Flexible affine cones and flexible coverings
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Abstract
We provide a new criterion for flexibility of affine cones over varieties covered by flexible affine varieties. We apply this criterion to prove flexibility of affine cones over secant varieties of Segre–Veronese embeddings and over certain Fano threefolds. We further prove flexibility of total coordinate spaces of Cox rings of del Pezzo surfaces.
Keywords
Automorphism group Transitivity Flexibility Affine cone Cox ring Segre–Veronese embedding Secant variety Del Pezzo surfaceMathematics Subject Classification
14R20 14J50In this article we study affine algebraic varieties with the following property: the (special) automorphism group acts infinitely transitively on the regular locus. The systematic study of this remarkable property and its complex analytic counterpart is presented in Arzhantsev et al. [1].
Let X be an affine variety over an algebraically closed field \(\mathbf {k}\) of characteristic zero. We consider actions of the additive group \(\mathbb {G}_a = (\mathbf {k},+)\) on X. The subgroup of \({{\mathrm{Aut}}}(X)\) generated by all \(\mathbb {G}_a\)actions is called the special automorphism group of X and will be denoted by \({{\mathrm{SAut}}}(X)\). We are interested in transitivity of the \({{\mathrm{SAut}}}(X)\)actions on the regular locus \(X_\mathrm {reg}\). Recall that an action of a group G on a set M is mtransitive if for every two mtuples \((x_1, \ldots , x_m)\) and \((x'_1, \ldots , x'_m)\) of pairwise distinct elements of M there exists \(g\in G\) such that \(g\cdot x_i = x_i'\) for \(i=1, \ldots m\).
We have the following result.
Theorem 0.1
 (i)
The group \({{\mathrm{SAut}}}(X)\) acts transitively on the regular locus \(X_{\mathrm {reg}}\).
 (ii)
The group \({{\mathrm{SAut}}}(X)\) acts mtransitively on \(X_{\mathrm {reg}}\) for every \(m > 0\).
 (iii)
The tangent space of every \(x \in X_{\mathrm {reg}}\) is spanned by tangent vectors to orbits of \(\mathbb {G}_a\)actions.
We say that X is flexible if these conditions are fulfilled.
As examples of flexible varieties, let us mention affine cones over del Pezzo surfaces of degree \(\ge 4\) (see [30]), over flag varieties, and affine toric varieties without torus factors [5]. It is also possible to construct new flexible varieties from a given flexible one, e.g. via suspensions [5] or open subsets with complement of codimension \(\ge 2\) [14].
Our first main result provides a new criterion for flexibility of affine cones, see Sect. 1 for the proof. A similar, but independent, criterion using the notions of cylinders was provided in Perepechko [30, Th. 5].
Theorem 1.4
Let Y be a normal projective variety covered by flexible affine open subsets \(U_i\), \(i\in I\), and H be a very ample divisor on Y. If each subset \(Y{\setminus } U_i\) is the support of an effective \(\mathbb {Q}\)divisor \(D_i\) linearly equivalent to H, then the affine cone \(X= {{\mathrm{AffCone}}}_H Y\) is flexible.
Recall that a Segre–Veronese variety is an embedding of the direct product \(\mathbb {P}^{d_1}\times \dots \times \mathbb {P}^{d_n}\) of projective spaces by the very ample line bundle of the form \({\mathcal {O}}(s_1)\boxtimes \dots \boxtimes {\mathcal {O}}(s_n)\). Further, given a projective variety \(X\subset \mathbb {P}^n\), the Zariski closure of the union of the secant (resp. tangent) lines to X is called a secant (resp. tangential) variety of X. As the first application of Theorem 1.4, we deduce the following result, see Sect. 2 for details.
Theorem 2.20
Let \(X=v_{s_1}(\mathbb {P}(V_1))\times \dots \times v_{s_n}(\mathbb {P}(V_n))\) be a Segre–Veronese variety. Then the affine cone over the secant variety of X is flexible. Further, if \(s_1=\dots =s_n=1\), then also the affine cone over the tangential variety of X is flexible.
Our proof technique relies on triangular transformations of the affine charts of the ambient projective space. They are inspired by algebraic statistics, precisely by computation of cumulants [10, 29, 34, 41, 42].
Section 3 contains preliminaries on smooth rational Tvarieties of complexity one. These are varieties X with an effective action of a torus T, where \(\dim X = \dim T+1\). Section 4 is devoted to flexibility of affine cones over such varieties.
In Arzhantsev et al. [6] it was shown that smooth varieties of this type admit a toric covering and for certain affine cones over these varieties we, indeed, obtain flexibility. For example, this applies to all known Fano threefolds with 2torus action. We use below the list of Fano threefolds in Mori–Mukai’s classification [26].
Theorem 4.5
All the affine cones over the Fano threefolds Q, 2.29, 2.30, 2.31, 2.32, 3.8, 3.18, 3.19, 3.23, 3.24, 4.4, and certain elements of the families 2.24, 3.10 admitting a 2torus action in Mori–Mukai’s classification are flexible.
The main tool to obtain these results is the combinatorial description of Tvarieties developed in Altmann and Hausen [2] and Altmann et al. [3], which in the case of complexity one allows to study (torus equivariant) coverings as in Theorem 1.4.
While Theorems 2.20 and 4.5 are concerned with projective coordinate rings, in Sect. 5 of the paper we obtain related results for total coordinate rings or Cox rings.
Theorem 5.4
The total coordinate spaces of smooth del Pezzo surfaces are flexible.
This was known so far only for the toric del Pezzo surfaces (i.e. those of degree 9, 8, 7 and 6) and by Arzhantsev et al. [5, Thm. 0.2] for the case of degree 5, where the total coordinate space is known to be the affine cone over the Grassmannian G(2, 5). On the other hand, this extends a result of Arzhantsev et al. [6], where flexibility was proved only outside a subset of codimension 2.
Theorem 5.9
The total coordinate space of a complete smooth rational Tvariety of complexity one is flexible.
1 Flexibility of affine cones
Lemma 1.1

