Views on level $\ell$ curves, K3 surfaces and Fano threefolds

Abstract

An analogue of the Mukai map $m_g: {\mathcal {P}}_g \rightarrow {\mathcal {M}}_g$ is studied for the moduli ${\mathcal {R}}_{g, \ell }$ of genus g curves C with a level $\ell$ structure. Let ${\mathcal {P}}^{\perp }_{g, \ell }$ be the moduli space of 4-tuples $(S, {\mathcal {L}}, {\mathcal {E}}, C)$ so that $(S, {\mathcal {L}})$ is a polarized K3 surface of genus g, ${\mathcal {E}}$ is orthogonal to ${\mathcal {L}}$ in ${{\,\mathrm{Pic}\,}}S$ and defines a standard degree $\ell$ K3 cyclic cover of S, $C \in \vert {\mathcal {L}} \vert$. We say that $(S, {\mathcal {L}}, {\mathcal {E}})$ is a level $\ell$ K3 surface. These exist for $\ell \le 8$ and their families are known. We define a level $\ell$ Mukai map $r_{g, \ell }: {\mathcal {P}}^{\perp }_{g, \ell } \rightarrow {\mathcal {R}}_{g, \ell }$, induced by the assignment of $(S, {\mathcal {L}}, {\mathcal {E}}, C)$ to $(C, {\mathcal {E}} \otimes {\mathcal {O}}_C)$. We investigate a curious possible analogy between $m_g$ and $r_{g, \ell }$, that is, the failure of the maximal rank of $r_{g, \ell }$ for $g = g_{\ell } \pm 1$, where $g_{\ell }$ is the value of g such that $\dim {\mathcal {P}}^{\perp }_{g, \ell } = \dim {\mathcal {R}}_{g,\ell }$. This is proven here for $\ell = 3$. As a related open problem we discuss Fano threefolds whose hyperplane sections are level $\ell$ K3 surfaces and their classification.

1 Introduction

Our aim is to convince the reader, showing a program and new results, of the interest represented by some complex projective varieties whose curvilinear sections are canonical curves C of genus g, endowed with a distinguished nonzero $\ell$-torsion element $\eta \in {{\,\mathrm{Pic}\,}}C$. Often one says that $(C, \eta )$ is a level $\ell$ curve of genus g, cfr. [7]. Fixing $(g, \ell )$ the moduli space of these pairs is integral, quasi projective and denoted by ${\mathcal {R}}_{g, \ell }$.

To enter further in the matter let us mention two other names from the title: K3 surface and Fano threefold. The K3 surfaces S we consider are very special: they admit a non split cyclic cover of degree $\ell$, still birational to a K3 surface. This is defined by a line bundle ${\mathcal {O}}_S(E):= {\mathcal {E}}$ such that $h^0({\mathcal {O}}_S(\ell E)) = 1$ and $h^0({\mathcal {O}}_S(mE)) = 0$ for $m < \ell$. The study of these surfaces stems from Nikulin’s classification of K3 surfaces with an order $\ell$ symplectic automorphism and the classification implies $\ell \le 8$, [24]. Since then several foundational results, in use here, did follow, cfr. [13,14,15,16, 26].

Now let ${\mathcal {L}} \in {{\,\mathrm{Pic}\,}}S$ be a genus g polarization orthogonal to ${\mathcal {E}}$. Let $\eta := {\mathcal {O}}_C(E)$, where $C \in \vert {\mathcal {L}} \vert$ is smooth, then it turns out that $(C, \eta )$ is a level $\ell$ curve. We say that the triple $(S, {\mathcal {L}}, {\mathcal {E}})$ is a level $\ell$ K3 surface of genus g, see definition (3.1) for some precision. Fixing $\ell$ the moduli of these triples are reducible for infinitely many values of g. However a distinguished irreducible component exists for every g, namely the moduli space of triples $(S, {\mathcal {L}}, {\mathcal {E}})$ such that ${{\,\mathrm{Pic}\,}}S$ is the sum of ${\mathbb {Z}}{\mathcal {L}}$ and its orthogonal lattice. We denote it by

$$\begin{aligned} {\mathcal {F}}^{\perp }_{g, \ell }. \end{aligned}$$

(1)

Finally we come to the moduli space ${\mathcal {P}}^{\perp }_{g, \ell }$ of 4-tuples $(S, {\mathcal {L}}, {\mathcal {E}}, C)$ such that $C \in \vert {\mathcal {L}} \vert$ and $(S, {\mathcal {L}}, {\mathcal {E}})$ defines a point in ${{\mathcal {F}}}^{\perp }_{g, \ell }$. Such a space is strictly related with the first topic considered in our paper. To introduce it let us define the level $\ell$ Mukai map. This is the rational map

$$\begin{aligned} r_{g, \ell }: {\mathcal {P}}^{\perp }_{g, \ell } \rightarrow {\mathcal {R}}_{g, \ell }, \end{aligned}$$

(2)

assigning the moduli point of the 4-tuple $(S, {\mathcal {L}}, {\mathcal {E}}, C)$ to the moduli point of the pair $(C, \eta )$, where $\eta$ is ${\mathcal {O}}_C(E)$. Let ${\mathcal {P}}_g$ be the moduli space of triples $(S, {\mathcal {L}}, C)$, where $(S, {\mathcal {L}})$ is a polarized K3 surface of genus g and $C \in \vert {\mathcal {L}} \vert$, then the previous name is motivated by the well known Mukai map

$$\begin{aligned} m_g: {\mathcal {P}}_g \rightarrow {\mathcal {M}}_g, \end{aligned}$$

(3)

assigning the moduli point of the triple $(S, {\mathcal {L}}, C)$ to the moduli point of the curve C. Some famous connections between canonical curves of genus g, K3 surfaces and Fano threefolds are well represented by $m_g$ and, in particular, by a curious variation of its rank. We recall that a rational map $f: X \rightarrow Y$ of integral varieties has maximal rank if $\dim f(X) = \min \lbrace \dim X, \dim Y \rbrace$.

Considering $m_g$ we recall that $\dim {\mathcal {P}}_g = 19 + g$ and $\dim {\mathcal {M}}_g = 3g-3$, therefore $\dim {\mathcal {P}}_g = \dim {\mathcal {M}}_g$ iff $g = 11$. Now $m_{11}$ is birational but, curiously, $m_g$ fails to be of maximal rank precisely before and after this transition value, that is, for $g = 11 \pm 1$. For the rest $m_g$ is dominant for $g \le 9$ and generically injective for $g \ge 13$. As is well known this anomaly is due to the presence behind the scene of some Fano varieties, whose curvilinear sections are general canonical curves of genus $11 \pm 1$, cfr. [8, 22, 23, 25].

A task of this paper is to point out the same possible anomalies for the level $\ell$ Mukai maps $r_{g, \ell }$. The case $\ell = 2$ has already been done and it is an experimental origin to this work. If $\ell = 2$ we have $\dim {\mathcal {P}}^{\perp }_{g,2} = \dim {\mathcal {R}}_{g,2}$ for $g = 7$. Then $r_{g,2}$ fails to be of maximal rank for $g = 7\pm 1$ and is birational for $g = 7$, [11, 19, 27]. The ’Fano varieties behind the scene’ for $g = 8$ and $g = 6$ are addressed or revisited in Sect. 7.

In Sect. 5 we summarize the question for each $\ell$. Let $g_{\ell }$ be the unique value of g such that $\dim {\mathcal {P}}^{\perp }_{g, \ell } = \dim {\mathcal {R}}_{g, \ell }$, for $l = 2, \ 3, \ 4, \ 5, \ 6, \ 7, \ 8$ we respectively have:

$$\begin{aligned} g_{\ell } = 7 \ , \ 5 \ , \ 4 \ , \ 3 \ , \ 2 \ , \ 2 \ , \ 2. \end{aligned}$$

(4)

In this paper we present the following theorem, solving the question for $\ell = 3$.

Theorem 1.1

Let $r_{g, 3}: {\mathcal {P}}^{\perp }_{g, 3} \rightarrow {\mathcal {R}}_{g,3}$ be the level 3 Mukai map then:

• (1) $r_{4,3}$ has not maximal rank,

• (2) $r_{5,3}$ is birational,

• (3) $r_{6,3}$ has not maximal rank.

The image of $r_{4,3}$ is contained in a divisor of ${\mathcal {R}}_{4,3}$, parametrizing pairs $(C, \eta )$ such that the multiplication map $\mu : H^0(\omega _C \otimes \eta ) \otimes H^0(\omega _C \otimes \eta ^{-1}) \rightarrow H^0(\omega _C^{\otimes 2})$ is not an isomorphism. This case seems interestingly related to the $G_2$-variety, see [23] and Sect. 7.

The proof of (3) is sketched here and it will appear elsewhere. The image of $r_{6,3}$ parametrizes pairs $(C, \eta )$, where C is a curvilinear section of a suitable Gushel–Mukai threefold singular along a rational normal sextic curve, see Sect. 7.

Let $(S, {\mathcal {L}}, {\mathcal {E}})$ be a level $\ell$ K3 surface of genus g and $\phi : S \rightarrow {\mathbb {P}}^g$ the morphism defined by ${\mathcal {L}}$, we assume for simplicity that $\phi$ is birational onto ${\overline{S}} := \phi (S)$. Then we close this introduction with few lines addressing the classification of Fano threefolds

$$\begin{aligned} {\overline{X}} \subset {\mathbb {P}}^{g+1} \end{aligned}$$

whose general hyperplane sections are projective models ${\overline{S}}$ as above. The problem sounds similar to that of classifying threefolds $T \subset {\mathbb {P}}^g$ whose hyperplane sections are Enriques surfaces, that is, Enriques–Fano threefolds. It seems however quite neglected.

Some examples of threefolds ${\overline{X}}$ appear in this paper, most are normal and ${{\,\mathrm{Sing}\,}}{\overline{X}}$ is a curve. Moreover ${\overline{X}}$ admits a cyclic cover $\pi : {\tilde{X}} \rightarrow {\overline{X}}$, branched exactly on ${{\,\mathrm{Sing}\,}}{\overline{X}}$. A basic notion of level $\ell$ polarized projective variety $(X, {\mathcal {L}}, {\mathcal {E}})$ is introduced in the next section, since it is useful in the cases we want to consider.

We wish to thank the referee for the careful reading and the useful advice.

2 Some preliminaries

In what follows X is a smooth, irreducible complex projective variety and ${\mathcal {L}}$ is a big and nef line bundle on X, we say that $(X, {\mathcal {L}})$ is a polarized projective variety. On the other hand we are interested, along this paper, in some families of cyclic coverings

$$\begin{aligned} \pi : {\tilde{X}} \rightarrow X. \end{aligned}$$

(5)

Then we fix our conventions about, [10, 21, I p.242]. By definition $\pi$ is a finite morphism of degree $\ell \ge 2$ and it is the quotient map of the action of an automorphism of order $\ell$ of ${\tilde{X}}$. We assume that ${\tilde{X}}$ is normal, up to composing $\pi$ with the normalization map. Hence ${\tilde{X}}$ is reduced with irreducible connected components. Starting from $\pi$, we briefly review the recipe for its construction. Notice that $\pi _* {\mathcal {O}}_{{\tilde{X}}} \cong {\mathcal {A}}$, where

$$\begin{aligned} {\mathcal {A}} = {\mathcal {O}}_X \oplus {\mathcal {E}}^{-1} \oplus \cdots \oplus {\mathcal {E}}^{-\ell +1} \end{aligned}$$

(6)

and ${\mathcal {E}} \in {{\,\mathrm{Pic}\,}}X$. Assume ${\tilde{X}}$ is connected and hence irreducible. Then $\pi$ defines the field extension $\pi ^*: k(X) \rightarrow k({\tilde{X}})$ and its trace map induces the exact sequence

$$\begin{aligned} 0 \rightarrow {{\mathcal {E}}}^{- \ell } {\mathop {\rightarrow }\limits ^{ s}}{\mathcal {O}}_X \rightarrow {\mathcal {O}}_B \rightarrow 0, \end{aligned}$$

(7)

for some $s \in H^0({\mathcal {E}}^{ \ell })$. The multiplication by s defines a structure of ${\mathcal {O}}_X$-Algebra on ${\mathcal {A}}$. We have ${\tilde{X}} = {{\,\mathrm{Spec}\,}}{\mathcal {A}}$, moreover $\pi$ factors through the projection $u: {\mathbb {P}}({\mathcal {A}}) \rightarrow X$. The branch divisor of $\pi$ is $\mathrm{div}(s)$ and will be denoted by B. For B we fix the notation

$$\begin{aligned} B = m_1B_1 + \cdots + m_rB_r, \end{aligned}$$

(8)

where $B_1, \ldots , B_r$ are prime divisors. Conversely, a pair $({\mathcal {E}}, B)$ such that $B \in \vert {\mathcal {E}}^{\ell } \vert$ defines on ${\mathcal {A}}$ an ${\mathcal {O}}_X$- Algebra structure as above and a cyclic cover $\pi$. Notice that the condition $g.c.d. (\ell , m_1, \ldots , m_r) = 1$ implies the irreducibility of ${\tilde{X}}$.

Now let C be a reduced curve and $\eta \in {{\,\mathrm{Pic}\,}}C$ a nontrivial $\ell$-torsion element. Then $(C, \eta )$ uniquely defines, using a nonzero vector $s \in H^0(\eta ^{\ell })$, a nonramified cyclic cover

$$\begin{aligned} \pi : {\tilde{C}} \rightarrow C, \end{aligned}$$

which is nontrivial. To give a pair $(C, \pi )$ is equivalent to give a singular level $\ell$ curve $(C, \eta )$. Now recall that a curve $C \subset X$ is mobile if moves in an irreducible algebraic family covering X, with integral general member. In the Néron–Severi group $N_1(X) \otimes _{{\mathbb {Z}}} {\mathbb {R}}$ the mobile classes of such curves generate an important convex cone, [5, 1.3 (vi)], [21, II p. 307]. Finally we introduce the following definition.

