $ is base point free on Z.
(ii)
The set $\mathrm{conv}(\nu \in {{\mathbb {Z}}}_{\geqslant 0}^r; \, Q(\nu ) = \mu )$ is a Dolgachev polytope.

We combine the concepts of algebraic modifications [5, Section 4.1.2] and $\Sigma $-homogenizations [21] to provide further factoriality criterions for graded hypersurface rings. These will apply to several cases where the relation degree lies on the boundary of the moving cone. Let us first briefly recall the notion of polynomials arising from Laurent polynomials by homogenization with respect to a lattice fan from [21].

Remark 5.7

Let $\Sigma $ be a complete lattice fan in ${{\mathbb {Z}}}^n$ and $v_1, \dotsc , v_r$ the primitive lattice vectors generating the rays of $\Sigma $. Consider the mutually dual exact sequences

This induces a pointed K-grading on the polynomial algebra $S = {{\mathbb {K}}}[T_1, \dotsc , T_r]$ via . For any $w \in K$ we denote $S_w \subseteq S$ for the finite-dimensional vector space of homogeneous polynomials of degree w.

Moreover, fix a lattice polytope $B \subseteq {{\mathbb {Q}}}^n$ and set

We call $\mu = Q(a(\Sigma )) \in K$ the $\Sigma $-degree of B. Besides regarded as a divisor class is base point free if $\Sigma $ refines the normal fan of B. The $\Sigma $-homogenization of a Laurent polynomial with Newton polytope B(f) equal to B is the $\mu $-homogeneous polynomial $g = T^{a(\Sigma )} p^* f \in S$ where $p:{{\mathbb {T}}}^r \rightarrow {{\mathbb {T}}}^n$ is the homomorphism of tori associated with P. Each spread polynomial $g \in S_{\mu }$ arises as $\Sigma $-homogenization of a Laurent polynomial f with $B(f) = B$.

Let $\Sigma _1$, $\Sigma _2$ be lattice fans refining the normal fan $\Sigma (B)$ of B. The vector space V(B) of all Laurent polynomials of the form $\sum _{\nu \in B \cap {{\mathbb {Z}}}^r} a_\nu T^\nu $ fits into the following commutative diagram of vector space isomorphisms:

Moreover, if $g \in S_{\mu _1}$ is spread, then $\varphi (g) \in S_{\mu _2}$ is spread as well and g, $\varphi (g)$ are homogenizations of a common Laurent polynomial with respect to different fans $\Sigma _i$.

We state an adapted version of [5, Theorem 4.1.2.2]; see also [5, Proposition 4.1.2.4].

Theorem 5.8

Let be a Laurent polynomial and $\Sigma _2 \preceq \Sigma _1$ a refinement of fans in . Moreover, let $g_i \in {{\mathbb {K}}}[T_1, \dotsc , T_{r_i}]$ be the respective $\Sigma _i$-homogenization of f and consider the $K_i$-graded algebra

$$\begin{aligned} R_{g_i} = {{\mathbb {K}}}[T_1, \dotsc , T_{r_i}] / \langle g_i \rangle . \end{aligned}$$

Assume that $g_1, g_2$ are prime polynomials, $T_1, \dotsc , T_{r_1}$ define $K_1$-primes in $R_{g_1}$ and $T_1, \dotsc , T_{r_2}$ define $K_2$-primes in $R_{g_2}$. Then the following statements are equivalent:

(i)
The algebra $R_{g_1}$ is factorially $K_1$-graded.
(ii)
The algebra $R_{g_2}$ is factorially $K_2$-graded.

Now let us bring this theorem in the context of general hypersurface rings. We observe that factoriality is inherited between general hypersurface rings with relation degrees stemming from a common lattice polytope.

Proposition 5.9

Let $B \subseteq {{\mathbb {Q}}}^n$ be a lattice polytope, $\Sigma _2 \preceq \Sigma _1 \preceq \Sigma (B)$ a refinement of fans in , and $\mu _i \in K_i$ the respective $\Sigma _i$-degree. Assume that for $i = 1, 2$ there is a $\mu _i$-homogeneous prime polynomial $g_i$ and a non-empty open subset $U_i \subseteq S_{\mu _i}$ such that for all $g_i \in U_i$ the variables $T_1, \dotsc , T_{r_i}$ define $K_i$-primes in the $K_i$-graded algebra

$$\begin{aligned} R_{g_i} = {{\mathbb {K}}}[T_1, \dotsc , T_{r_i}] / \langle g_i \rangle . \end{aligned}$$

Then the following statements are equivalent:

(i)
There is a non-empty open subset of polynomials $g_1 \in S_{\mu _1}$ such that $R_{g_1}$ is $K_1$-factorial.
(ii)
There is a non-empty open subset of polynomials $g_2 \in S_{\mu _2}$ such that $R_{g_2}$ is $K_2$-factorial.

Proof

We know that the subset of spread $\mu _i$-homogeneous polynomials is open and non-empty. According to Remark 5.7 there is an isomorphism of vector spaces such that g and $\varphi (g)$ arise as $\Sigma _i$-homogenization of the same Laurent polynomial whenever $g \in U_{\mu _1}$. Besides, by [20, Lemma 4.9] the $\mu _i$-homogeneous prime polynomials form an open subset of $S_{\mu _i}$, which is non-empty by assumption. Therefore, by suitably shrinking $U_1$ and $U_2$ we achieve

$\varphi (U_1) = U_2$,
$g_1$ and are respective $\Sigma _i$-homogenizations of a common Laurent polynomial whenever $g_1 \in U_1$,
for every $g_1 \in U_1$ the ring $R_{g_1}$ is integral and are $K_1$-prime,
for every $g_2 \in U_2$ the ring $R_{g_2}$ is integral and are $K_2$-prime.