the canonical \(\mathbb {G}_m\)action on X by homotheties sends Gorbits to Gorbits,

all Gorbits are locally closed, and

there is an orbit \(Gx\subset X{\setminus }\{0\}\), whose image \(Y^*\) under the projection \(\pi :X{\setminus }\{0\}\rightarrow Y\) is an open subset in Y with complement of codimension \(\ge 2\).
Proof
Since the Gaction is \(\mathbb {G}_m\)equivariant, \(X^*=\pi ^{1}(Y^*)\) is a union of Gorbits, whose projections coincide with \(Y^*\). Hence \(X^*=\bigcup _{\lambda \in \mathbb {G}_m}\lambda Gx\), where all Gorbits are closed in \(X^*\).
Let us show that \(Gx=X^*\). Assume the contrary. Then \(\dim Gx = \dim Y\) and the stabilizer \(S\subset \mathbb {G}_m\) of the orbit Gx is finite. Indeed, two points \(v,v'\in Gx\) lie in the same \(\mathbb {G}_m\)orbit if and only if \(v = \lambda \cdot v'\) for some \(\lambda \in \mathbb {G}_m\). The latter is equivalent to \(\lambda Gx = Gx\), i.e., \(\lambda \in S\). So, \(Gv\cap \mathbb {G}_m v= Sv\) for any \(v\in X^*\).
On the other hand, under our assumptions \(D\subseteq Y{\setminus } Y^*\) is of codimension at least 2. Therefore, D is empty and \(\mu (Gx)\) is a global section of \(X' \rightarrow Y\) disjoint with the zerosection. Since Y is of positive dimension \(X'\rightarrow Y\) is nontrivial. This gives a contradiction.
So, the group G acts on \(X^*\) transitively. \(\square \)
Corollary 1.2
Under the setup of Lemma 1.1, if Y is smooth in codimension one and \(\pi (Gx)=Y^*\) coincides with the regular locus \(Y_\mathrm {reg}\), then \(Gx=X_\mathrm {reg}{\setminus }\{0\}\).
Lemma 1.3

\(\phi (\mathbb {G}_a\times \{\pi (x)\})=\pi ({\hat{\phi }}(\mathbb {G}_a\times \{x\}))\) for any \( x\in \pi ^{1}(U)\).