Definition 2.1

Let ${\mathcal {E}} \in {{\,\mathrm{Pic}\,}}X$, the pair $(X, {\mathcal {E}})$ is a level $\ell$ structure on X if:

$\circ$ $\vert {\mathcal {E}}^{\ell } \vert \ne \emptyset$ and a general $B \in \vert {\mathcal {E}}^{\ell } \vert$ defines an integral cyclic cover,

$\circ$ there exists a mobile curve C in X such that $C B = 0$.

Assume $\dim X = 1$ then X is the smooth, integral curve C and ${\mathcal {E}}$ is a line bundle of degree 0 such that ${\mathcal {E}}^{\ell } \cong {\mathcal {O}}_C$. Moreover we are assuming that the cover $\pi : {\tilde{C}} \rightarrow C$ defined by ${\mathcal {E}}$ is integral. Hence ${\mathcal {E}}$ is a nontrivial $\ell$-torsion element. Then, for curves, the definition is the traditional one. In higher dimension the next property is clear.

Proposition 2.1

Let $(X, {\mathcal {E}})$ be a level $\ell$ structure on X and $C \subset X$ a mobile curve such that $CE = 0$, where ${\mathcal {O}}_X(E) \cong {\mathcal {E}}$. Then ${\mathcal {O}}_C(E)$ is an $\ell$-torsion element of ${{\,\mathrm{Pic}\,}}C$.

Proof

Consider $D \in \vert {\mathcal {E}}^{\ell } \vert$. Since C is movable we can assume that C is not a component of D. Then $C \cap D$ is empty because $CE= 0$. This implies that ${\mathcal {E}}^{\ell } \otimes {\mathcal {O}}_C \cong {\mathcal {O}}_C(D) \cong {\mathcal {O}}_C$. $\square$

Remark 2.1

Nevertheless we may have a trivial ${\mathcal {O}}_C(E)$ even when ${\mathcal {E}}$ is not, and even generically when C moves in its family. This is obvious if C is smooth and rational. Furthermore consider a curve F and the projection $p: F \times X \rightarrow X$. Then $(F \times X, p^* {\mathcal {E}})$ is a level $\ell$-structure on $F \times X$ and $p^* {\mathcal {E}}$ is trivial on the mobile curve $p^*(x)$, $x \in X$.

Then, to address the concrete topics of our paper, we turn to polarized pairs $(X, {\mathcal {L}})$ and we denote by d the dimension of X. We assume that $\vert {\mathcal {L}}^{m} \vert$ is globally generated for $m>>0$ and observe that a general complete intersection of $d-1$ elements of $\vert {\mathcal {L}}^{m} \vert$ is a smooth, integral mobile curve, which moves in an irreducible family ${\mathcal {C}}_m$ of transversal complete intersections in X.

Proposition 2.2

Let $X, \ {\mathcal {L}}, \ {\mathcal {E}}$ be as above. Assume $C E = 0$, where $C \in {\mathcal {C}}_m$ and ${\mathcal {O}}_X(E) \cong {\mathcal {E}}$. Then ${\mathcal {O}}_C(E)$ is a nontrivial $\ell$-torsion element of ${{\,\mathrm{Pic}\,}}C$, moreover

$$\begin{aligned} h^0({\mathcal {O}}_X(kE)) = 0, \ \ k \not \equiv 0 \mod \ell . \end{aligned}$$

Proof

By induction on $d = \dim X$. Let $d = 1$ then $X = C$ and $\lbrace C \rbrace = {\mathcal {C}}_m$. Since ${\mathcal {E}}$ defines an integral cover, the statement follows. Let $d \ge 2$ and $C = D_1 \cdot \cdots \cdot D_{d-1}$, where $D_1, \ldots , D_{d-1} \in \vert {\mathcal {L}}^{m} \vert$, then a general D in the linear system generated by $D_1 \cdots D_{d-1}$ is smooth. ${\mathcal {O}}_D(D)$ is nef, big and globally generated. Let $\pi : {\tilde{X}} \rightarrow X$ be the cyclic cover, branched on B, since C is mobile and $CB = 0$ we can assume $C \cap B = \emptyset$. Now let $f: X \rightarrow {\mathbb {P}}^n$ be the morphism defined by $\vert D \vert$, then f is generically finite onto its image and the same is true for $f \circ \pi : {\tilde{X}} \rightarrow {\mathbb {P}}^n$. Then ${\tilde{C}} = \pi ^{-1}(C)$ is connected by the connectedness theorem and ${\mathcal {O}}_C(E)$ is non trivial of $\ell$-torsion in ${{\,\mathrm{Pic}\,}}C$. Moreover $(D, {\mathcal {O}}_D(E))$ is a level $\ell$ structure and the second statement follows by induction on d. $\square$

Keeping this notation we finally come to the following definition.

Definition 2.2

A level $\ell$ polarized variety is a triple $(X, {\mathcal {L}}, {\mathcal {E}})$ such that $(X, {\mathcal {E}})$ is a level $\ell$ structure on X and $CE = 0$, where $C \in {\mathcal {C}}_m$.

Actually the triples $(X, {\mathcal {L}}, {\mathcal {E}})$ we will consider always satisfy the additional property:

$\vert {\mathcal {L}} \vert$ is base point free and defines a birational morphism onto its image

$$\begin{aligned} f: X \rightarrow {\mathbb {P}}^n. \end{aligned}$$

(9)

Hence we assume $C = H_1 \cap \cdots \cap H_{d-1} \in {\mathcal {C}}_1$, where $H_1 \cdots H_{d-1} \in \vert f^* {\mathcal {O}}_{{\mathbb {P}}^n}(1) \vert$. So C shows the distinguished line bundles $\eta _C := {\mathcal {E}} \otimes {\mathcal {O}}_C$ and ${\mathcal {L}}_C := {\mathcal {L}} \otimes {\mathcal {O}}_C$ and these lead us to the varieties we are interested in. For these ${\mathcal {L}}_C$ is the canonical sheaf $\omega _C$. For the triples considered, we will also have that the restriction $r: H^0({\mathcal {L}}) \rightarrow H^0(\omega _C)$ is surjective and that ${\overline{X}} := f(X)$ is normal. So we are going to deal with projective varieties ${\overline{X}}$ whose curvilinear sections are canonical curves C, endowed with the étale cover defined by $\eta _C$. This includes K3 surfaces and Fano threefolds with a prescribed level $\ell$ structure.

3 Level $\ell$ K3 surfaces

We begin discussing the families of level $\ell$ polarized K3 surfaces $(S, {\mathcal {L}}, {\mathcal {E}})$ and the chances that $C \in \vert {\mathcal {L}} \vert$ be a curve with general moduli. We say that $C^2 = 2g - 2$ is the degree of $(S, {\mathcal {L}})$ and g its genus. As usual the moduli space of $(S, {\mathcal {L}})$ is denoted by

$$\begin{aligned} {\mathcal {F}}_g, \end{aligned}$$

(10)

it is an integral quasi projective variety of dimension 19. Let $[S, {\mathcal {L}}] \in {\mathcal {F}}_g$ be a general point, we recall that then ${{\,\mathrm{Pic}\,}}S \cong {\mathbb {Z}} {\mathcal {L}}$ and $\vert {\mathcal {L}} \vert$ defines an embedding

$$\begin{aligned} f: S \rightarrow {\mathbb {P}}^g \end{aligned}$$

(11)

for $g \ge 3$. Coming to level $\ell$ structures $(S, {\mathcal {L}}, {\mathcal {E}})$, these properties are no longer satisfied, as we are going to recall. We fix our notation as follows, the map

$$\begin{aligned} \pi ': {\tilde{S}}' \rightarrow S \end{aligned}$$

(12)

is the covering morphism defined by ${\mathcal {E}}$. As already established its branch divisor is

$$\begin{aligned} B = m_1B_1 + \cdots + m_rB_r, \end{aligned}$$

where $B_1, \ldots , B_r$ are the irreducible components of ${{\,\mathrm{Supp}\,}}B$. Of course, since ${{\,\mathrm{Pic}\,}}S$ has no torsion, B is not zero. We fix the following convention:

$\circ$:

r is the number of irreducible components of ${{\,\mathrm{Supp}\,}}B$,

$\circ$:

t is the number of its connected components.

Moreover we set

$$\begin{aligned} B_1 + \cdots + B_r = B_{\mathsf {red}} = N_1 + \cdots + N_t, \end{aligned}$$

(13)

where $N_1 \cdots N_t$ denote the connected components of ${{\,\mathrm{Supp}\,}}B$. Notice that $CB_i = 0$ for $i = 1 \cdots r$. Indeed C is integral and $\dim \vert C \vert \ge 1$ so that $CB_i \ge 0$. Since $B \in \vert \ell E \vert$ then $CB = 0$ and this implies $CB_i = 0$. Then, applying the Hodge Index Theorem, $B_i$ is an integral curve on S with $B^2_i < 0$. Hence $B^2_i = -2$ and $B_i$ is ${\mathbb {P}}^1$. The same argument applies to $N_j$ which is a reduced connected curve of arithmetic genus 0. In particular each $N_j$ is contracted by f to a quadratic singularity and ${{\,\mathrm{Pic}\,}}S$ is not isomorphic to ${\mathbb {Z}}$.

It is not difficult to see that the Kodaira dimension of ${\tilde{S}}'$ is zero, moreover, with some elaboration, one has the following property, cfr. [14, 24].

Proposition 3.1

Either ${\tilde{S}}'$ is birational to a K3 surface or to an abelian surface.

Definition 3.1

Let $(S, {\mathcal {L}}, {\mathcal {E}})$ be a level $\ell$ K3 surface, we say that:

• (1) $(S, {\mathcal {L}}, {\mathcal {E}})$ is of K3 type if ${\tilde{S}}'$ is birational to a K3 surface,

• (2) $(S, {\mathcal {L}}, {\mathcal {E}})$ is of abelian type if ${\tilde{S}}'$ is birational to an abelian surface.

Case (2) is scarcely interesting for our purposes. We aim indeed to use the curves $C \in \vert {\mathcal {L}} \vert$ in order to parametrize the moduli space ${\mathcal {R}}_{g, \ell }$ of level $\ell$ curves in low genus. But in case (2) C has not enough moduli for $g \ge 3$.

We assume since now that $(S, {\mathcal {L}}, {\mathcal {E}})$ is a level $\ell$ K3 surface of K3 type. Then, to ameliorate the expositon, we just say with some abuse that $(S, {\mathcal {L}}, {\mathcal {E}})$ is a level $\ell$ K3 surface. We say that two triples $(S_n, {\mathcal {L}}_n, {\mathcal {E}}_n)$, ($n = 1,2$), are isomorphic if there exists a biregular map $\beta :S_1 \rightarrow S_2$ such that $\beta ^* {\mathcal {L}}_2 \cong {\mathcal {L}}_1$ and $\beta ^* {\mathcal {E}}_2 \cong {\mathcal {E}}_1$, $i = 1,2$.

As mentioned the classification of these triples is due to Nikulin and originates from his paper [24]. The part of interest here is the classification of pairs $({\tilde{S}}, G)$, where ${\tilde{S}}$ is a K3 surface and G is a finite group of symplectic automorphisms of ${\tilde{S}}$. There exist 14 classes of pairs $({\tilde{S}}, G)$ such that G is commutative and G is ${\mathbb {Z}} / \ell {\mathbb {Z}}$ exactly for $2 \le \ell \le 8$. After the classification, several papers addressed the description of the moduli and the projective models of these K3 surfaces. It is due to mention here [13,14,15,16, 26].

The triple $(S, {\mathcal {L}}, {\mathcal {E}})$ determines an associated triple $({\tilde{S}},\tilde{ {\mathcal {L}}}, \gamma )$, where $\gamma \in {{\,\mathrm{Aut}\,}}{\tilde{S}}$ is a symplectic automorphisms of order $\ell$ and $({\tilde{S}}, \tilde{ {\mathcal {L}}})$ is a polarized K3 surface of degree $\ell (2g-2)$. We have indeed $B_{\mathsf {red}} = N_1 + \cdots + N_t$, where the summands are the connected components and $-2$-curves. Let $\nu : S \rightarrow {\overline{S}}$ be their contraction morphism, then the Cartesian square

(14)

is the Stein factorization of $\nu \circ \pi '$. In it $\nu '$ is a birational morphism. Let $G \subset {{\,\mathrm{Aut}\,}}{\tilde{S}}'$ be the group whose quotient map is $\pi '$. As we will see ${\pi '} ^*H^0({\mathcal {L}}(-E))$ sits in $H^0(\tilde{{\mathcal {L}}})$ as an eigenspace of the natural representation of G and defines a generator $\gamma$ of G. Moreover $\pi$ is the quotient map of the induced action of G on ${\tilde{S}}$. Conversely, starting from $\pi$ and the minimal desingularization $\nu$, $\pi '$ is reconstructed from the fibre product $\pi \times _{{\overline{S}}}\nu$.

In order to describe the rational singularities occurring in ${{\,\mathrm{Sing}\,}}{\overline{S}}$ we use the notation

$$\begin{aligned} {\mathsf {T}} := n_1 {\mathsf {T}}_1 + \dots + n_s {\mathsf {T}}_s, \end{aligned}$$

(15)

where ${\mathsf {T}}_j$ is the singularity type and $n_j$ the number of points of type ${\mathsf {T}}_j$ in ${{\,\mathrm{Sing}\,}}{\overline{S}}$.

Theorem 3.2

Let $(S, {\mathcal {E}}, {\mathcal {L}})$ be a level $\ell$ K3 surface of genus g, then one has $2 \le \ell \le 8$ and $(S, {\mathcal {E}})$ satisfies one of the following conditions:

• (1) $\ell = 2$. One has $t = 8$, $r = 8$ and ${\mathsf {T}} = 8 {\mathsf {A}}_1$.

• (2) $\ell = 3$. One has $t = 6$, $r = 12$ and ${\mathsf {T}}= 6{\mathsf {A}}_2$.

• (3) $\ell = 4$. One has $t = 6$, $r = 14$ and ${\mathsf {T}}= 4{\mathsf {A}}_3 + 2{\mathsf {A}}_1$.