In this situation Theorem 5.8 tells us that for any $g_1 \in U_1$ and we have

$$\begin{aligned} R_{g_1} \text { is } K_1\text {-factorial} \quad \Longleftrightarrow \quad R_{g_2} \text { is } K_2\text {-factorial}. \end{aligned}$$

Now let $V_1 \subseteq S_{\mu _1}$ be a non-empty open subset such that $R_{g_1}$ is factorially graded for each $g_1 \in V_1$. Then is a non-empty open subset of $S_{\mu _2}$ and $R_{g_2}$ is $K_2$-factorial for all $g_2 \in V_2$. This proves “(i) $\Rightarrow $ (ii)”. The inverse implication is shown analogously.$\square $

Remark 5.10

In the situation of Construction 5.1, assume that Z is a fake weighted projective space, i.e., Z is ${{\mathbb {Q}}}$-factorial and is of rank one. Then is base point free if and only if there is an $l_i \in {{\mathbb {Z}}}_{\geqslant 1}$ with $\mu = l_i w_i$ for all $1 \leqslant i \leqslant 6$.

According to [21, Remark 5.8] general base point free hypersurfaces in fake weighted projective spaces of dimension at least four always stem from Cox ring embeddings.

Proposition 5.11

In the situation of Construction 5.1, suppose that K is of rank one, $r \geqslant 5$ holds and that for any $i = 1, \dotsc , r$ there is an $l_i \in {{\mathbb {Z}}}_{\geqslant 1}$ with $\mu = l_i w_i$. Then there is a non-empty open subset of polynomials $g \in S_\mu $ such that the ring $R_g$ is normal and K-factorial, and $T_1, \dotsc , T_r \in R_g$ are prime.

Corollary 5.12

Let $n \geqslant 4$, $B \subseteq {{\mathbb {Q}}}^n$ an integral n-simplex, $\Sigma $ a fan in ${{\mathbb {Z}}}^n$ refining the normal fan of B, and $\mu \in K$ the $\Sigma $-degree of B. Assume that there is a $\mu $-homogeneous prime polynomial g and a non-empty open subset $U \subseteq S_{\mu }$ such that for all $g \in U$ the variables $T_1, \dotsc , T_{r}$ define K-primes in the K-graded algebra

$$\begin{aligned} R_{g} = {{\mathbb {K}}}[T_1, \dotsc , T_{r}] / \langle g \rangle . \end{aligned}$$

Then there is a non-empty open subset of polynomials $g \in S_{\mu }$ such that $R_{g}$ is K-factorial.

Proof

Since B is a simplex, the toric variety associated with $\Sigma (B)$ is a fake weighted projective space. Now we apply Proposition 5.9 to the refinement $\Sigma \preceq \Sigma (B)$ and the suitable open subset of polynomials provided by Proposition 5.11.$\square $

In many situations we encounter it can be read of straight from the specifying data whether the conditions from Corollary 5.12 are met.

Corollary 5.13

Situation as in Construction 5.1. Assume that we have $r \geqslant 5$, $K = {{\mathbb {Z}}}^2$ and the degree matrix is of the form

$$\begin{aligned} Q = [w_1, \dotsc , w_{r+1}] = \begin{bmatrix} x_1 &{} \cdots &{} x_{r} &{} 0 \\ -d_1 &{} \cdots &{} -d_{r} &{} 1 \end{bmatrix}, \quad x_i \in {{\mathbb {Z}}}_{\geqslant 1},\;\; d_i \in {{\mathbb {Z}}}_{\geqslant 0}. \end{aligned}$$

Then for any $\mu = (\mu _1, \mu _2) \in K = {{\mathbb {Z}}}^2$ satisfying the subsequent conditions there is a non-empty open subset of polynomials $g \in S_\mu $ such that $R_g$ is factorial:

(i):: for each i there exists some $l_i \in {{\mathbb {Z}}}_{\geqslant 1}$ with $\mu = l_i x_i$,
(ii):: $\mu _2 = {}- \min _{\nu } \nu _1 d_1 + \cdots + d_{r} \nu _{r}$ where the minimum runs over all lattice points $\nu = (\nu _1, \dotsc , \nu _r) \in {{\mathbb {Z}}}^r_{\geqslant 0}$ with $\nu _1 x_1 + \cdots + \nu _{r} x_{r} = \mu _1$,
(iii):: there is some $g \in S_\mu $ such that $T_1, \dotsc , T_{r+1}$ define primes in $R_g$.

Proof

Observe that each $r-1$ of $x_1, \dotsc , x_r$ generate ${{\mathbb {Z}}}$ as a group since the first coordinate of $w_{r+1}$ vanishes and the ${{\mathbb {Z}}}^2$-grading associated with Q is almost free according to the assumptions made in Construction 5.1. Consider the weighted projective space

Condition (i) ensures that regarded as a divisor class on $Z'$ is ample and base point free. Choose some representative of $\mu _1$. The associated divisorial polytope is a full-dimensional integral simplex.

The normal fan $\Sigma '$ of B is a lattice fan in ${{\mathbb {Z}}}^{r-1}$ corresponding with . Write $v_1, \dotsc , v_r \in {{\mathbb {Z}}}^{r-1}$ for the primitive ray generators of . Observe that the maps

fit into a mutually dual pair of exact sequences as shown in Remark 5.7. Now set

The second row of Q encodes the relation satisfied by $v_1, \dotsc , v_{r+1}$ thus indicates that the following maps constitute a pair of mutually dual sequences as well:

Since the first r columns of Q generate , the vector $v_{r+1} \in {{\mathbb {Z}}}^{r-1}$ is primitive; see [5, Lemma 2.1.4.1]. This allows us to consider the stellar subdivision $\Sigma $ of $\Sigma '$ along $v_{r+1}$.