• (4) $\ell = 5$. One has $t = 4$, $r = 16$ and ${\mathsf {T}}= 4{\mathsf {A}}_4$.

• (5) $\ell = 6$. One has $t = 6$, $r = 16$ and ${\mathsf {T}}= 2 {\mathsf {A}}_5 + 2{\mathsf {A}}_2 + 2{\mathsf {A}}_1$.

• (6) $\ell = 7$. One has $t = 3$, $r = 18$ and ${\mathsf {T}}= 3{\mathsf {A}}_6$.

• (7) $\ell = 8$. One has $t = 4$, $r = 18$ and ${\mathsf {T}} = 2{\mathsf {A}}_7 + {\mathsf {A}}_3 + {\mathsf {A}}_1$.

See [24]. It is also useful to observe that always one has

$$\begin{aligned} E^2 = \frac{B^{ 2 }}{{\ell }^2} = - 4. \end{aligned}$$

(16)

Now, in view of the concrete applications in this paper, we mention some relevant properties of the structure of ${{\,\mathrm{Pic}\,}}S$ and of the moduli of the above triples.

Definition 3.2

${\mathcal {F}}_{g, \ell }$ is the moduli space of level $\ell$ K3 surfaces of genus g.

As in the case of $(S, {\mathcal {L}})$, the construction of ${\mathcal {F}}_{g, \ell }$ relies on the usual notion of lattice polarized variety, see [3, 9, 18, 24] for this K3 case. In particular, for every $g \ge 2$, ${\mathcal {F}}_{g,\ell }$ has a standard irreducible component to be constructed as follows. We may have

$$\begin{aligned} {\mathbb {Z}}[{\mathcal {L}}] \oplus {\mathbb {M}}_S \subseteq {{\,\mathrm{Pic}\,}}S, \end{aligned}$$

(17)

where the sum is orthogonal. Moreover ${\mathbb {M}}_S$ has rank r and it is generated by the classes $[B_1], \ldots , [B_r], [E]$, with ${\mathcal {E}} \cong {\mathcal {O}}_S(E)$, so that the relation $\ell [E] - [B] = 0$ is satisfied in ${{\,\mathrm{Pic}\,}}S$. We can see the inclusion as the image of a primitive embedding of lattices

$$\begin{aligned} \upsilon : {\mathbb {Z}} c \oplus {\mathbb {M}}_{\ell } \rightarrow {{\,\mathrm{Pic}\,}}S, \end{aligned}$$

(18)

where $\upsilon (c) := [{\mathcal {L}} ]$ and $\upsilon ({\mathbb {M}}_{\ell }) = {\mathbb {M}}_S$. The lattice ${\mathbb {M}}_{\ell }$ is given with the set of generators $\lbrace e , b_1, \ldots , b_r \rbrace$ so that $\upsilon (e) = [E]$, $\upsilon (b_1) = [B_1], \ldots , \upsilon (b_r) = [B_r]$. Notice also that

$$\begin{aligned} c^2 = 2g -2 \ , \ e^2 = -4 \ , \ b_1^2 = \cdots = b_r^2 = -2, \end{aligned}$$

(19)

cfr. [24]. Fixing these data, the moduli space of triples $(S, {\mathcal {L}}, {\mathcal {E}})$ endowed with an embedding $\upsilon$, can be constructed as a moduli space of lattice polarized K3 surfaces $(S, \upsilon )$. In our case S is M-polarized with $M := \mathbb Zc \oplus {\mathbb {M}}_{\ell }$ and the induced embedding $M \subset L := H^2(S, {\mathbb {Z}})$ is unique up to isometries, [24]. Then the moduli space is constructed as quotient of the period domain of these surfaces S. In particular its dimension is $19 - r$, [9, Section 4.1 and Theorem 1.4.8], [4, Section 2.4 and Proposition 2.6]. Moreover a unique irreducible component of it is the closure of the moduli points of pairs $(S, \upsilon )$ such that

$$\begin{aligned} {{\,\mathrm{Pic}\,}}S = {\mathbb {Z}}[{\mathcal {L}}] \oplus {\mathbb {M}}_S. \end{aligned}$$

(20)

In this case we will say that $(S, {\mathcal {L}}, {\mathcal {E}})$ is a standard triple of genus g and level $\ell$. Let us fix our notation:

Definition 3.3

${\mathcal {F}}^{\perp }_{g, \ell }$ is the moduli space of standard triples of genus g and level $\ell$.

${\mathcal {F}}^{\perp }_{g, \ell }$ exists for any $g \ge 2$ and $\ell = 2 \cdots 8$. Fixing $\ell$, ${\mathcal {F}}^{\perp }_{g, \ell }$ is the unique irreducible component of ${\mathcal {F}}_{g, \ell }$ along a proper countable set of values $g \in {\mathbb {N}}$.

Remark 3.1

Let $(S, {\mathcal {L}}, {\mathcal {E}})$ be a non standard triple and $C \in \vert {\mathcal {L}} \vert$. Then, at least experimentally for $\ell = 2$, C is never general in moduli for $g \ge 4$. This is true even when the parameter count makes that possible in low genus, see [20]. The situation is quite different for standard triples. This paper studies indeed the modular properties of C in this case: standard behavior or peculiarities of C.

4 A standard projective model

Given a standard triple $(S, {\mathcal {L}}, {\mathcal {E}})$, let us construct a projective realization of S useful to our purposes. Consider $C \in \vert {\mathcal {L}} \vert$ such that $C \cap B = \emptyset$ and ${\tilde{C}}' = \pi '^*C$. Then the curve ${\tilde{C}} = \nu '_* {\tilde{C}}'$ is biregular to ${\tilde{C}}'$ via the contraction $\nu ': {\tilde{S}}' \rightarrow {\tilde{S}}$ and the linear map

$$\begin{aligned} \nu '_*: H^0({\mathcal {O}}_{{\tilde{S}}'}({\tilde{C}}')) \rightarrow H^0({\mathcal {O}}_{{\tilde{S}}}({\tilde{C}})) \end{aligned}$$

(21)

is an isomorphism, we identify the two spaces under it. Then, using ${\tilde{C}}$, it is easy to remind of the action of the group ${\mathbb {Z}} / \ell {\mathbb {Z}}$ on this space and of its eigenspaces. Let

$$\begin{aligned} 0 \rightarrow {\mathcal {O}}_{{\tilde{S}}'} \rightarrow {\mathcal {O}}_{{\tilde{S}}'}({\tilde{C}}') \rightarrow \omega _{{\tilde{C}}} \rightarrow 0\nonumber \\ \end{aligned}$$

(22)

be the standard exact sequence, then ${\mathbb {Z}} / \ell {\mathbb {Z}}$ acts on its associated long exact sequence

$$\begin{aligned} 0 \rightarrow H^0({\mathcal {O}}_{{\tilde{S}}'}) \rightarrow H^0({\mathcal {O}}_{{\tilde{S}}'}({\tilde{C}}')) \rightarrow H^0 (\omega _{{\tilde{C}}}) \rightarrow 0. \end{aligned}$$

As is well known the ${\mathbb {Z}} / \ell {\mathbb {Z}}$-decomposition of $H^0(\omega _{{\tilde{C}} })$ is as follows

$$\begin{aligned} H^0(\omega _{{\tilde{C}} }) = \bigoplus _{k = 1 \cdots \ell -1} {\pi '}^* H^0(\omega _C \otimes \eta ^{-k}) \bigoplus {\pi '}^* H^0(\omega _C). \end{aligned}$$

(23)

and this implies that $H^0({\mathcal {O}}_{{\tilde{S}}}({\tilde{C}}' ))$ decomposes as

$$\begin{aligned} H^0({\mathcal {O}}_{{\tilde{S}}}({\tilde{C}}' )) = \bigoplus _{k = 1 \dots \ell -1} {\pi '}^* H^0({\mathcal {O}}_S(H_k)) \bigoplus {\pi '}^* H^0({\mathcal {O}}_S(C)), \end{aligned}$$

(24)

where ${\mathcal {O}}_S(H_1) \dots {\mathcal {O}}_S(H_{\ell - 1}) \in {{\,\mathrm{Pic}\,}}S$ and ${\mathcal {O}}_C(H_k) \cong \omega _C \otimes \eta ^{\otimes -k}$, up to reindexing. Since ${\tilde{C}}$ has genus ${\tilde{g}} = g + (\ell - 1)(g-1)$ it follows $\dim H^0({\mathcal {O}}_{{\tilde{S}}}( {\tilde{C}})) = g+1 + (\ell - 1)(g-1)$. In particular the above decomposition immediately implies that

$$\begin{aligned} \dim H^0( {\mathcal {O}}_S(H_k)) = \dim H^0(\omega _C \otimes \eta ^{- k}) = g - 1, \quad k = 1 \cdots \ell - 1. \end{aligned}$$

(25)

In what follows, it is also useful to recall the mentioned fact that $E^ 2 = - 4$.

Lemma 4.1

It holds $h^i({\mathcal {O}}_S(E)) = h^i({\mathcal {O}}_S(-E)) = 0$, for $i \ge 0$.

Proof

By assumption E is not effective. The same is true for $-E$, since $\ell E \sim B$ and $B > 0$. This implies $h^0({\mathcal {O}}_S(E)) = 0$ and $h^2({\mathcal {O}}_S(E)) = h^0({\mathcal {O}}_S(-E)) = 0$. Since $E^2 = -4$ we have $\chi ({\mathcal {O}}_S(E)) = 0$ and then $h^1({\mathcal {O}}_S(E)) = 0$. The same argument applies to $-E$. $\square$

Now we consider the line bundle ${\mathcal {O}}_S(C - E)$ and the standard exact sequence

$$\begin{aligned} 0 \rightarrow {\mathcal {O}}_S(-E) \rightarrow {\mathcal {O}}_S(C - E) \rightarrow {\mathcal {O}}_C(C - E) \rightarrow 0. \end{aligned}$$

Lemma 4.2

Let $g \ge 2$ then the associated long exact sequence is

$$\begin{aligned} 0 \rightarrow H^0({\mathcal {O}}_S(C - E)) \rightarrow H^0 (\omega _C \otimes \eta ^{-1} )\rightarrow 0, \end{aligned}$$

in particular it follows $\dim \vert C - E \vert = g - 2$ and $h^i({\mathcal {O}}_S(C-E)) = 0, \ i \ge 1$.

Proof

By the previous lemma $h^i({\mathcal {O}}_S(E)) = h^i({\mathcal {O}}_S(-E)) = 0$, for $i \ge 0$. Moreover we have $h^0(\omega _C \otimes \eta ^{- 1}) = g- 1$ and $h^1(\omega _C \otimes \eta ^{- 1}) = 0$. Then the statement follows. $\square$

Now we observe that the pull-back by $\pi '$ defines a linear embedding

$$\begin{aligned} {\pi '}^*: H^0({\mathcal {O}}_S(C - E)) \rightarrow H^0({\mathcal {O}}_{{\tilde{S}}'}({\tilde{C}}')). \end{aligned}$$

We have indeed ${\mathcal {O}}_{{\tilde{S}}'}({\tilde{C}}') \otimes {\pi '}^* {\mathcal {O}}_S(E - C) \cong {\mathcal {O}}_{{\tilde{S}}'}({\pi '}^*E)$ and finally

$$\begin{aligned} h^0 ( {\mathcal {O}}_{{\tilde{S}}'}({\pi '}^*E)) = h^0({\pi '}_* {\mathcal {O}}_{{\tilde{S}}'}({\pi '}^*E)) = h^0({\mathcal {A}}(E)) = 1, \end{aligned}$$

(26)

with ${\mathcal {A}} = {\mathcal {O}}_S \oplus {\mathcal {O}}_S(-E) \oplus \cdots \oplus {\mathcal {O}}_S((1-\ell )E)$. The equality defines, up to a nonzero constant factor, the linear embedding ${\pi '}^*$. Then Im ${\pi '}^*$ is the ${\mathbb {Z}}/ \ell {\mathbb {Z}}$-invariant space

$$\begin{aligned} {\pi '}^* H^0({\mathcal {O}}_S(C - E)). \end{aligned}$$

Proposition 4.3

Let $g \ge 3$ and ${{\,\mathrm{Pic}\,}}S \cong \mathbb Zc \oplus {\mathbb {M}}_{\ell }$, then $\vert C - E \vert$ is base point free.

Proof

Since S is a K3 surface, it suffices to prove that $\vert C - E \vert$ has no fixed component. Let F be an integral fixed component of $\vert C - E \vert$, set $f = F \cdot C$ for a general C. Then f is a fixed divisor of $\vert \omega _C \otimes \eta ^{-1} \vert$. Applying Riemann-Roch to C it follows $\dim \vert \eta (f) \vert = \deg f - 1$. Since $g \ge 3$ then $\deg f \le 2$. Hence F is a line, a conic or $FC = 0$. We have $F \sim xC + \sum y_jB_j +zE$ in ${{\,\mathrm{Pic}\,}}S$. Assume $\deg f > 0$ then $0 < CF = (2g-2)x \le 2$ with $x \in {\mathbb {Z}}$: a contradiction for $g \ge 3$. Let $CF = 0$ then $F^2 = -2$ by the Hodge Index Theorem and F is a ${\mathbb {P}}^1$ contracted by $f_{\vert C \vert }: S \rightarrow {\mathbb {P}}^g$. By Lemma 4.2, $h^0(C-E)=g-1=(C-E)^2/2+2$. Let M be the moving part of the linear system $|C-E|$, then $\dim \vert M \vert \ge 1$ and $MF \ge 0$. Moreover we have $C - E \sim M + kF + R$, where R is a curve not containing F and $k \ge 1$. Let $G \in \vert M + F \vert$ be general then G contains F: otherwise the curve kF could’nt be a component of the element $G + (k-1)F + R \in \vert C - E \vert$. Hence F is a fixed component of $\vert M + F \vert$. Now observe that $MF \ge 0$ and then consider the standard exact sequence

$$\begin{aligned} 0 \rightarrow {\mathcal {O}}_S(M) \rightarrow {\mathcal {O}}_S(M+F) \rightarrow {\mathcal {O}}_F(M) \rightarrow 0. \end{aligned}$$

We claim that, passing to the associated long exact sequence, it follows

$$\begin{aligned} \chi ({\mathcal {O}}_S(M)) = \chi ({\mathcal {O}}_S(M+F)) \end{aligned}$$

and $\chi ({\mathcal {O}}_F(M)) = 0$. Since $F = {\mathbb {P}}^1$ this implies $MF < 0$: a contradiction. To prove the claim consider a smooth $D \in \vert M \vert$. Then either D is integral of genus $g - 2$ and $h^1({\mathcal {O}}_S(M)) = 0$ or $M \sim (g-2)N$ and N is a smooth integral elliptic curve. Via Serre duality we have $h^2({\mathcal {O}}_S(M)) = h^2({\mathcal {O}}_S(M+F)) = 0$. Moreover $MF \ge 0$ implies $h^1({\mathcal {O}}_F(M)) = 0$. Then, in the former case, $h^1({\mathcal {O}}_S(M)) = 0$ implies $h^1({\mathcal {O}}_S(M+F)) = 0$ and the claim follows. In the latter case replace M by N. Then the equality and the same contradiction follow by the same type of arguments. $\square$

Now we introduce a second linear system associated with E. At first let us set

$$\begin{aligned} B_{\mathsf {red}} := B_1 + \cdots + B_r, \end{aligned}$$

(27)

where the summands are the irreducible components of ${{\,\mathrm{Supp}\,}}B$. Then we recall that

$$\begin{aligned} E = \frac{1}{\ell } (m_1 B_1 + \cdots + m_r B_r), \quad \text{ with } m_1 \cdots m_r \in [1 \cdots \ell -1]\text{. } \end{aligned}$$

Definition 4.1

Set $\mathring{E} = B_{\mathsf {red}} - E = \frac{1}{\ell } ( \mathring{m}_1B_1 + \cdots + \mathring{m}_rB_r)$, where $\mathring{m}_i := \ell - m_i$.

Let us denote by $n_i$ the coefficients of the curves $B_i$ in $-\ell E$. Then $n_i\equiv \mathring{m}_i\mod \ell$. More precisely, E is a generator of $\mathbb {Z}/\ell \mathbb {Z}=\langle B_i, E\rangle /\langle B_i\rangle$ and $\mathring{E}$ is its opposite in $\mathbb {Z}/\ell \mathbb {Z}$; in particular it is a different generator of the same group. Hence $\mathring{{\mathcal {E}}}:= {\mathcal {O}}_S(\mathring{E})$ is a level $\ell$ structure, with the same properties of ${\mathcal {E}}$ . We notice that $\mathring{E}$ defines a cover $\mathring{\pi }': {\tilde{S}}' \rightarrow S$ so that $\mathring{\pi }' = \pi ' \circ a$ and $a^{\ell } = id_{{\tilde{S}}'}$. Then we define

$$\begin{aligned} \vert H \vert := \vert C - E \vert , \quad \mathring{H} := \vert C - \mathring{E} \vert . \end{aligned}$$

(28)

The rational maps associated with these linear systems respectively will be

$$\begin{aligned} p: S \rightarrow {\mathbb {P}},\quad \mathring{p}: S \rightarrow \mathring{{\mathbb {P}}}, \end{aligned}$$

(29)

where ${\mathbb {P}} := \vert H \vert ^*$ and $\mathring{{\mathbb {P}}} := \vert \mathring{H} \vert ^*$ are the projective space ${\mathbb {P}}^{g-2}$. Let $\iota$ be the inclusion

$$\begin{aligned} {\mathbb {P}} \times \mathring{ {\mathbb {P}}} \subset {\mathbb {P}}^{(g-1)^2-1} \end{aligned}$$

(30)

defined by the Segre embedding, we set $f:= \iota \circ (p \times \mathring{p})$ and fix the notation

$$\begin{aligned} f: S \rightarrow {\mathbb {P}} \times \mathring{{\mathbb {P}}} \subset {\mathbb {P}}^{(g-1)^2-1}. \end{aligned}$$

(31)

Definition 4.2

The morphism f is the main projective model of $(S, {\mathcal {L}}, {\mathcal {E}})$.

The next two remarks are simple but relevant in order to discuss f, (the second one follows by a direct computation of $E\cdot \mathring{E}$, where the class E is explicitly given in [24]):

• (1) $f^* {\mathcal {O}}_{{\mathbb {P}}^{(g-1)^2 -1}}(1) \cong {\mathcal {O}}_S(H + \mathring{H}) \cong {\mathcal {O}}_S(2C - B_\mathsf{red})$,

• (2) $H \mathring{H} = 2g + 2 - t$.

Proposition 4.4

The divisors $[H - \mathring{H}]$ and $[\mathring{H} - H]$ are not effective classes for $\ell \ge 3$ and

$$\begin{aligned} h^1({\mathcal {O}}_S(H - \mathring{H})) = h^1({\mathcal {O}}_S(\mathring{H} - H)) = 6-t.\end{aligned}$$

(32)

Proof

We have $H(H - \mathring{H}) = \mathring{H} (\mathring{H} - H) = t - 8$. Since the general elements of $\vert H \vert$ and $\vert \mathring{H} \vert$ are irreducible curves, the first statement follows for $\ell \ge 3$ because then $t \le 6$. The second statement just follows from Riemann-Roch. $\square$

Now let us consider, for a general $C \in \vert {\mathcal {L}} \vert$, the standard exact sequence

$$\begin{aligned} 0 \rightarrow {\mathcal {O}}_S(C - B_\mathsf{red}) \rightarrow {\mathcal {O}}_S(2C - B_\mathsf{red}) \rightarrow {\mathcal {O}}_C(2C - B_\mathsf{red}) \rightarrow 0. \end{aligned}$$

(33)

Since C is smooth and disjoint from $B_\mathsf{red}$, then ${\mathcal {O}}_C(- B_\mathsf{red})$ is trivial and $\vert 2C - B_\mathsf{red} \vert$ cuts on C a linear system of bicanonical divisors. Moreover we know that both $\vert H \vert$ and $\vert \mathring{H} \vert$ are base point free. Hence the same is true for $\vert H + \mathring{H} \vert = \vert 2C - B_{\mathsf {red}} \vert$. Notice that

$$\begin{aligned} (2C - B_{\mathsf {red}})^2 = 8(g-1) - 2t, \end{aligned}$$

which is $\ge 0$ for $g \ge 3$ and any of the prescribed values of $t, \ell$. Actually the zero value is only reached in the known situation $g = 3$, $\ell = 2$. Hence we assume $g \ge 4$ for $\ell = 2$. Then a general $D \in \vert H + \mathring{H} \vert$ is a smooth integral curve such that $D^2 > 0$. As is well known, this implies $h^i({\mathcal {O}}_S(H + \mathring{H})) = 0$ for $i \ge 1$ and the next property follows.

Proposition 4.5

Let g be as above then $\dim \vert 2C - B_{\mathsf {red}} \vert = 4g - t - 3$ and the long exact sequence associated with the exact sequence (33) is as follows:

$$\begin{aligned} 0 \rightarrow H^0({\mathcal {O}}_S(C-B_\mathsf{red})) \rightarrow H^0({\mathcal {O}}_S (2C - B_\mathsf{red})) \rightarrow H^0(\omega _C^{\otimes 2}) \rightarrow H^1({\mathcal {O}}_S(C-B_\mathsf{red})) \rightarrow 0. \end{aligned}$$

The linear system $\vert C - B_{\mathsf {red}} \vert$ also deserves some observations. Since we are dealing with a general standard triple $(S, {\mathcal {L}}, {\mathcal {E}})$, we know that $\vert C \vert$ defines a morphism

$$\begin{aligned} f_{\vert C \vert }: S \rightarrow {\mathbb {P}}^g \end{aligned}$$

which is the contraction $\nu : S \rightarrow {\overline{S}}$, composed with the embedding ${\overline{S}} \subset {\mathbb {P}}^g$ defined by $\vert \nu _* C \vert$. Since a general C is disjoint from B, $\vert \nu _*C \vert$ is a linear system of Cartier divisors. Let ${\mathcal {I}}_{{{\,\mathrm{Sing}\,}}{\overline{S}}}$ be the ideal sheaf of ${{\,\mathrm{Sing}\,}}{\overline{S}}$, it is clear that the natural map

$$\begin{aligned} f^*_{\vert C \vert }: H^0({\mathcal {I}}_{{{\,\mathrm{Sing}\,}}{\overline{S}}}(1)) \rightarrow H^0({\mathcal {O}}_S(C - B_{\mathsf {red}})) \end{aligned}$$

is an isomorphism. Then, considering the above exact sequence (33), we have

$$\begin{aligned} h^0({\mathcal {O}}_S(C - B_{\mathsf {red}})) - h^1({\mathcal {O}}_S(C - B_{\mathsf {red}})) = \chi ({\mathcal {O}}_S(2C - B_{\mathsf {red}})) - \chi (\omega ^{\otimes 2}_C) = g + 1 - t.\nonumber \\ \end{aligned}$$

(34)

This implies the next property.

Proposition 4.6

It holds $h^1({\mathcal {O}}_S(C - B_{\mathsf {red}})) = 0$ if and only if $h^0({\mathcal {O}}_S(C - B_{\mathsf {red}})) = g + 1 - t$, that is, the points of ${{\,\mathrm{Sing}\,}}{\overline{S}}$ are linearly independent in ${\mathbb {P}}^g$.

On the other hand consider the commutative diagram

(35)

where $\mu _S$ and $\mu _C$ are the multiplication maps and the vertical arrows are the restriction maps. It follows from Lemma (4.2) that $\rho _H \otimes \rho _{\mathring{H}}$ is an isomorphism. The next property is clear.

Proposition 4.7

If $\mu _C$ is surjective then $h^1({\mathcal {O}}_S(C - B_{\mathsf {red}})) = 0$ i.e. $\rho _C$ is surjective.

Since $\chi ({\mathcal {O}}_S (C - B_{\mathsf {red}}) = g + 1 - t$ let us point out that $\mu _C$ is not surjective if

$$\begin{aligned} g < t - 1. \end{aligned}$$

(36)

We do not further investigate the diagram, for our applications these results suffice.

5 Views on the Mukai maps in level $\ell$

In this section we only put in large the picture we have outlined in the introduction. This picture concerns the maps in (3) and (2), that is, the Mukai map

$$\begin{aligned} m_g: {\mathcal {P}}_g \rightarrow {\mathcal {M}}_g \end{aligned}$$

and the level $\ell$ Mukai maps

$$\begin{aligned} r_{g,\ell }: {\mathcal {P}}^{\perp }_{g,\ell } \rightarrow {\mathcal {R}}_{g, \ell }. \end{aligned}$$

These maps, and the involved moduli spaces, have been previously considered. We recall that the points of ${\mathcal {P}}_g$ are the elements $[S, {\mathcal {L}}, C]$ such that $[S, {\mathcal {L}}] \in {\mathcal {F}}_g$ and $C \in \vert {\mathcal {L}} \vert$. The Mukai map $m_g$ is the natural forgetful map. We have

• (1) $m_g$ is dominant for $g \le 9$,

• (2) $m_g$ is not dominant for $g = 10$,

• (3) $m_g$ is birational for $g = 11$,

• (4) $m_g$ has 1-dimensional fibre for $g = 12$.

• (5) $m_g$ is generically injective for $g \ge 13$.

Thus $m_g$ has not maximal rank for $g = 10, 12$. It is indeed known that a general $[C] \in m_{10}({\mathcal {P}}_{10})$ is a linear section C of the $G_2$ variety $W \subset {\mathbb {P}}^{13}$, [23]. Hence the family of 2-dimensional linear sections of W through C is a ${\mathbb {P}}^3$. It turns out from this fact that the fibre of $m_{10}$ at [C] is 3-dimensional. Then $m_{10}({\mathcal {P}}_{10})$ has codimension 1. Genus 12 Fano threefolds play a similar role, then a general fibre of $m_{12}$ is a rational curve.

In this perspective, asking about the connections between the moduli space ${\mathcal {F}}^{\perp }_{g, \ell }$, of level $\ell$ K3 surfaces of genus g, and ${\mathcal {R}}_{g, \ell }$ is, as observed, natural. For a general point $[S, {\mathcal {L}}, {\mathcal {E}}] \in {\mathcal {F}}^{\perp }_{g, \ell }$ one can ask if $(C, \eta )$, with $C \in \vert {\mathcal {L}} \vert$ and $\eta = {\mathcal {E}} \otimes {\mathcal {O}}_C$, defines a general point of ${\mathcal {R}}_{g, \ell }$. More precisely recall that ${\mathcal {P}}^{\perp }_{g, \ell }$ is the moduli space of 4-tuples $(S, {\mathcal {L}}, {\mathcal {E}}, C)$ such that $[S, {\mathcal {L}}, {\mathcal {E}}] \in {\mathcal {F}}^{\perp }_{g,\ell }$ and $C \in \vert {\mathcal {L}} \vert$. The level $\ell$ Mukai map $r_{g,\ell }: {\mathcal {P}}^{\perp }_{g, \ell } \rightarrow {\mathcal {R}}_{g,\ell }$ is the morphism sending $[S, {\mathcal {L}}, {\mathcal {E}}, C] \in {\mathcal {P}}^{\perp }_{g, \ell }$ to the point $[C, \eta _C] \in {\mathcal {R}}_{g, \ell }$, where $\eta _C$ is ${\mathcal {E}} \otimes {\mathcal {O}}_C$. About the possible dominance of the map $r_{g,\ell }$ we have:

• (1) $3g-3 = \dim {\mathcal {R}}_{g,2} \le \dim {\mathcal {P}}^{\perp }_{g, 2 } = 11 + g$ iff $g \le 7$.

• (2) $3g-3 = \dim {\mathcal {R}}_{g,3} \le \dim {\mathcal {P}}^{\perp }_{g, 3 } = \ 7 + g$ iff $g \le 5$.