We show that $\mu \in {{\mathbb {Z}}}^2$ is the $\Sigma $-degree $\mu _B$ of B. First note that $\mu _1$ is the $\Sigma '$-degree of B by construction. Consider

from Remark 5.7. Since $\Sigma $ arises from $\Sigma '$ by introducing an $(r\,{+}\,1)$-th ray, we have $a_i = a_i'$ for $i = 1, \dotsc , r$. From this we infer

$$\begin{aligned} \mu _1 = Q'(a') = a_1 x_1 + \cdots + a_r x_r, \quad \mu _B = Q(a) = a_1 w_1 + \cdots + a_{r+1} w_{r+1}. \end{aligned}$$

As the first coordinate of $w_{r+1}$ vanishes, we conclude that the first coordinate of $\mu _B$ equals $\mu _1$. It remains to investigate the second coordinate of $\mu _B$. We have

$$\begin{aligned} a_{r+1} = {}- \min _{u \in B} \,\langle u,\, v_{r+1} \rangle = {}- \min _{u \in B} \,\langle u,\, P'(d) \rangle ={} - \min _{u \in B} \,\langle (P')^*u,\, d \rangle . \end{aligned}$$

Using this presentation of $a_{r+1}$, the second coordinate of $\mu _B$ is given as

From condition (ii) and the fact that the lattice points $\nu \in {{\mathbb {Z}}}^r_{\geqslant 0}$ with $Q'(\nu ) = \mu _1$ are precisely those of the form , where , follows that the second coordinate of $\mu _B$ equals $\mu _2$. Altogether we have verified $\mu = \mu _B$.

Note that condition (iii) ensures the existence of a $\mu $-homogeneous prime polynomial; see Proposition 5.5. The above discussion combined with condition (iii) ensures that we may apply Corollary 5.12 to B and $\Sigma $, which finishes the proof.$\square $

6 Proof of Theorem 1.1: verification

The second mission in the proof of Theorem 1.1 is to ensure that the list of specifying data given there does not contain any superfluous items. So we have to verify that all items from Theorem 1.1 are realized by pairwise non-isomorphic smooth Calabi–Yau threefolds having a (general) hypersurface Cox ring.

Let us highlight generator and relation degrees of a graded algebra as invariants of hypersurface Cox rings that distinguish varieties with different specifying data; see [19, Section 2] for details.

Remark 6.1

Let $R = \bigoplus _{w \in K} R_w$ be an integral pointed K-graded algebra. We denote $S(R) = \{w \in K;\, R_w \ne 0\}$. An important invariant of R is the set of generator degrees

where $R_{< w}$ denotes the subalgebra of R spanned by all homogeneous components $R_{w'}$ such that $w = w' + w_0$ holds for some $0 \ne w_0 \in S(R)$. In the situation of Setting 3.1 the set of generator degrees is given as

$$\begin{aligned} \Omega _{R} = \{w_1, \dotsc , w_r\} \subseteq K. \end{aligned}$$

The set of generator degrees is unique and does not depend on a graded presentation of R. From this emerges another invariant: Choose pairwise different $u_1, \dotsc , u_l \in K$ such that $\Omega _R = \{u_1, \dotsc , u_l\}$ and set . By suitably reordering $u_1, \dotsc , u_l$ we achieve $d_1 \leqslant \dotsc \leqslant d_l$. We call $(d_1, \dotsc , d_l)$ the generator degree dimension tuple of R. If two graded algebras are isomorphic, then they have the same generator degree dimension tuple.

Moreover, if R admits an irredundant graded presentation $R = {{\mathbb {K}}}[T_1, \dotsc , T_r] / \langle g \rangle $, then the relation degree is unique and does not depend on the choice of the minimal graded presentation.

Lemma 6.2

Consider n-dimensional varieties $X_1,X_2$ with hypersurface Cox rings having relation degree $\mu _1$ resp. $\mu _2$. If $X_1$ and $X_2$ are isomorphic, then $\mu _1^n = \mu _2^n$ where $\mu _i^n$ is the self-intersection number of $\mu _i$ regarded as a divisor class on $X_i$.

Proof

Let $\varphi :X_1 \rightarrow X_2$ be an isomorphism. Then the induced pullback maps

form an isomorphism of -graded algebras. From this we deduce that the pullback of the relation degree of ${\mathscr {R}}(X_2)$ is the unique relation degree of ${\mathscr {R}}(X_1)$; see also Remark 6.1. Hence $\mu _1^n = {\tilde{\varphi }}^*(\mu _2)^n = \mu _2^n$.$\square $

Proof of Theorem 1.1: Verification

We show that each item from Theorem 1.1 indeed stems from a smooth Calabi–Yau threefold with a general hypersurface Cox ring.

Let $(Q, \mu , u)$ be specifying data as presented in Theorem 1.1. Consider the linear K-grading on $S = {{\mathbb {K}}}[T_1, \dotsc , T_6]$ given by $Q:{{\mathbb {Z}}}^6 \rightarrow K$. We run Construction 5.1 with the unique GIT-chamber $\tau \in \Lambda (S)$ containing u in its relative interior . In doing so guarantees . In what follows we construct a non-empty open subset $U \subseteq U_\mu $ of polynomials satisfying the conditions from Remark 5.4, thereby obtaining a smooth general Calabi–Yau hypersurface Cox ring. This is done by starting with and shrinking U successively.

Since $\mu \ne w_i$ holds for all i, Remark 5.4 ensures that $T_1, \dotsc , T_6$ form a minimal system of generators for $R_g$, whenever $g \in U_\mu $. We want to achieve K-primeness of $T_1, \dotsc , T_6 \in R$. Here Numbers 2 and 7 have to be treated separately. For all remaining items from Theorem 1.1 and any $1 \leqslant i \leqslant 6$ we find in Table 1 a $\mu $-homogeneous prime binomial $T^{\kappa } - T^{\nu } \in S$ not depending on $T_i$. Thus, Proposition 5.5 allows us to shrink U such that $T_1, \dotsc , T_6$ define primes in $R_g$ for all $g \in U$.