• (3) $3g-3 = \dim {\mathcal {R}}_{g,4} \le \dim {\mathcal {P}}^{\perp }_{g, 4 } = \ 5 + g$ iff $g \le 4$.

• (4) $3g-3 = \dim {\mathcal {R}}_{g,5} \le \dim {\mathcal {P}}^{\perp }_{g, 5 } = \ 3 + g$ iff $g \le 3$.

• (5) $3g-3 = \dim {\mathcal {R}}_{g,6} \le \dim {\mathcal {P}}^{\perp }_{g, 6 } =$$3 + g$ iff $g \le 3$.

• (6) $3g-3 = \dim {\mathcal {R}}_{g,7} \le \dim {\mathcal {P}}^{\perp }_{g, 7 } = \ 1 + g$ iff $g \le 2$.

• (7) $3g-3 = \dim {\mathcal {R}}_{g,8} \le \dim {\mathcal {P}}^{\perp }_{g, 8 } = \ 1 + g$ iff $g \le 2$.

These issues have not been systematically considered but for $\ell = 2$. We close this expository section with a summary on what happens for $\ell = 2, 3$.

5.1 The picture for $\ell = 2$

We have $3g-3 = \dim {\mathcal {M}}_g \le \dim {\mathcal {P}}^{\perp }_{g,2} = 11 + g$ iff $g \le 7$. Again, $r_{g,2}$ behaves unexpectedly near the value of transition, which is now $g = 7$.

• (1) $r_{g,2}$ is dominant for $g \le 5$,

• (2) $r_{g,2}$ is not dominant for $g = 6$,

• (3) $r_{g,2}$ is birational for $g = 7$,

• (4) $r_{g,2}$ has not finite fibres for $g = 8$.

• (5) $r_{g,2}$ is generically injective for $g \ge 9$.

These surfaces are known as (standard) Nikulin surfaces. Cases (1), (2), (3) are treated in [11, 12], the remaining ones, (standard and non standard), in [19, 20]. Notice that $r_{g,2}$ is not of maximal rank for $g = 6,8$. In genus 6 the condition $C \subset S$ implies that the following multiplication map is not an isomorphism as expected:

$$\begin{aligned} \mu : \mathrm{Sym}^2 H^0(\omega _C \otimes \eta _C) \rightarrow H^0(\omega _C^{\otimes 2}). \end{aligned}$$

(37)

Then $(C, \eta _C)$ does not define a general point of ${\mathcal {R}}_{g,2}$, see [3]. We point out that, studying the two cases where $r_{g,2}$ has not maximal rank, two families of singular Fano threefolds appear. Their hyperplane sections are singular models ${\overline{S}}$ of general Nikulin surfaces S. The existence of these threefolds implies the failure of the maximal rank.

5.2 The picture for $\ell = 3$

We will prove that $r_{g,3}$ behaves unexpectedly near $g = 5$:

• (1) $r^s_{g,3}$ is dominant for $g \le 3$,

• (2) $r^s_{g,3}$ has not maximal rank for $g = 4$,

• (3) $r^s_{g,3}$ is birational for $g = 5$,

• (4) $r^s_{g,3}$ has not maximal rank for $g = 6$.

Remark 5.1

The case $g \ge 7$ should be considered for further investigation, addressing the generic injectivity. The (uni)rationality of ${\mathcal {R}}_{g,3}$ is known, or elementary, for $g \le 5$, cfr. [1, 2, 28]. We recall that ${\mathcal {R}}_{g,3}$ is of general type for $g \ge 12$ and of Kodaira dimension $\ge 19$ for $g = 11$, [7]. Bruns proved in [6] that ${\mathcal {R}}_{8,3}$ is of general type. The cases $g = 6, 7, 9, 10$ and partially $g =11$ are open.

6 The Mukai map in level 3

6.1 The case of genus 4

Let $[S, {\mathcal {L}}, {\mathcal {E}}, C] \in {\mathcal {P}}^{\perp }_{g, \ell }$ be general and $\ell = 3$, as in Sect. 2, (35) we consider the commutative diagram

(38)

Since $\ell = 3$ we have $t = 6$ connected components of ${{\,\mathrm{Supp}\,}}B$. Then, by proposition (4.7), $\mu _C$ is not surjective if $g < t - 1 = 5$. This is obvious for $g \le 3$. For $g = 4$ the dimension count suggests that in ${\mathcal {R}}_{4,3}$ the map $\mu _C$ is not surjective in codimension 1 .

Proposition 6.1

Let $[C, \eta ] \in {\mathcal {R}}_{4,3}$ be a general point then $\mu _C$ is surjective, moreover the locus of points such that $\mu _C$ is not surjective is an effective Cartier divisor in ${\mathcal {R}}_{4,3}$.

Indeed, for $g = 4$ and $\ell = 3$, this locus turns out to be the locus ${\mathcal {D}}_{g,\ell }$ defined in [7, p. 77]. There, for low level $\ell \ge 3$ and for $g \le 16$, the so defined Torsion bundle conjecture B is proven, which implies that ${\mathcal {D}}_{4,3}$ is an effective Cartier divisor in ${\mathcal {R}}_{4,3}$. Then the next theorem follows. Notice also that, for $g = 4$, theorem 1.7 of [2] implies that $\mu _C$ is an isomorphism for a general $(C, \eta )$.

Theorem 6.2

The map $r_{4,3}: {\mathcal {P}}^{\perp }_{4,3} \rightarrow {\mathcal {R}}_{4, 3}$ fails to be dominant.

Remark 6.1

The case $g = 4$ turns out to be of special interest. See the last section for a natural, presently conjectural, geometric interpretation.

6.2 The case of genus 5

Differently from the case $g \le 4$ the multiplication map

$$\begin{aligned} \mu _C: H^0(\omega _C \otimes \eta ) \otimes H^0(\omega _C \otimes \eta ^{-1}) \rightarrow H^0(\omega _C ^{\otimes 2}) \end{aligned}$$

can be surjective for $g \ge 5$ and a general point $[C, \eta ] \in {\mathcal {R}}_{g,3}$. This property occurs in genus $g = 5$ and makes possible the proof of the next birationality theorem.

Theorem 6.3

The Mukai map $r_{5,3}: {\mathcal {P}}^{\perp }_{5,3} \rightarrow {\mathcal {R}}_{5,3}$ is birational.

Before proving it we cannot avoid a long series of preliminaries. We will always assume that $[S, {\mathcal {L}}, {\mathcal {E}}, C] \in {\mathcal {P}}^{\perp }_{5,3}$ is a general point, in particular ${{\,\mathrm{Pic}\,}}S \cong {\mathbb {Z}} c \oplus {\mathbb {M}}_3$. Let

$$\begin{aligned} 0 \rightarrow {\mathcal {O}}_S(H + \mathring{H} - C) \rightarrow {\mathcal {O}}_S(H + \mathring{H}) \rightarrow \omega _C^{\otimes 2} \rightarrow 0 \end{aligned}$$

(39)

be the standard exact sequence, at first we point out the following fact.

Proposition 6.4

The associated long exact sequence is

$$\begin{aligned} 0 \rightarrow H^0({\mathcal {O}}_S(H + \mathring{H})) {\mathop {\rightarrow }\limits ^{\rho _C}}H^0(\omega _C^{\otimes 2}) \rightarrow 0. \end{aligned}$$

(40)

Since $H + \mathring{H} - C \sim C - B_{\mathsf {red}}$, the next lemma implies the previous statement.

Lemma 6.5

It holds $h^i({\mathcal {O}}_S(C - B_{\mathsf {red}})) = 0$ for $i \ge 0$.

Proof

Since $C(B_{\mathsf {red}} - C) < 0$, $h^0({\mathcal {O}}_S(B_{\mathsf {red}} - C)) = 0$. Hence $h^2({\mathcal {O}}_S(C - B_{\mathsf {red}}))$ is zero by Serre duality. Since $(C- B_{\mathsf {red}})^2 = - 4$ then $\chi ({\mathcal {O}}_S(C - B_{\mathsf {red}})) = 0$ and the statement follows if $h^0({\mathcal {O}}_S(C - B_{\mathsf {red}})) = 0$. Assume $A \in \vert C - B_{\mathsf {red}} \vert$ then A is not connected. This follows from $\chi ({\mathcal {O}}_S(A)) = h^0({\mathcal {O}}_S(A)) - h^1({\mathcal {O}}_S(A)) = 0$ and the standard exact sequence

$$\begin{aligned} 0 \rightarrow {\mathcal {O}}_S(-A) \rightarrow {\mathcal {O}}_S \rightarrow {\mathcal {O}}_A \rightarrow 0. \end{aligned}$$

This implies $A = A_1 + A_2$, where $A_1$ is a connected component and $A_2 = A - A_1$ is a curve. We have $C(A_1+ A_2) = C(C - B_{\mathsf {red}})= 8$ and we can choose $A_1$ so that $CA_1 > 0$. Assume $CA_2 = 0$ then the morphism $\phi : S \rightarrow {\mathbb {P}}^5$, defined by $\vert C \vert$, maps birationally $A_1 + A_2 + B_{\mathsf {red}}$ onto a degree 8 hyperplane section of ${\overline{S}} = \phi (S)$. This is the curve $\phi _*A_1$, singular at the points of $\phi (B_{\mathsf {red}}) = {{\,\mathrm{Sing}\,}}{\overline{S}}$. These points are the images by $\phi$ of the six connected components of $B_{red}$ and are exactly six. Indeed each fibre of $\phi$ is connected and hence two connected components $V_1, V_2$ of $B_{red}$, contracted to the same point, are connected by an effective divisor W orthogonal to C. On the other hand, under our generality assumption, we have ${{\,\mathrm{Pic}\,}}S \cong \mathbb Zc \oplus {\mathbb {M}}_3$. Moreover a direct computation shows that, in the negative definite lattice ${\mathbb {M}}_3$, ${{\,\mathrm{Supp}\,}}W$ is union of irreducible components of $B_{red}$. Actually one computes that the only classes of irreducible $(-2)$-curves are the classes of $B_1 \cdots B_{12}$. This implies $W = 0$ and $V_1 = V_2$. But then $\phi _*A_1$ is not integral, because it is a hyperplane section of $\phi (S)$ with six singular points. Then there exists an irreducible component R of it such that $0< CR < 8$. The same is obvious if $CA_2 > 0$. Since ${{\,\mathrm{Pic}\,}}S \cong \mathbb Zc \oplus {\mathbb {M}}_3$ we have $[R] =x[C] + \sum y_i [B_i] + z[E]$, with $x, y_i, z \in {\mathbb {Z}}$. But this implies $0< CR = x8 < 8$ with $x \notin {\mathbb {Z}}$: a contradiction. $\square$

Proposition 6.6

The linear systems $\vert H \vert$ and $\vert \mathring{H} \vert$ are not hyperelliptic.

Proof

Let $\vert H \vert$ be hyperelliptic, then $\vert H \vert$ defines a 2 : 1 morphism $\psi : S \rightarrow {\mathbb {P}}^3$ onto a quadric surface $Q := \psi (S)$. As is well known the pull-back of a ruling of lines of Q defines a pencil $\vert F_2 \vert$ of curves such that $F_2^2 = 0$ and $HF_2 = 2$. Moreover $\vert F_1 \vert := \vert H - F_2 \vert$ is a pencil of irreducible elliptic curves. The same is true for the moving part of $\vert F_2 \vert$. Since $H \sim F_1 + F_2$ and $C \sim H + E$ we have $C (F_1 + F_2) = 8$ and also $CF_i \ge 2$, $i = 1,2$. Let $\vert F \vert$ be the moving part of the pencil $\vert F_i \vert$ such that $CF_i$ is minimal, then it follows $2 \le CF \le 4$. On the other hand we have $F \sim xC + \sum y_j B_j + zE$ in ${{\,\mathrm{Pic}\,}}S$. This implies $2 \le CF = 8x \le 4$ and $x \notin {\mathbb {Z}}$: a contradiction. The same argument works for $\vert \mathring{H} \vert$. $\square$

Lemma 6.7

It holds $h^i({\mathcal {O}}_S(2H - \mathring{H})) = h^i({\mathcal {O}}_S(2 \mathring{H} - H)) = 0$ for $i \ge 0$.

Proof

From $H \sim C - E$ and $\mathring{H} \sim C - \mathring{E}$ we have $2H - \mathring{H} \sim C - 2E + \mathring{E}$, moreover

$$\begin{aligned} \mathring{H} (\mathring{H} - 2H) = -8 \Rightarrow h^0 ({\mathcal {O}}_S(\mathring{H} -2H)) = 0 \Rightarrow h^2({\mathcal {O}}_S(2H - \mathring{H})) = 0. \end{aligned}$$

Since $(2H - \mathring{H})^2 = -4$ then $\chi ({\mathcal {O}}_S(2H - \mathring{H})) = 0$. Hence the statement follows for $2H - \mathring{H}$ if we prove $h^0({\mathcal {O}}_S(2H - \mathring{H})) = 0$. For this we observe that the well known descriptions of E and $\mathring{E}$ are as follows. For $i = 1 \cdots 6$ consider $N_i = B_i + B'_{i}$, that is, the i-th connected component of $B_{\mathsf {red}} = \sum _{i = 1 \cdots 6} B_i + B'_{i}$. Then in ${{\,\mathrm{Pic}\,}}S$ we have

$$\begin{aligned}{}[E] = \sum _{i = 1 \cdots 6} \frac{1}{3} [B_i + 2B'_i] , \quad [\mathring{E}] = \sum _{i = 1 \cdots 6} \frac{1}{3} [2B_i + B'_i] \end{aligned}$$

(41)

up to exchanging E with $\mathring{E}$. Since $2H - \mathring{H} \sim C - 2E + \mathring{E}$, it follows that

$$\begin{aligned} 2H - \mathring{H} \sim C - \sum _{i = 1 \cdots 6} B'_i. \end{aligned}$$

(42)

This implies that $[2H - \mathring{H}]$ is not an effective class. Indeed let $B' := B'_1 + \cdots + B'_6$, observe that $(C - B')B_i = - 1$, $i = 1 \cdots 6$. Assume $C - B' \sim F$ where F is an effective divisor. Then $FB_i = -1$ implies $B_i \subset F$ and $F = F' + B_1 + \cdots + B_6$ where $F'$ is effective. Hence $C - B_{\mathsf {red}}\sim F' > 0$: a contradiction to the above lemma (6.5). $\square$

We will profit of genus 3 curves of the non hyperelliptic linear systems $\vert H \vert$ or $\vert \mathring{H} \vert$.

Lemma 6.8

It holds $\forall \ D \in \vert H \vert , \ h^0({\mathcal {O}}_D(\mathring{H} - H)) = 0$ and $\forall \ \mathring{D} \in \vert \mathring{H} \vert , \ h^0({\mathcal {O}}_{\mathring{D}}(H - \mathring{H})) = 0$.

Proof

Let $D \in \vert H \vert$, once more consider the standard exact sequence

$$\begin{aligned} 0 \rightarrow {\mathcal {O}}_S(\mathring{H} - 2H) \rightarrow {\mathcal {O}}_S (\mathring{H} - H) \rightarrow {\mathcal {O}}_D(\mathring{H} - H) \rightarrow 0 \end{aligned}$$

and its long exact sequence. We have $h^1({\mathcal {O}}_S(\mathring{H} - 2H)) = h^1({\mathcal {O}}_S(2H - \mathring{H})) = 0$ by the previous lemma and $h^0({\mathcal {O}}_S(\mathring{H} - 2H)) = 0$ because $H(\mathring{H} - 2H) = -2$. Then it follows $h^0({\mathcal {O}}_D(\mathring{H} - H)) = h^0({\mathcal {O}}_S(\mathring{H} - H))$. Finally the latter is zero by Proposition (4.4). $\square$

Let $D \in \vert H \vert$ be smooth then ${\mathcal {O}}_D(\mathring{H} - H) \cong {\mathcal {O}}_D(b)$, where $\deg b = 2$. We fix the notation b for such a divisor and the notation $\mu _D$ for the following multiplication map:

$$\begin{aligned} \mu _D: H^0(\omega _D) \otimes H^0(\omega _D(b)) \rightarrow H^0(\omega ^{\otimes 2}_D(b)). \end{aligned}$$

(43)

Let us also point out that $h^0({\mathcal {O}}_D(b)) = 0$ by the above lemma. Moreover we fix the notation

$$\begin{aligned}&\nu _D: H^0({\mathcal {O}}_S(H)) \rightarrow H^0(\omega _D), \quad \mathring{\nu }_D: H^0({\mathcal {O}}_S(\mathring{H})) \rightarrow H^0(\omega _D(b)) \ ,\nonumber \\&\rho _D: H^0({\mathcal {O}}_S(H + \mathring{H})) \rightarrow H^0(\omega _D^{\otimes 2}(b)) \end{aligned}$$

(44)

for the natural restriction maps. Then we consider the commutative diagram:

(45)

which is similar to our main diagram (35)

Proposition 6.9

The vertical arrows and the horizontal arrow $\mu _D$ are surjective.

Proof

Let $p: S \rightarrow {\mathbb {P}}^3$ be the map defined by $\vert H \vert$, then $p \vert D: D \rightarrow {\mathbb {P}}^2 = \vert \omega _D \vert ^*$ is the canonical map and $\vert \omega _D(b) \vert$ is cut on D by $\vert {\mathcal {I}}_{d \vert S}(3H) \vert$, where d is any element of $\vert \omega ^{\otimes 2}(-b)\vert$ and ${\mathcal {I}}_{d \vert S}$ is its ideal sheaf. Moreover the map $p^*: \vert {\mathcal {O}}_{{\mathbb {P}}^2}(3) \vert \rightarrow \vert \omega _D^{\otimes 3} \vert$ is an isomorphism and $\vert {\mathcal {I}}_{d \vert S}(3H) \vert$ $=$ $p^* \vert {\mathcal {I}}_{Z \vert {\mathbb {P}}^2}(3) \vert$, where $Z = p_*d$ and ${\mathcal {I}}_{Z \vert {\mathbb {P}}^2}$ is its ideal sheaf. Hence it follows $h^0({\mathcal {I}}_{Z \vert {\mathbb {P}}^2}(2)) = h^0(\omega _D^{\otimes 2}(- b)) = h^0({\mathcal {O}}_D(b)) = 0$ and $h^1({\mathcal {O}}_D(b)) = h^0({\mathcal {O}}_D(b)) = 0$. This easily implies $h^i({\mathcal {I}}_{Z \vert {\mathbb {P}}^2}(3-i)) = 0$ for $i > 0$, that is, ${\mathcal {I}}_{Z \vert {\mathbb {P}}^2}$ is 3-regular. Hence, by Castelnuovo-Mumford regularity theorem, the multiplication map

$$\begin{aligned} \mu : H^0({\mathcal {O}}_{{\mathbb {P}}^2}(1)) \otimes H^0({\mathcal {I}}_{Z \vert {\mathbb {P}}^2}(3)) \rightarrow H^0({\mathcal {I}}_{Z \vert {\mathbb {P}}^2}(4))) \end{aligned}$$

(46)

is surjective. Now consider the standard exact sequence of ideal sheaves

$$\begin{aligned} 0 \rightarrow {\mathcal {I}}_{p(D) \vert {\mathbb {P}}^2}(4) \rightarrow {\mathcal {I}}_{Z \vert {\mathbb {P}}^2}(4) {\mathop {\rightarrow }\limits ^{\rho }}{\mathcal {I}}_{Z \vert p(D)}(4) \rightarrow 0 \end{aligned}$$

and its associated long exact sequence. Since ${\mathcal {I}}_{p(D) \vert {\mathbb {P}}^2}(4) \cong {\mathcal {O}}_{{\mathbb {P}}^2}$ it follows that

$$\begin{aligned} h^0(\rho ): H^0({\mathcal {I}}_{Z \vert {\mathbb {P}}^2}(4)) \rightarrow H^0(\omega ^{\otimes 2}_D(b)) \end{aligned}$$

is surjective. On the other hand we have $\mu _D \circ \lambda = h^0(\rho ) \circ \mu$, where $\lambda$ is the tensor product

$$\begin{aligned} \lambda _1 \otimes \lambda _2: H^0({\mathcal {O}}_{{\mathbb {P}}^2}(1)) \otimes H^0({\mathcal {I}}_{Z \vert {\mathbb {P}}^2}(3)) \rightarrow H^0(\omega _D) \otimes H^0(\omega _D(b)) \end{aligned}$$

of the natural isomorphisms $\lambda _1: H^0({\mathcal {O}}_{{\mathbb {P}}^2}(1)) \rightarrow H^0(\omega _D)$ and $\lambda _2: H^0({\mathcal {I}}_{Z \vert {\mathbb {P}}^2}(3)) \rightarrow H^0(\omega _D(b))$. Since $\lambda$ is an isomorphism and $h^0(\rho )$ and $\mu$ are surjective, then $\mu _D$ is surjective. The surjectivity of $\rho _D$ follows from the vanishing of $h^1({\mathcal {O}}_S(\mathring{H}))$ and the standard exact sequence

$$\begin{aligned} 0 \rightarrow {\mathcal {O}}_S(\mathring{H}) \rightarrow {\mathcal {O}}_S(H + \mathring{H}) \rightarrow \omega _D^{\otimes 2}(b) \rightarrow 0. \end{aligned}$$

Since $\omega _D^{\otimes 2}(b)$ is ${\mathcal {O}}_D(H + \mathring{H})$, the surjectivity of $\nu _D$ follows from the above exact sequence twisted by $-\mathring{H}$. Finally the exact sequence

$$\begin{aligned} 0 \rightarrow {\mathcal {O}}_S(\mathring{H} - H) \rightarrow {\mathcal {O}}_S(\mathring{H}) \rightarrow \omega _D(b) \rightarrow 0 \end{aligned}$$

implies that $\mathring{\nu }_D$ is an isomorphism. Indeed we have $h^0({\mathcal {O}}_S(\mathring{H} - H )) = h^1({\mathcal {O}}_S(\mathring{H} - H))$ $= 0$ in its long exact sequence by (32). Hence $\nu _D \otimes \mathring{\nu }_D$ is surjective too. $\square$

Proposition 6.10

The map $\mu _S: H^0({\mathcal {O}}_S( H)) \otimes H^0({\mathcal {O}}_S(\mathring{H})) \rightarrow H^0({\mathcal {O}}_S( H+\mathring{H}))$ is surjective.

Proof

Let us consider again the commutative diagram (45), that is,

Counting dimensions we have $\dim {{\,\mathrm{Ker}\,}}\mu _S \ge 4$, hence it suffices to show that the equality holds. Now we know that $\mu _D$ and $\nu _D \otimes \mathring{\nu }_{D}$ are surjective. Let ${\mathbb {K}}$ be the Kernel of $\mu _D \circ (\nu _D \otimes \mathring{\nu }_D)$, then the dimension count gives $\dim {\mathbb {K}} = 8$ and, of course, we have ${{\,\mathrm{Ker}\,}}\mu _S \subseteq {\mathbb {K}}$. Therefore, to prove $\dim {{\,\mathrm{Ker}\,}}\mu _S = 4$, it suffices to produce a 4-dimensional subspace $V \subset {\mathbb {K}}$ such that $V \cap {{\,\mathrm{Ker}\,}}\mu _S = (0)$. To this purpose consider the space of decomposable vectors $V := \langle s \rangle \otimes H^0({\mathcal {O}}_S(\mathring{H}))$, where s is nonzero and ${ \mathrm div } (s) = D$. Then we have $(\nu _D \otimes \mathring{\nu }_D) (V) =(0)$ and hence $V \subset {\mathbb {K}}$. On the other hand let $t \in H^0({\mathcal {O}}_S(\mathring{H}))$, then $\mu _S(s \otimes t) = st$ and this is zero iff $t = 0$. Hence $V \cap {{\,\mathrm{Ker}\,}}\ \mu _S = (0)$. $\square$

Now we go back, in genus 5, to our usual diagram (35) in Sect. 2. This is

(47)

Proposition 6.11

$\mu _C: H^0(\omega _C \otimes \eta ) \otimes H^0(\omega _C \otimes \eta ^{-1}) \rightarrow H^0(\omega _C^{\otimes 2})$ is surjective.

Proof

We have already shown that $\mu _S$ and $\rho _H \otimes \rho _{\mathring{H}}$ are surjective. By (40) and its related lemma the same is true for $\rho _C$. Hence the surjectivity of $\mu _C$ follows. $\square$

Let ${\mathbb {P}}^{15}: = {\mathbb {P}}( H^0({\mathcal {O}}_S( H))^* \otimes H^0({\mathcal {O}}_S(\mathring{H}))^*)$ and let ${\mathbb {P}}^3 \times {\mathbb {P}}^3 := \iota ( \vert H \vert ^* \times \vert \mathring{H} \vert ^*)$ be the image in ${\mathbb {P}}^{15}$ of the Segre embedding $\iota$. Now we study the morphism defined in (4.2)

$$\begin{aligned} f: S \rightarrow {\mathbb {P}}^3 \times {\mathbb {P}}^3 \subset {\mathbb {P}}^{15}, \end{aligned}$$

that is, $f = \iota \circ (p \times \mathring{p})$. Since the map $\mu _S$ is surjective it follows that

$$\begin{aligned} (p \times \mathring{p})^* H^0({\mathcal {O}}_{{\mathbb {P}}^3 \times {\mathbb {P}}^3}(1,1)) = H^0({\mathcal {O}}_S(H + \mathring{H})). \end{aligned}$$

(48)

Let ${\mathbb {P}}^{11} \subset {\mathbb {P}}^{15}$ be the linear embedding of ${\mathbb {P}}( \mathrm{Im} \mu ^*_S)$ defined by $\mu ^*_S$, then we have

$$\begin{aligned} f(S) \subseteq {\mathbb {P}}^{11} \cdot ({\mathbb {P}}^3 \times {\mathbb {P}}^3) \subset {\mathbb {P}}^{15}, \end{aligned}$$

(49)

In other words f is just the morphism defined by the complete linear system $\vert H + \mathring{H} \vert$ composed with the linear embedding ${\mathbb {P}}^{11} \subset {\mathbb {P}}^{15}$.

Proposition 6.12

The map $p \times \mathring{p}$ is an embedding for a general point $[S, {\mathcal {L}}, {\mathcal {E}}] \in {\mathcal {F}}^{\perp }_{5,3}$.

Proof

The linear systems $\vert H \vert$ and $\vert \mathring{H} \vert$ are non hyperelliptic. Hence p, $\mathring{p}$ are generically injective and the same is true for f. In particular $f: S \rightarrow f(S)$ is biregular over $f(S) - {{\,\mathrm{Sing}\,}}f(S)$ and ${{\,\mathrm{Sing}\,}}f(S)$ is a finite set of rational double points. Let $R \subset S$ be an integral curve contracted by f then R is biregular to ${\mathbb {P}}^1$ but it is not $B_i$. Indeed R is contracted by p and $\mathring{p}$ while $B_i$ is not, as one can directly compute. Notice also that $C \sim \frac{1}{2}(H + \mathring{H} + B_{red})$. Therefore, since $RC \ge 0$, it follows

$$\begin{aligned} RC = \frac{1}{2} \sum _{i = 1 \dots 12} RB_i \ge 0 \end{aligned}$$

with $RB_i \ge 0$. Assume $RB_i = 0$ for each i, then $RC=0$. Since the Picard group of S is ${\mathbb {Z}}[\mathcal {L}]\oplus \mathbb {M}_3$, R is necessarily contained in $\mathbb {M}_3=\mathbb {Z}[\mathcal {L}]^{\perp }$. By [14] the unique $(-2)$-curves contained in $\mathbb {M}_3$ are the $B_i$’s, which contradicts the fact that R cannot be a $B_i$. Now assume that $RB_i \ge 2$ for some $B_i$ and consider, among the maps p and $\mathring{p}$, the one not contracting $B_i$, say p. Then p embeds $B_i$ as a line. On the other hand p contracts $R \cdot B_i$, which is a divisor of degree $\ge 2$ in $B_i$: a contradiction. This implies $RB_i = 1$ for each i. Finally consider two distinct curves as above, say $B_1$ and $B_2$, which are contracted by p. Let us also claim that $p(B_1)$ and $p(B_2)$ are distinct points for a general $(S, {\mathcal {L}}, {\mathcal {E}})$. Since $RB_1 = RB_2 = 1$ then p(R) is not a point: a contradiction.