Number 2. For Number 2 observe that all the generator degrees are indecomposable in the weight monoid

$$\begin{aligned} S(R) = \{ u \in K;\, R_u \ne 0 \} = \mathrm{Pos}_{{\mathbb {Z}}}(w_1, \dotsc , w_6) \subseteq K. \end{aligned}$$

Thus every $T_i \in R_g$ is K-irreducible. As soon as we know that $R_g$ is K-factorial, we may conclude that $T_i$ is K-prime.

Number 7. Table 1 shows $\mu $-homogeneous prime binomials $T^{\kappa } - T^{\nu } \in S$ not depending on $T_i$ for $i = 1, \dotsc , 5$. Thus, Proposition 5.5 allows us to shrink U such that $T_1, \dotsc , T_5$ define primes in $R_g$ for all $g \in U$.

Observe that $T_6$ defines a K-prime in $R_g$ if and only if is K-prime. Since S is a UFD, thus K-factorial, the latter is equivalent to $h \in S$ being K-irreducible. The only monomials of degree $\mu $ not depending on $T_6$ are $T_4^3$ and $T_5^3$, hence $h = aT_4^3 - b T_5^3$. Note that $T_4^3,T_5^3$ are vertices of the polytope

From g being spread we infer . For degree reasons, any non-trivial factorization of h has a linear form $\ell = a'T_4 + b'T_5$ with among the factors. From $w_4 \ne w_5$ we deduce that such $\ell $ is not homogeneous w.r.t. the K-grading. We conclude that h admits no non-trivial presentation as product of homogeneous elements, i.e., $h \in S$ is K-irreducible. This implies that $T_6 \in R_g$ is K-prime.

We take the next step, that is to make sure that each $R_g$ is normal and factorially graded. For example this holds when $R_g$ admits unique factorization. Whenever K is torsion-free the converse is also true. Here we encounter different classes of candidates.

Numbers 1, 2, 5, 6, 10–22, and 26–28. One directly checks that the convex hull over the $\nu \in {{\mathbb {Z}}}_{\geqslant 0}^6$ with $Q(\nu ) = \mu $ is Dolgachev polytope; we have used the Magma function IsDolgachevPolytope from [28] for this task. Proposition 5.6 (ii) ensures that $R_g$ is factorial after suitably shrinking U.

Numbers 3, 4, and 30. Here, the cone satisfies . Thus, Construction 5.1 gives raise to a toric variety . We have $\mu \in (\tau ')^\circ $ and one directly verifies that $\mu $ is basepoint free for . Hence Proposition 5.6 (i) shows that after shrinking U suitably, $R_g$ admits unique factorization for all $g \in U$.

Number 7. We are aiming to apply Corollary 5.12. For this purpose we have to verify that $\mu $ occurs as degree associated with a simplex in the sense of Remark 5.7. The following polytope does the job:

$$\begin{aligned} B = \mathrm{conv}\bigl ( (0,0,0,0),\, (0,0,0,3)\, (0,0,9,-3),\, (3,0,3,-1),\, (3,3,3,-2) \bigr ) \subseteq {\mathbb {Q}}^4. \end{aligned}$$

The rays of its normal fan $\Sigma (B)$ are given as the columns of the following matrix:

$$\begin{aligned} P_1 = \begin{bmatrix} -2 &{} 0 &{} -1 &{} 0 &{} 1 \\ -1 &{} 1 &{} 0 &{} 1 &{} -1 \\ -2 &{} 1 &{} 1 &{} 0 &{} 0 \\ -3 &{} 3 &{} 0 &{} 0 &{} 0 \end{bmatrix}. \end{aligned}$$

Now consider the stellar subdivision $\Sigma _2$ of $\Sigma (B)$ along $(-1, 0, 0, 0)$. The associated data of $\Sigma _2$ is and

$$\begin{aligned} P_2 = \begin{bmatrix} -2 &{} 0 &{} -1 &{} 0 &{} 1 &{} -1 \\ -1 &{} 1 &{} 0 &{} 1 &{} -1 &{} 0 \\ -2 &{} 1 &{} 1 &{} 0 &{} 0 &{} 0\\ -3 &{} 3 &{} 0 &{} 0 &{} 0 &{} 0 \end{bmatrix}, \quad Q_2 = \begin{bmatrix} 1 &{} 1 &{} 1 &{} 0 &{} 0 &{} -3 \\ 0 &{} 0 &{} 0 &{} 1 &{} 1 &{} 1 \\ \overline{0} &{} \overline{1} &{} \overline{2} &{} \overline{1} &{} \overline{2} &{} \overline{0} \end{bmatrix}. \end{aligned}$$

We compute the $\Sigma _2$-degree $\mu _2$ of B. Observe $a(\Sigma _2) = (9, 0, 0, 0, 3)$. From this we infer $\mu _2 = Q_2(a(\Sigma _2)) = (0, 3, \overline{0})$. Note that $(Q_2, \mu _2)$ coincides with the specifying data $(Q, \mu )$ for which we run the verification process. In the previous step of this process we have ensured that $U \subseteq S_\mu $ is a non-empty open subset of prime polynomials such that $T_1, \dotsc , T_6$ define K-primes in $R_g$ whenever $g \in U$. According to Corollary 5.12 we may shrink U such that $R_g$ is K-factorial for each $g \in U$.

Finally, Bechtold’s criterion [8, Corollary 0.6], [18, Proposition 4.1] directly implies that $R_g$ is normal since each five of $w_1, \dotsc , w_6$ generate K as a group.

Numbers 8, 9, 10, and 24, 25. By applying a suitable coordinate change we achieve that the degree matrix Q and the relation degree $\mu $ are as in the following table.

No.	Q	$\mu $
8	$ \begin{bmatrix} 1 &{} 1 &{} 1 &{} 6 &{} 9 &{} 0 \\ -1 &{} -1 &{} -1 &{} -4 &{} -6 &{} 1 \end{bmatrix}$	$(18, -12)$
9	$ \begin{bmatrix} 1 &{} 1 &{} 1 &{} 6 &{} 9 &{} 0 \\ -1 &{} -1 &{} -1 &{} -4 &{} -6 &{} 1 \end{bmatrix}$	$(8, -4)$
10	$ \begin{bmatrix} 0 &{} 2 &{} 2 &{} 2 &{} 1 &{} 1 \\ 1 &{} -1 &{} -1 &{} -1 &{} -1 &{} -1 \end{bmatrix}$	$(8, -4)$
24	$ \begin{bmatrix} 0 &{} 2 &{} 2 &{} 4 &{} 3 &{} 1 \\ 1 &{} -1 &{} -1 &{} -2 &{} -2 &{} -1 \end{bmatrix}$	$(12, -6)$
25	$ \begin{bmatrix} 0 &{} 2 &{} 2 &{} 2 &{} 7 &{} 1 \\ 1 &{} -1 &{} -1 &{} -1 &{} -4 &{} -1 \end{bmatrix}$	$(14, -7)$

We apply Corollary 5.13. In the last three cases it is necessary to reorder the variables so that Q has precisely the shape requested by Corollary 5.13. Now the conditions from there can be directly checked. As a result, we may shrink U so that each $R_g$ is a factorial ring.

Number 26. Again we want to use Corollary 5.12 thus we have to present $\mu $ as degree associated with a simplex in the sense of Remark 5.7. Consider

$$\begin{aligned} B = \mathrm{conv}\bigl ((0,0,0,0),\, (0,0,0,8)\, (0,8,0,0),\, (0,0,4,0),\, (2,2,1,2)\bigr ) \subseteq {\mathbb {Q}}^4. \end{aligned}$$

Its normal fan $\Sigma _1 = \Sigma (B)$ has the rays given by the columns of the matrix

$$\begin{aligned} P_1 = \begin{bmatrix} 0 &{} 0 &{} 1 &{} -1 &{} 3 \\ 1 &{} 0 &{} 2 &{} -1 &{} 1 \\ 0 &{} 1 &{} 2 &{} -1 &{} 1 \\ 0 &{} 0 &{} 3 &{} -1 &{} 1 \end{bmatrix}. \end{aligned}$$

Now consider the stellar subdivision $\Sigma _2$ of $\Sigma (B)$ along (1, 0, 0, 0). Here associated data of $\Sigma _2$ is given by $K_2 = {{\mathbb {Z}}}^2$ and

$$\begin{aligned} P_2 = \begin{bmatrix} 1 &{} 0 &{} 0 &{} 1 &{} -1 &{} 3 \\ 0 &{} 1 &{} 0 &{} 2 &{} -1 &{} 1 \\ 0 &{} 0 &{} 1 &{} 2 &{} -1 &{} 1 \\ 0 &{} 0 &{} 0 &{} 3 &{} -1 &{} 1 \end{bmatrix}, \quad Q_2 = \begin{bmatrix} 2 &{} 1 &{} 1 &{} 1 &{} 3 &{} 0 \\ -2 &{} 0 &{} 0 &{} 0 &{} 1 &{} 1 \\ \end{bmatrix}. \end{aligned}$$

We compute the $\Sigma _2$-degrees $\mu _2$ of B. Observe $a(\Sigma _2) = (0, 8, 0, 0, 0)$. From this we infer $\mu _2 = Q_2(a(\Sigma _2)) = (8, 0)$. Here $(Q_2, \mu _2)$ equals $(Q, \mu )$ from the specifying data for which we run the verification process. In the previous step of this process we have ensured that $U \subseteq S_\mu $ is a non-empty open subset such that $T_1, \dotsc , T_6$ define primes in $R_g$ whenever $g \in U$. Now Corollary 5.12 shows that we may shrink U such that $R_g$ is factorial for each $g \in U$.

At this point we have that U defines a general hypersurface Cox ring. Note that Proposition 2.4 immediately yields that the corresponding varieties $X_g$ are weakly Calabi–Yau. The next step is to attain $X_g$ being smooth. Checking the condition from Proposition 5.2 with the help of the Magma program IsMuAmbientSmooth from [28] shows that $Z_\mu $ is smooth in all 30 cases. Observe that we have $\mu \in \tau $ except for Numbers 12, 16, 18, and 27. Whenever $\mu \in \tau $ holds we may apply Corollary 5.3 allowing us to shrink U once more such that $X_g$ is smooth for all $g \in U$. The four exceptional cases turn out to be small quasimodifications of smooth weakly Calabi–Yau threefolds, hence are smooth by Proposition 4.1. Eventually Remark 2.5 (ii) ensures that $X_g$ is Calabi–Yau.

The last task in the proof of Theorem 1.1 is to make sure that two varieties from different families from Theorem 1.1 are non-isomorphic. Note that if two varieties from Theorem 1.1 are isomorphic, then their Cox rings are isomorphic as graded rings. For each family from Theorem 1.1 we give the number l of generator degrees, the entries of the generator degree dimension tuple $(d_1, \dotsc , d_l)$ and the self-intersection number $\mu ^3$ of the relation degree in the following table.