We now prove that $p(B_1)\ne p(B_2)$ for a general $(S, {\mathcal {L}}, {\mathcal {E}})$. If two curves are contracted by a map p to the same point, there is a tree of $(-2)$-curves connecting these curves which is contracted by p. Since p is defined by |H|, the $(-2)$-curves contracted by p are orthogonal to H in ${\mathbb {Z}}[\mathcal {L}]\oplus \mathbb {M}_3$, which is the Picard group of a general S. By a direct computation one observes that the negative defined lattice orthogonal to H contains exactly 12 $(-2)$-classes, which are $\pm B_i$ for $i=1,\ldots ,6$. Since $B_iB_j=0$ if $i,j\in \{1,\ldots ,6\}$ and $i\ne j$, $p(B_1)\ne p(B_2)$. $\square$

At this point the special geometry determined by $\mu _S$ appears, we have

$$\begin{aligned} {{\,\mathrm{Ker}\,}}\mu _S = H^0({\mathcal {I}}(1,1)), \end{aligned}$$

(50)

where ${\mathcal {I}}$ is the ideal sheaf of ${\mathbb {P}}^{11} \cdot ({\mathbb {P}}^3 \times {\mathbb {P}}^3)$ in ${\mathbb {P}}^3 \times {\mathbb {P}}^3$ and $\dim {{\,\mathrm{Ker}\,}}\mu _S = 4$. Let

$$\begin{aligned} \Sigma := {\mathbb {P}}^{11} \cdot ({\mathbb {P}}^3 \times {\mathbb {P}}^3), \end{aligned}$$

(51)

then f(S) sits in ${\mathbb {P}}^{11}$ as a K3 surface of degree 20 and $f(S) \subseteq \Sigma$. Now assume that the intersection scheme $\Sigma$ is proper, then $\Sigma$ is a K3 surface of degree 20 and hence

$$\begin{aligned} f(S) = \Sigma . \end{aligned}$$

(52)

Postponing its proof, we therefore assume the following claim.

Claim For a general triple $(S, {\mathcal {L}}, {\mathcal {E}})$ the intersection scheme $\Sigma$ is proper. Then we prove the birationality of the Mukai map $r_{5,3}: {\mathcal {P}}^{\perp }_{5,3} \rightarrow {\mathcal {R}}_{5,3}$.

Proof

(Proof of the birationality) Since ${\mathcal {P}}^{\perp }_{5,3}$ and ${\mathcal {R}}_{5,3}$ are irreducible of the same dimension, it suffices to show that $r_{5,3}$ is birational onto ${\mathcal {M}} := r_{5,3}({\mathcal {P}}^{\perp }_{5,3})$. Let $x = [S, {\mathcal {L}}, {\mathcal {E}}, C]$ be general in ${\mathcal {P}}^{\perp }_{5,3}$ and $y = r_{5,3}(x)$, then $y = [C, \eta ]$ with $\eta := {\mathcal {E}} \otimes {\mathcal {O}}_C$. Let $y \in {\mathcal {M}}$ be general, we prove that a unique $x = [S, {\mathcal {L}}, {\mathcal {E}}, C]$ exists so that $[C, {\mathcal {E}} \otimes {\mathcal {O}}_C]= y$. We already know, for a general $y = [C, \eta ] \in {\mathcal {M}}$, the surjectivity of the multiplication map

$$\begin{aligned} \mu _C: H^0(\omega _C \otimes \eta ) \otimes H^0(\omega _C \otimes \eta ^{-1}) \rightarrow H^0(\omega _C^{\otimes 2}), \end{aligned}$$

because this condition is open and non empty on ${\mathcal {M}}$. Then, applying to $\mu _C$ the same construction applied to $\mu _S$, one obtains

$$\begin{aligned} C \subseteq \Sigma := {\mathbb {P}}^{11} \cdot ({\mathbb {P}}^3 \times {\mathbb {P}}^3) \subset {\mathbb {P}}^{15}. \end{aligned}$$

(53)

Let $V = H^0(\omega _C \otimes \eta )^*$ and $\mathring{V} = H^0(\omega _C \otimes \eta ^{-1})^*$, here C is bicanonically embedded in ${\mathbb {P}}^{11} := {\mathbb {P}}(\mathrm{Im} \ \mu _C)^*$ and the inclusion is the Segre embedding ${\mathbb {P}}(V) \times {\mathbb {P}}(\mathring{V}) \subset {\mathbb {P}}(V \otimes \mathring{V})$. Now the properness of $\Sigma$ is an open condition on ${\mathcal {M}}$, not empty under our claim. Then $(\Sigma , {\mathcal {O}}_{\Sigma }(1))$ is a polarized K3 surface as above. Since $y = r_{5,3}(x)$ for some $x = [S, {\mathcal {L}}, {\mathcal {E}}, C]$, the commutative diagram (47) implies that $[\Sigma , {\mathcal {O}}_{\Sigma }(1)]= [S, {\mathcal {L}}]$. Therefore $\mu _C$ defines a rational map, sending $y = [C, \eta ] \in {\mathcal {M}}$ to $x \in {\mathcal {P}}^{\perp }_{5,3}$, which is inverse to $r_{5,3}$. $\square$

Proof

(Proof of the claim) Since each component of $\Sigma$ has dimension $\ge 2$, it suffices to construct one ${\mathbb {D}} \in \vert {\mathcal {O}}_{{\mathbb {P}}^3 \times {\mathbb {P}}^3}(1,1) \vert$ so that ${\mathbb {D}} \cdot \Sigma = {\mathbb {D}} \cdot S$. We choose the hyperplane section

$$\begin{aligned} {\mathbb {D}} = (P \times {\mathbb {P}}^3) + ({\mathbb {P}}^3 \times \mathring{P}), \end{aligned}$$

(54)

where P and $\mathring{P}$ are general planes. Then we have ${\mathbb {D}} \cdot S = D + \mathring{D}$, where $D \in \vert H \vert$ and $\mathring{D} \in \vert \mathring{H} \vert$ are smooth, non hyperelliptic curves of genus 3. We show, only for D, that

$$\begin{aligned} D = {\mathbb {P}}^{11} \cdot (P \times {\mathbb {P}}^3) \ , \ \mathring{D} = {\mathbb {P}}^{11} \cdot ({\mathbb {P}}^3 \times \mathring{P}). \end{aligned}$$

(55)

The map $p: D \rightarrow P$ is the canonical map; we fix on P coordinates $(x) = (x_1:x_2:x_3)$. The map $\mathring{p}: D \rightarrow {\mathbb {P}}^3$ is defined by $\vert \omega _D(b) \vert$, where $\deg b = 2$ and $h^0({\mathcal {O}}_D(b)) = 0$. This implies that $\omega _D(b)$ is very ample, we fix coordinates $(y) = (y_1: \dots : y_4)$ on ${\mathbb {P}}^3$. The resolution of ${\mathcal {O}}_{\mathring{p}(D)}(1) \cong \omega _D(b)$ is definitely well known, [17]. We have the exact sequence

$$\begin{aligned} 0 \rightarrow {\mathcal {O}}_{{\mathbb {P}}^3}(-1)^{\oplus 3} {\mathop {\rightarrow }\limits ^{A}}{\mathcal {O}}_{{\mathbb {P}}^3}^{\oplus 4} \rightarrow \omega _D(b) \rightarrow 0, \end{aligned}$$

(56)

$A = (a_{ij})$ being a $4 \times 3$ matrix of linear forms in (y). Then $\mathring{p}(D)$ is a determinantal curve defined by the cubic minors of A. In particular A has rank 3 on ${\mathbb {P}}^3 - \mathring{p}(D)$ and, since $\mathring{p}: D \rightarrow \mathring{p}(D)$ is biregular and $\mathring{p}(D)$ is smooth, it also follows that $\mathring{p}(D)$ is the set of points $y \in {\mathbb {P}}^3$ such that A has exactly rank 2. This implies that the equations $a_{i1}x_1 + a_{i2}x_2 + a_{i3}x_3 = 0, \ i = 1 \dots 4,$ define a complete intersection ${\hat{D}} \subset P \times {\mathbb {P}}^3$ such that ${{\,\mathrm{Supp}\,}}{\hat{D}} = D$. Finally one easily computes that ${\hat{D}}$ and D have the same degree 10 with respect to ${\mathcal {O}}_{{\mathbb {P}}^3 \times {\mathbb {P}}^3}(1,1)$. This implies ${\hat{D}} = D$ and the claim follows. $\square$

6.3 The case of genus 6

Theorem 6.13

The Mukai map $r_{6,3}: {\mathcal {P}}^{\perp }_{6,3} \rightarrow {\mathcal {R}}_{6,3}$ has not maximal rank.

In this paper we only sketch the proof of this theorem and its geometric motivation: see Sect. 7 and also [28]. We postpone some details to further investigation on ${\mathcal {R}}_{6,3}$. We conclude that the mentioned analogies are confirmed for $\ell = 3$: the Mukai maps

$$\begin{aligned} m_{_{11 \pm 1}} \ , \ r_{_{7 \pm 1, 2}} \ , \ r_{_{5 \pm 1, 3}} \end{aligned}$$

(57)

have not maximal rank, while they are birational for $g = 11, \ 7, \ 5$. These maps are not dominant for $g = 10, \ 6, \ 4$ and they have positive dimensional fibre for $g = 12, \ 8, \ 6$.

7 Views on Fano threefolds with sections of level 2 or 3

We close this paper discussing some families of Fano threefolds ${\overline{X}} \subset {\mathbb {P}}^{g+1}$, whose general hyperplane sections are singular K3 surfaces ${\overline{S}}$ of the considered types. Then ${\overline{S}}$ is endowed with a degree $\ell$ cyclic cover $\pi : {\tilde{S}} \rightarrow {\overline{S}}$ with branch locus ${{\,\mathrm{Sing}\,}}{\overline{S}}$. Moreover its minimal desingularization $\nu : S \rightarrow {\overline{S}}$ fits in a standard level $\ell$ K3 surface $(S, {\mathcal {L}}, {\mathcal {E}})$, so that ${\mathcal {L}} \cong \nu ^* {\mathcal {O}}_{{\overline{S}}}(1)$ and ${\mathcal {E}}$ induces $\pi : {\tilde{S}} \rightarrow {\overline{S}}$. We have $\ell = 2, 3$.

For some families a natural cyclic cover $\pi _{{\overline{X}}}: {\tilde{X}} \rightarrow {\overline{X}}$ is visible, with branch locus the curve ${{\,\mathrm{Sing}\,}}{\overline{X}}$. However we do not address it here. The existence of these families implies that $r_{g, \ell }$ has not maximal rank. They correspond to the peculiar values

$$\begin{aligned} (g, \ell ) = (6,3), (6,2), (8,2), (4,3). \end{aligned}$$

(58)

For $\ell = 2$ these families are known, [11, 19, 27]. The case (6, 2) is revisited here with emphasis on a singular quadratic complex of the Grassmannian G(2, 5). This implies that $r_{6, 2}$ is not of maximal rank. For (6, 3) we introduce a family of Gushel - Mukai threefolds singular along a rational normal sextic curve. This is responsible for the failure of the maximal rank of $r_{6,3}$. The case (8, 2) is similar and not treated here, [27]. Finally we point out the plausible relation of the case (4, 3) to the $G_2$-variety.

7.1 A singular Gushel–Mukai threefold: $\ell = 3$ and $g = 6$

We sketch the geometric construction implying theorem (6.13). Let $g = 6$ and $\ell = 3$, keeping our notation we consider $p \times \mathring{p}: S \rightarrow {\mathbb {P}}^4 \times {\mathbb {P}}^4$. Then p is defined by the linear system

$$\begin{aligned} \vert H \vert = \vert C - \frac{1}{3} \sum _{i = 1 \cdots 6} (B_i+ 2B_i') \vert , \end{aligned}$$

(59)

where $B_i+ B_i'$, are the connected components of $B_{\mathsf {red}}$. Let $x_0 := [S, {\mathcal {L}}, {\mathcal {E}}, C] \in {\mathcal {P}}^{\perp }_{6,3}$ be a general point, then a standard analysis shows that $p: S \rightarrow p(S)$ is the contraction of $\sum B_i$ to six points and that $p(B_i')$ is a line. Moreover we have

$$\begin{aligned} p(S) = F_0 \cap Q, \end{aligned}$$

(60)

where $F_0$ is a cubic and Q a smooth quadric. Notice that $p \vert C$ is the embedding defined by $\omega _C \otimes \eta ^{-1}$, since $CB_i = 0$ then $p(C) \cap {{\,\mathrm{Sing}\,}}p(S) = \emptyset$. Let $C' := p(C)$ and let

$$\begin{aligned} 0 \rightarrow {\mathcal {I}}_{p(S)}(3) \rightarrow {\mathcal {I}}_{p(C)}(3) \rightarrow {\mathcal {I}}_{C' \vert p(S)}(3) \rightarrow 0 \end{aligned}$$

(61)

be the standard exact sequence of ideal sheaves of Q, we notice the isomorphisms ${\mathcal {I}}_{p(S)}(3) \cong {\mathcal {O}}_Q$ and $p_*: H^0({\mathcal {O}}_S(3H-C)) \rightarrow H^0({\mathcal {I}}_{p(C) \vert p(S)}(3))$. This implies that

$$\begin{aligned} 0 \rightarrow H^0( {\mathcal {O}}_Q) \rightarrow H^0({\mathcal {I}}_{C'}(3)) \rightarrow H^0({\mathcal {O}}_S(3H - C)) \rightarrow 0 \end{aligned}$$