No.	l	$d_1$	$d_2$	$d_3$	$d_4$	$d_5$	$d_6$	$\mu ^3$	No.	l	$d_1$	$d_2$	$d_3$	$d_4$	$d_5$	$\mu ^3$
1	2	3	3	–	–	–	–	486	16	4	1	2	4	8	–	512
2	6	1	1	1	1	1	1	162	17	4	1	2	5	9	–	539
3	3	2	2	6	–	–	–	512	18	4	1	2	5	9	–	512
4	4	2	2	5	31	–	–	864	19	4	1	3	4	10	–	567
5	3	1	3	5	–	–	–	513	20	4	1	3	4	32	–	896
6	6	1	1	1	1	2	3	243	21	4	1	3	7	35	–	992
7	3	1	3	8	–	–	–	594	22	5	1	2	3	4	28	784
8	4	1	3	29	66	–	–	1944	23	5	1	2	4	7	32	912
9	3	1	2	6	–	–	–	512	24	5	1	1	3	4	8	432
10	4	1	2	5	31	–	–	864	25	4	1	1	4	21	–	686
11	3	2	3	7	–	–	–	513	26	4	1	1	3	14	–	512
12	3	2	3	7	–	–	–	512	27	5	1	2	3	6	31	864
13	3	2	3	31	–	–	–	864	28	5	1	2	3	6	31	872
14	4	1	2	4	31	–	–	864	29	5	1	1	3	4	29	808
15	4	1	2	4	8	–	–	520	30	5	1	1	3	4	4	432

Most of the varieties from Theorem 1.1 are distinguished by the generator degree dimension tuple. Note that the pairs having the same generator dimension degree tuple are precisely Numbers 11 & 12, 15 & 16, 17 & 18 and 27 & 28 as they share the same Cox ring. These pairs can be distinguished by the relation degree self-intersection number; see Lemma 6.2.$\square $

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Acknowledgements

The author would like to thank Jürgen Hausen for his constant support and very valuable advice. In addition, the author is grateful to Antonio Laface for helpful discussions and his interest in this work.

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Mauz, C. Calabi–Yau 3-folds of Picard number 2 with hypersurface Cox rings. European Journal of Mathematics 8 (Suppl 2), 533–573 (2022). https://doi.org/10.1007/s40879-022-00553-5

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Received: 07 May 2021
Revised: 16 March 2022
Accepted: 28 April 2022
Published: 07 July 2022
Issue Date: November 2022
DOI: https://doi.org/10.1007/s40879-022-00553-5

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No.	\([w_1, \dotsc , w_6]\)	\( \mu \)	u	No.	\([w_1, \dotsc , w_6]\)	\( \mu \)	u
1	\(\left[ \begin{array}{cccccc} 1 &{} 1 &{} 1 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 1 &{} 1 &{} 1 \end{array}\right] \)	\(\left[ \begin{array}{c} 3 \\ 3 \end{array}\right] \)	\(\left[ \begin{array}{c} 1 \\ 1 \end{array}\right] \)	15	\(\left[ \begin{array}{cccccc} 1 &{} 1 &{} 1 &{} 1 &{} 0 &{} 0 \\ -1 &{} 0 &{} 0 &{} 1 &{} 1 &{} 1 \end{array}\right] \)	\(\left[ \begin{array}{c} 4 \\ 2 \end{array}\right] \)	\( \left[ \begin{array}{c} 2 \\ 1 \end{array}\right] \)
2	\(\left[ \begin{array}{cccccc} 1 &{} 1 &{} 1 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 1 &{} 1 &{} 1 \\ \overline{0} &{} \overline{1} &{} \overline{2} &{} \overline{0} &{} \overline{1} &{} \overline{2} \end{array}\right] \)	\(\left[ \begin{array}{c} 3 \\ 3 \\ \overline{0} \end{array}\right] \)	\(\left[ \begin{array}{c} 1 \\ 1 \\ \overline{0} \end{array}\right] \)	16	\(\left[ \begin{array}{cccccc} 1 &{} 1 &{} 1 &{} 1 &{} 0 &{} 0 \\ -1 &{} 0 &{} 0 &{} 1 &{} 1 &{} 1 \end{array}\right] \)	\(\left[ \begin{array}{c} 4 \\ 2 \end{array}\right] \)	\( \left[ \begin{array}{c} 1 \\ 2 \end{array}\right] \)
3	\( \left[ \begin{array}{cccccc} 1 &{} 1 &{} 1 &{} 1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 1 &{} 1 &{} 1 &{} 1 \end{array}\right] \)	\(\left[ \begin{array}{c} 4 \\ 4 \end{array}\right] \)	\( \left[ \begin{array}{c} 2 \\ 1 \end{array}\right] \)	17	\(\left[ \begin{array}{cccccc} 1 &{} 1 &{} 1 &{} 1 &{} 0 &{} 0 \\ -2 &{} 0 &{} 0 &{} 1 &{} 1 &{} 1 \end{array}\right] \)	\(\left[ \begin{array}{c} 4 \\ 1 \end{array}\right] \)	\(\left[ \begin{array}{c} 2 \\ 1 \end{array}\right] \)
4	\(\left[ \begin{array}{cccccc} 1 &{} 1 &{} 1 &{} 3 &{} 0 &{} 0 \\ 0 &{} 0 &{} 1 &{} 3 &{} 1 &{} 1 \end{array}\right] \)	\(\left[ \begin{array}{c} 6 \\ 6 \end{array}\right] \)	\( \left[ \begin{array}{c} 2 \\ 1 \end{array}\right] \)	18	\(\left[ \begin{array}{cccccc} 1 &{} 1 &{} 1 &{} 1 &{} 0 &{} 0 \\ -2 &{} 0 &{} 0 &{} 1 &{} 1 &{} 1 \end{array}\right] \)	\(\left[ \begin{array}{c} 4 \\ 1 \end{array}\right] \)	\( \left[ \begin{array}{c} 1 \\ 2 \end{array}\right] \)
5	\(\left[ \begin{array}{cccccc} 1 &{} 1 &{} 1 &{} 0 &{} 0 &{} -1 \\ 0 &{} 0 &{} 0 &{} 1 &{} 1 &{} 1 \end{array}\right] \)	\(\left[ \begin{array}{c} 2 \\ 3 \end{array}\right] \)	\( \left[ \begin{array}{c} 1 \\ 1 \end{array}\right] \)	19	\(\left[ \begin{array}{cccccc} 1 &{} 1 &{} 1 &{} 2 &{} 1 &{} 0 \\ 0 &{} 0 &{} 0 &{} 1 &{} 1 &{} 1 \end{array}\right] \)	\(\left[ \begin{array}{c} 6 \\ 3 \end{array}\right] \)	\( \left[ \begin{array}{c} 3 \\ 1 \end{array}\right] \)
6	\(\left[ \begin{array}{cccccc} 1 &{} 1 &{} 1 &{} 0 &{} 0 &{} -2 \\ 0 &{} 0 &{} 0 &{} 1 &{} 1 &{} 1 \end{array}\right] \)	\(\left[ \begin{array}{c} 1 \\ 3 \end{array}\right] \)	\( \left[ \begin{array}{c} 1 \\ 1 \end{array}\right] \)	20	\(\left[ \begin{array}{cccccc} 1 &{} 1 &{} 1 &{} 4 &{} 1 &{} 0 \\ 0 &{} 0 &{} 0 &{} 2 &{} 1 &{} 1 \end{array}\right] \)	\(\left[ \begin{array}{c} 8 \\ 4 \end{array}\right] \)	\( \left[ \begin{array}{c} 3 \\ 1 \end{array}\right] \)
7	\(\left[ \begin{array}{cccccc} 1 &{} 1 &{} 1 &{} 0 &{} 0 &{} -3 \\ 0 &{} 0 &{} 0 &{} 1 &{} 1 &{} 1 \\ \overline{0} &{} \overline{1} &{} \overline{2} &{} \overline{1} &{} \overline{2} &{} \overline{0} \end{array}\right] \)	\(\left[ \begin{array}{c} 0 \\ 3 \\ \overline{0} \end{array}\right] \)	\( \left[ \begin{array}{c} 1 \\ 1 \\ \overline{0} \end{array}\right] \)	21	\(\left[ \begin{array}{cccccc} 1 &{} 1 &{} 1 &{} 5 &{} 2 &{} 0 \\ 0 &{} 0 &{} 0 &{} 2 &{} 1 &{} 1 \end{array}\right] \)	\(\left[ \begin{array}{c} 10 \\ 4 \end{array}\right] \)	\( \left[ \begin{array}{c} 3 \\ 1 \end{array}\right] \)
8	\(\left[ \begin{array}{cccccc} 1 &{} 1 &{} 1 &{} 0 &{} 0 &{} -3 \\ 0 &{} 0 &{} 0 &{} 2 &{} 3 &{} 1 \end{array}\right] \)	\(\left[ \begin{array}{c} 0 \\ 6 \end{array}\right] \)	\( \left[ \begin{array}{c} 1 \\ 1 \end{array}\right] \)	22	\(\left[ \begin{array}{cccccc} 1 &{} 1 &{} 2 &{} 5 &{} 1 &{} 0 \\ 0 &{} 0 &{} 0 &{} 2 &{} 1 &{} 1 \end{array}\right] \)	\(\left[ \begin{array}{c} 10 \\ 4 \end{array}\right] \)	\( \left[ \begin{array}{c} 3 \\ 1 \end{array}\right] \)
9	\(\left[ \begin{array}{cccccc} 1 &{} 1 &{} 1 &{} 1 &{} 0 &{} 0 \\ -2 &{} 0 &{} 0 &{} 0 &{} 1 &{} 1 \end{array}\right] \)	\(\left[ \begin{array}{c} 4 \\ 0 \end{array}\right] \)	\( \left[ \begin{array}{c} 1 \\ 1 \end{array}\right] \)	23	\(\left[ \begin{array}{cccccc} 1 &{} 1 &{} 2 &{} 7 &{} 3 &{} 0 \\ 0 &{} 0 &{} 0 &{} 2 &{} 1 &{} 1 \end{array}\right] \)	\(\left[ \begin{array}{c} 14 \\ 4 \end{array}\right] \)	\( \left[ \begin{array}{c} 4 \\ 1 \end{array}\right] \)
10	\(\left[ \begin{array}{cccccc} 1 &{} 1 &{} 1 &{} 3 &{} 0 &{} 0 \\ -2 &{} 0 &{} 0 &{} 0 &{} 1 &{} 1 \end{array}\right] \)	\(\left[ \begin{array}{c} 6 \\ 0 \end{array}\right] \)	\( \left[ \begin{array}{c} 1 \\ 1 \end{array}\right] \)	24	\(\left[ \begin{array}{cccccc} 1 &{} 1 &{} 1 &{} 2 &{} 1 &{} 0 \\ -2 &{} 0 &{} 0 &{} 0 &{} 1 &{} 1 \end{array}\right] \)	\(\left[ \begin{array}{c} 6 \\ 0 \end{array}\right] \)	\(\left[ \begin{array}{c} 2 \\ 1 \end{array}\right] \)
11	\(\left[ \begin{array}{cccccc} 1 &{} 1 &{} 1 &{} 1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 1 &{} 1 &{} 1 \end{array}\right] \)	\(\left[ \begin{array}{c} 4 \\ 3 \end{array}\right] \)	\( \left[ \begin{array}{c} 2 \\ 1 \end{array}\right] \)	25	\(\left[ \begin{array}{cccccc} 1 &{} 1 &{} 1 &{} 1 &{} 3 &{} 0 \\ -2 &{} 0 &{} 0 &{} 0 &{} 1 &{} 1 \end{array}\right] \)	\(\left[ \begin{array}{c} 7 \\ 0 \end{array}\right] \)	\(\left[ \begin{array}{c} 4 \\ 1 \end{array}\right] \)
12	\(\left[ \begin{array}{cccccc} 1 &{} 1 &{} 1 &{} 1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 1 &{} 1 &{} 1 \end{array}\right] \)	\(\left[ \begin{array}{c} 4 \\ 3 \end{array}\right] \)	\( \left[ \begin{array}{c} 1 \\ 2 \end{array}\right] \)	26	\(\left[ \begin{array}{cccccc} 2 &{} 1 &{} 1 &{} 1 &{} 3 &{} 0 \\ -2 &{} 0 &{} 0 &{} 0 &{} 1 &{} 1 \end{array}\right] \)	\(\left[ \begin{array}{c} 8 \\ 0 \end{array}\right] \)	\(\left[ \begin{array}{c} 4 \\ 1 \end{array}\right] \)
13	\(\left[ \begin{array}{cccccc} 1 &{} 1 &{} 1 &{} 3 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 2 &{} 1 &{} 1 \end{array}\right] \)	\(\left[ \begin{array}{c} 6 \\ 4 \end{array}\right] \)	\( \left[ \begin{array}{c} 1 \\ 1 \end{array}\right] \)	27	\(\left[ \begin{array}{cccccc} 1 &{} 1 &{} 2 &{} 5 &{} 1 &{} 0 \\ 0 &{} 0 &{} 1 &{} 3 &{} 1 &{} 1 \end{array}\right] \)	\(\left[ \begin{array}{c} 10 \\ 6 \end{array}\right] \)	\( \left[ \begin{array}{c} 3 \\ 1 \end{array}\right] \)
14	\(\left[ \begin{array}{cccccc} 1 &{} 1 &{} 1 &{} 3 &{} 0 &{} 0 \\ -1 &{} 0 &{} 0 &{} 1 &{} 1 &{} 1 \end{array}\right] \)	\(\left[ \begin{array}{c} 6 \\ 2 \end{array}\right] \)	\( \left[ \begin{array}{c} 1 \\ 1 \end{array}\right] \)	28	\(\left[ \begin{array}{cccccc} 1 &{} 1 &{} 2 &{} 5 &{} 1 &{} 0 \\ 0 &{} 0 &{} 1 &{} 3 &{} 1 &{} 1 \end{array}\right] \)	\(\left[ \begin{array}{c} 10 \\ 6 \end{array}\right] \)	\( \left[ \begin{array}{c} 3 \\ 2 \end{array}\right] \)
				29	\(\left[ \begin{array}{cccccc} 1 &{} 1 &{} 1 &{} 4 &{} 1 &{} 0 \\ -1 &{} 0 &{} 0 &{} 1 &{} 1 &{} 1 \end{array}\right] \)	\(\left[ \begin{array}{c} 8 \\ 2 \end{array}\right] \)	\( \left[ \begin{array}{c} 5 \\ 1 \end{array}\right] \)
				30	\(\left[ \begin{array}{cccccc} 1 &{} 2 &{} 1 &{} 1 &{} 1 &{} 0 \\ -1 &{} -1 &{} 0 &{} 0 &{} 1 &{} 1 \end{array}\right] \)	\(\left[ \begin{array}{c} 6 \\ 0 \end{array}\right] \)	\( \left[ \begin{array}{c} 2 \\ 1 \end{array}\right] \)