(62)

is its associated long exact sequence. It easily follows that $C'$ is projectively normal. A second standard step is the remark that ${\mathcal {O}}_S(3H-C)$ is a genus 3 polarization of S. Now let $M \in \vert 3H - C \vert$, then $p_*(C+M) \in \vert {\mathcal {I}}_{p(C) \vert p(S)}(3) \vert$ and it is cut on p(S) by a cubic hypersurface. Therefore we have in Q the complete intersection scheme

$$\begin{aligned} p_*(C + M) = F_0 \cap F_{\infty } \cap Q, \end{aligned}$$

(63)

where $F_0, F_{\infty }$ are cubics. Let $S'_0 = F_0 \cdot Q$ and $S'_{\infty } = F_{\infty } \cdot Q$. We consider the pencil

$$\begin{aligned} P_M = \lbrace S'_t, \ t \in {\mathbb {P}}^1 \rbrace , \end{aligned}$$

(64)

of cubic sections of Q generated by $S'_0$ and $S'_{\infty }$. We can assume $p(S) = S'_0$, notice that a general $S'_t$ is a possibly singular K3 surface, smooth along $C'$. Let $\sigma _t: S_t \rightarrow S'_t$ be its minimal desingularization and $C_t := \sigma ^*_t C'$, then $S_t$ is endowed with the line bundles

$$\begin{aligned} {\mathcal {H}}_t := \sigma ^*_t {\mathcal {O}}_Q(1), \quad {\mathcal {L}}_t := {\mathcal {O}}_{S_t}(C_t), \ {\mathcal {E}}_t := {\mathcal {L}}_t \otimes {\mathcal {H}}^{-1}_t. \end{aligned}$$

(65)

For $t = 0$ the fourtuple $(S_t, {\mathcal {L}}_t, {\mathcal {E}}_t, C_t)$ defines the point $x_0 = [S, {\mathcal {L}}, {\mathcal {E}}, C]$ of ${\mathcal {P}}^{\perp }_{6,3}$. For $t \ne 0$ we have constantly $C_t = C$. Now consider the family of fourtuples

$$\begin{aligned} \lbrace (S_t, {\mathcal {L}}_t, {\mathcal {E}}_t, C_t), \ t \in {\mathbb {P}}^1 \rbrace , \end{aligned}$$

(66)

then the assignment $t \rightarrow [{\mathcal {S}}_t, {\mathcal {L}}_t] \in {\mathcal {F}}_6$ defines a non constant rational map $m: {\mathbb {P}}^1 \rightarrow {\mathcal {F}}_6$. Assume $(S_t, {\mathcal {L}}_t, {\mathcal {E}}_t)$ is a K3 surface of level 3 for a general t. Then m lifts to a map ${\tilde{m}}: {\mathbb {P}}^1 \rightarrow {\mathcal {P}}^{\perp }_{6,3}$, sending t to $[S_t, {\mathcal {L}}_t, {\mathcal {E}}_t, C_t]$, and the next statement immediately follows.

Proposition 7.1

If $(S_t, {\mathcal {L}}_t, {\mathcal {E}}_t)$ is a K3 surface of level 3 for a general t, the curve ${\tilde{m}}({\mathbb {P}}^1)$ is in the fibre at the point $[C, \eta ]$ of the Mukai map $r_{6,3}$, which is therefore not of maximal rank.

The assumption mentioned in the statement depends on the choice of the element M in $\vert 3H - C \vert$ and in general it is not satisfied. However the assumption is satisfied choosing in $\vert M \vert$ the very special element

$$\begin{aligned} M_0 := 2A + \sum _{i = 1 \cdots 6} B_i, \end{aligned}$$

(67)

where A is the unique element of $\vert C - \sum _{i = 1 \cdots 6} (B_i+B_i') \vert$. The curve A is biregular to ${\mathbb {P}}^1$ and $p \vert A$ embeds it as a rational normal quartic curve. Let $A' = p(A)$, then the base scheme of $P_{M_0}$ is a non reduced, complete intersection curve and its 1-cycle is

$$\begin{aligned} p_*(M_0 + C) = 2A' + C'. \end{aligned}$$

(68)

In other words the surfaces $S'_t$ intersect along a contact curve $A'$ of multiplicity two and along $C'$. It turns out that a general ${{\,\mathrm{Sing}\,}}S'_t$ consists of six nodes moving in $A'$ and each node belongs to a line in $S'_t$. This can be shown using the special property that $\eta \cong \omega _{C'}(-1) \in {{\,\mathrm{Pic}\,}}C$ is of 3-torsion. Omitting further details of this construction, let us just say that $M_0$ defines a pencil of level 3 and genus 6 K3 surfaces as required.

To close geometrically this sketch let ${\mathsf {A}}$ be the non reduced component, supported on $A'$, of the base curve of $P_{M_0}$ and ${\mathcal {I}}_{{\mathsf {A}} \vert Q}$ its ideal sheaf. Consider the rational map

$$\begin{aligned} \phi : Q \rightarrow {\mathbb {P}}^7 \end{aligned}$$

(69)

defined by the linear system $\vert {\mathcal {I}}_{{\mathsf {A}} \vert Q}(3) \vert$. Let us notice the following property.

Proposition 7.2

The map $\phi$ is birational onto its image W, which is a singular Gushel–Mukai threefold whose general hyperplane sections are singular K3 surfaces ${\overline{S}}$ as above.

Therefore W is a complete intersection of type (1, 1, 2) in the Grassmannian G(2, 5). We notice that ${{\,\mathrm{Sing}\,}}W$ is a rational normal sextic curve. This completes our sketch.

7.2 The tangential quadratic complex of ${\mathbb {P}}^4$: $\ell = 2$ and $g = 6$

Let ${\mathbb {G}}_n$ be the Plücker embedding of the Grassmannian of lines of ${\mathbb {P}}^n$, a quadratic complex is just a quadratic section of ${\mathbb {G}}_n$. Let $Q \subset {\mathbb {P}}^n$ be a quadric, then the family ${\mathbb {T}}$ of tangent lines to Q is a quadratic complex, named sometimes the tangential quadratic complex. We assume Q is smooth, then ${\mathbb {T}}$ is a Fano variety. Notice that ${{\,\mathrm{Sing}\,}}{\mathbb {T}}$ is the Hilbert scheme of lines of Q, of codimension and multiplicity 2 in ${\mathbb {T}}$.

Now we assume n is even. Then ${\mathbb {T}}$ has a unique nontrivial quasi étale 2:1 cover

$$\begin{aligned} \pi : \tilde{{\mathbb {T}}} \rightarrow {\mathbb {T}}, \end{aligned}$$

(70)

whose branch locus is ${{\,\mathrm{Sing}\,}}{\mathbb {T}}$. Let us describe the known map $\pi$ in the case $n = 4$, since it is linked to the Mukai map $r_{6,2}: {\mathcal {P}}^{\perp }_{6,2} \rightarrow {\mathcal {R}}_6$ and its behavior. This is treated in [11]. For $n = 4$ the Hilbert scheme of lines of Q is the 2-Veronese embedding of ${\mathbb {P}}^3$, say

$$\begin{aligned} V \subset {\mathbb {G}}_4 \subset {\mathbb {P}}^9. \end{aligned}$$

(71)

Let $t \in {\mathbb {T}}$, consider the pencil $\lbrace H_p, p \in t \rbrace$, where $H_p$ is the polar hyperplane to Q at p. Its base locus is a plane $P_t$ and $Q_t := P_t \cdot Q$ is a conic. Since t is tangent to Q, a standard exercise shows that ${{\,\mathrm{Sing}\,}}Q_t = t \cap Q$. This defines a smooth, integral correspondence

$$\begin{aligned} \tilde{{\mathbb {T}}} := \lbrace (t,r) \in {\mathbb {T}} \times V \ \vert \ r \subset Q_t \rbrace . \end{aligned}$$

(72)

Notice that its projection onto ${\mathbb {T}}$ is a quasi étale 2 : 1 cover branched on V, say

$$\begin{aligned} \pi : \tilde{{\mathbb {T}}} \rightarrow {\mathbb {T}}. \end{aligned}$$

(73)

Indeed the fibre $\zeta _t := \pi ^*(t)$ is the Hilbert scheme of lines of $Q_t$ and is finite of length 2. Then $\zeta _t$ is smooth iff ${{\,\mathrm{rank}\,}}Q_t = 2$ iff $t \notin V$ and $\zeta _t$ has multiplicity 2 iff ${{\,\mathrm{rank}\,}}Q_t = 1$ iff $t \in V$.

Now it is well known that a general 2-dimensional linear section ${\overline{S}} = {\mathbb {T}} \cap {\mathbb {P}}^6$ is the model defined by $\vert {\mathcal {L}} \vert$ of S, where $[S, {\mathcal {L}}, {\mathcal {E}}] \in {\mathcal {F}}^{\perp }_6$ is general. In particular ${{\,\mathrm{Sing}\,}}{\overline{S}} = V \cap {\mathbb {P}}^6$ is an even set of 8 nodes, defining $\pi \vert {\tilde{S}}$ with ${\tilde{S}} = \pi ^{-1}({\overline{S}})$, cfr. [11, 19, 20]. For $\ell = 2$ and $[S, {\mathcal {L}}, {\mathcal {E}}] \in {\mathcal {F}}^{\perp }_g$, the surface S, or its model ${\overline{S}}$, is known as a standard Nikulin surface of genus g. Therefore we can say that a general 3-dimensional linear section of ${\mathbb {T}}$ is a Fano threefold whose hyperplane sections are standard Nikulin surfaces of genus 6. Let us denote such a section by

$$\begin{aligned} X = {\mathbb {T}} \cap {\mathbb {P}}^7, \end{aligned}$$

(74)

notice that ${{\,\mathrm{Sing}\,}}X$ is a curvilinear section of V, hence an elliptic curve of degree 8.

Finally let ${\mathcal {C}}$ and $\overline{{\mathcal {S}}}$ respectively be the family of general curvilinear sections C and that of general 2-dimensional linear sections ${\overline{S}}$ of ${\mathbb {T}}$. Consider the family of pairs

$$\begin{aligned} {\mathcal {P}} := \lbrace (C, {\overline{S}}) \in {\mathcal {C}} \times \overline{{\mathcal {S}}} \ \vert \ C \subset {\overline{S}} \rbrace . \end{aligned}$$

(75)

Let $(C, {\overline{S}}) \in {\mathcal {P}}$ then C is a canonical curve and $C \in \vert {\mathcal {O}}_{{\overline{S}}}(1) \vert$. Let $\nu : S \rightarrow {\overline{S}}$ be the desingularization then $\nu ^*C \in \vert {\mathcal {L}} \vert$ and $\eta := {\mathcal {E}} \otimes {\mathcal {O}}_{\nu ^*C}$ defines $\pi \vert {\tilde{C}}$, where ${\tilde{C}} = \pi ^{-1}(C)$. Then the assignment of $(C, {\overline{S}})$ to $[S, {\mathcal {L}}, {\mathcal {E}}, \nu ^*C]$ defines a dominant rational map

$$\begin{aligned} m: {\mathcal {P}} \rightarrow {\mathcal {P}}^{\perp }. \end{aligned}$$

We already know that the Mukai map $r_{6,2}$ fails to be of maximal rank. However we can now see this fact from a geometric perspective: the existence of the Fano variety ${\mathbb {T}}$ and its quasi finite 2 : 1 cover $\pi$. Indeed this implies that $C \in {\mathcal {C}}$ is contained in a higher dimensional family of sections ${\overline{S}}$ of ${\mathbb {T}}$, so that C cannot have general moduli.

More precisely the parameter space ${\mathcal {C}}$ is open in the Grassmannian G(5, 9), hence $\dim {\mathcal {C}} = 24$. Moreover ${{\,\mathrm{Aut}\,}}Q \subset {{\,\mathrm{Aut}\,}}{\mathbb {P}}^4$ has dimension 10 and acts faithfully on ${\mathcal {C}}$. Then we have $\dim {\mathcal {C}} /\!\!/{{\,\mathrm{Aut}\,}}Q = 14 < \dim {\mathcal {R}}_6 = 15$. Hence $r_{6,2}$ cannot be dominant.

Remark 7.1

Let $C \in {\mathcal {C}}$ then ${\tilde{C}} = \pi ^{-1}(C)$ is a smooth, integral curve of genus 11. We have ${\tilde{C}} \subset {\tilde{S}} \subset {\tilde{X}} \subset {\mathbb {P}}^{12}$, where ${\tilde{X}} = \pi ^{-1}(X)$ is a non prime Fano threefold of genus 11. We just mention that ${\tilde{C}}$ is the base locus of a pencil of hyperplane sections of ${\tilde{X}}$ and that the birational Mukai map $m_{11}: {\mathcal {P}}_{11} \rightarrow {\mathcal {M}}_{11}$ is not invertible at $[{\tilde{C}}]$.

7.3 The ${\mathsf {G}}_2$-variety: $\ell = 3$ and $g = 4$

A geometric interpretation seems plausible and it is possibly postponed to future work. It relates to the failure of the Mukai map in genus 10. As in (14) let $\pi : {\tilde{S}} \rightarrow {\overline{S}}$ be the cover induced by ${\mathcal {E}}$ and $\nu : S \rightarrow {\overline{S}}$ the desingularization map. For a general C the map $\nu : C \rightarrow {\overline{S}} {\setminus } {{\,\mathrm{Sing}\,}}{\overline{S}}$ is an embedding, then we set $C := \nu (C)$. Let ${\tilde{C}} := \pi ^{-1}(C)$ then $({\tilde{S}}, {\mathcal {O}}_{{\tilde{S}}}({\tilde{C}} ))$ is a K3 surface of genus 10. This suggests that ${\tilde{S}}$ embeds in the $G_2$-variety $W \subset {\mathbb {P}}^{13}$ as a linear section, [23]. Now a general curvilinear section of W is not general as a genus 10 curve. In the same way, if it is a triple cover of a genus 4 curve, it seems not a general genus 4 triple cover.