No.	Q	\(\mu \)
8	\( \begin{bmatrix} 1 &{} 1 &{} 1 &{} 6 &{} 9 &{} 0 \\ -1 &{} -1 &{} -1 &{} -4 &{} -6 &{} 1 \end{bmatrix}\)	\((18, -12)\)
9	\( \begin{bmatrix} 1 &{} 1 &{} 1 &{} 6 &{} 9 &{} 0 \\ -1 &{} -1 &{} -1 &{} -4 &{} -6 &{} 1 \end{bmatrix}\)	\((8, -4)\)
10	\( \begin{bmatrix} 0 &{} 2 &{} 2 &{} 2 &{} 1 &{} 1 \\ 1 &{} -1 &{} -1 &{} -1 &{} -1 &{} -1 \end{bmatrix}\)	\((8, -4)\)
24	\( \begin{bmatrix} 0 &{} 2 &{} 2 &{} 4 &{} 3 &{} 1 \\ 1 &{} -1 &{} -1 &{} -2 &{} -2 &{} -1 \end{bmatrix}\)	\((12, -6)\)
25	\( \begin{bmatrix} 0 &{} 2 &{} 2 &{} 2 &{} 7 &{} 1 \\ 1 &{} -1 &{} -1 &{} -1 &{} -4 &{} -1 \end{bmatrix}\)	\((14, -7)\)

Calabi–Yau 3-folds of Picard number 2 with hypersurface Cox rings

Abstract

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Calabi–Yau threefolds in \({\mathbb {P}}^6\)

Elliptic Calabi–Yau Threefolds over a del Pezzo Surface

Calabi-Yau threefolds with large h 2,1

1 Introduction

Theorem 1.1

Remark 1.2

2 Background on Mori dream spaces

Construction 2.1

Construction 2.2

Proposition 2.3

Proposition 2.4

Remark 2.5

Proposition 2.6

Lemma 2.7

Proof

Lemma 2.8

Proof

Proof of Proposition 2.6

3 Combinatorial constraints on smooth hypersurface Cox rings

Setting 3.1

Remark 3.2

Proposition 3.3

Proposition 3.4

Lemma 3.5

Lemma 3.6

Lemma 3.7

Lemma 3.8

Lemma 3.9

Lemma 3.10

Proof

Lemma 3.11

Proof

Lemma 3.12

Proof

Lemma 3.13

Proof

Lemma 3.14

Proof

Lemma 3.15

Proof

Lemma 3.16

Lemma 3.17

Proof

4 Proof of Theorem 1.1: collecting candidates

Proposition 4.1

Proof

Remark 4.2

Proposition 4.3

Proof

Remark 4.4

4.1 Part I

4.2 Part II

4.3 Part III

4.4 Part IV

4.5 Part VII

5 Constructing general hypersurface Cox rings

Construction 5.1

Proposition 5.2

Proposition 5.3

Remark 5.4

Proposition 5.5

Proposition 5.6

Remark 5.7

Theorem 5.8

Proposition 5.9

Proof

Remark 5.10

Proposition 5.11

Corollary 5.12

Proof

Corollary 5.13

Proof

6 Proof of Theorem 1.1: verification

Remark 6.1

Lemma 6.2

Proof

Proof of Theorem 1.1: Verification

Calabi-Yau threefolds with large h ^2,1