\({\hat{\phi }}\) is trivial on \(X{\setminus }\pi ^{1}(U)\).
Proof
Let Y be defined by a homogeneous ideal \(I\subset \mathbf {k}[x_0,\ldots ,x_n]\) which does not contain \(x_0\), \(\mathbf {k}[X]=\mathbf {k}[x_0,\ldots ,x_n]/I\), and \(U=\{x_0\ne 0\}\subset Y\).
There exists a natural embedding \(\rho {:} U\hookrightarrow X\), \(\rho (U)=\{x_0=1\}\subset X\). On the other hand, \(\pi :X{\setminus }\{x_0=0\}\rightarrow U\) is a trivial \(\mathbb {G}_m\)bundle. Therefore, we may extend the \(\mathbb {G}_a\)action \(\phi \) on U to a \(\mathbb {G}_a\)action \(\tilde{\phi }\) on \(X{\setminus }\{x_0=0\}\) defined by a homogeneous locally nilpotent derivation \(\tilde{\delta }\).
There exists \(d\in \mathbb {N}\) such that \(x_0^d\tilde{\delta }(x_i)\in \mathbf {k}[X]\) for \(i=1,\ldots ,n\). Since \(x_0\in \ker \tilde{\delta }\), a homogeneous derivation \({\hat{\delta }}=x_0^{d+1}\tilde{\delta }\) on X is locally nilpotent. The corresponding \(\mathbb {G}_a\)action \({\hat{\phi }}\) is normalised by the \(\mathbb {G}_m\)action, coincides with \(\phi \) on the hypeplane section \(\{x_0=1\} \cong U\) of X, and is trivial on \(\{x_0=0\}\subset X\). \(\square \)
Theorem 1.4
Let Y be a normal projective variety covered by flexible affine open subsets \(U_i\), \(i\in I\), and H be a very ample divisor on Y. If each subset \(Y{\setminus } U_i\) is the support of an effective \(\mathbb {Q}\)divisor \(D_i\) linearly equivalent to H, then the affine cone \(X= {{\mathrm{AffCone}}}_H Y\) is flexible.
Proof
The tangent space at 0 to X must contain all lines through 0 contained in X. As X is covered by such lines, the tangent space is equal to the linear span of X. If \(X=\mathbb {A}^{n+1}\) the theorem is trivial. Otherwise, the origin is a singular point of X, which we assume from now on.
For each subset \(U_i\) there exists a finite number of \(\mathbb {G}_a\)actions \(\{\phi _{ij}\}\) such that the orbit of the group generated by them is the regular locus of \(U_i\), see [1, Prop. 1.5]. Let \(k\in \mathbb {N}\) be such that \(kD_i\) is a \(\mathbb {Z}\)divisor for any \(i\in I\). For each action \(\phi _{ij}\) we can consider a lifted action \(\tilde{\phi }_{ij}\) on \({{\mathrm{AffCone}}}_{kH} Y\) as in Lemma 1.3. Since the Veronese map \(X \rightarrow {{\mathrm{AffCone}}}_{kH} Y\) is unramified outside the vertex, Theorem 1.3 of Masuda and Miyanishi [27] implies the existence of an action \({\hat{\phi }}_{ij}\) on X, whose orbits have the same image in Y as the orbits of \(\tilde{\phi }_{ij}\).
Let a subgroup \(G=\langle {\hat{\phi }}_{ij}\rangle \subset {{\mathrm{SAut}}}X\) be generated by the \(\mathbb {G}_a\)actions on X which correspond to all the open subsets \(U_i\). Then the image of the orbit Gx of a regular point \(x \in X_{\mathrm {reg}}\) under the projection \(X{\setminus }\{0\}\rightarrow Y\) is equal to \(Y_{\mathrm {reg}}\). Thus, the statement follows from Corollary 1.2, as the variety Y is normal, in particular smooth in codimension one, and the Gorbits are locally closed by Arzhantsev et al. [1, Prop. 1.3]. \(\square \)
Example 1.5
Consider \(\mathbb {P}^n\) with coordinates \(x_0, \ldots , x_n\) and a smooth subvariety \(Y=\mathbb {V}(x_0,f)\) of codimension 2, where f is an irreducible homogeneous polynomial of degree d. Let \(q :X \rightarrow \mathbb {P}^n\) be the blowup in Y. We apply Theorem 1.4 to show that all the affine cones over X are flexible. Notice that X is a Fano variety if \(d \le n\) holds.
2 Secant of Segre–Veronese variety
This section is based on Michalek et al. [29], where the toric covering of a Segre variety was constructed. Here we generalize that construction to a Segre–Veronese variety and hence prove Theorem 2.20. Throughout this section by a parameterization of a variety Z we mean a dominant morphism from a dense open subset of an affine space to Z.
Definition 2.1
We will be using an equivalent construction. Apart from (projective) Segre–Veronese varieties we will consider affine cones over them and refer to those as Segre–Veronese cones. They should not be confused with intersections of Segre–Veronese varieties with principal affine open subsets, which also play a crucial role.

\(e:(v_1,\dots ,v_n)\mapsto (v_1,\dots ,v_1,v_2,\dots ,v_2,v_3,\dots ,v_n)\) is the diagonal embedding,

\(\tilde{\psi }:(v_1^1,\dots ,v_1^{s_1},v_2^1,\dots ,v_2^{s_2},v_3^1,\dots ,v_n^{s_n})\mapsto v_1^1\otimes \dots \otimes v_n^{s_n}\) is the parameterization of the Segre cone, which is a nonlinear map,

\(\psi =\tilde{\psi }_{{{\mathrm{im}}}(e)}\circ e\) is the parameterization of the Segre–Veronese cone, and

\(s_{V_i}:V_i^{\otimes s_i}\rightarrow S^{s_i}(V_i)\) are the natural symmetrizing projections.
Notation 2.2
 (i)For a vector space \(V=V_{i_1}\otimes \cdots \otimes V_{i_k}\) we denoteand regard it as a vector space with basis \(\{e_{j_1}^{i_1}\otimes \cdots \otimes e_{j_k}^{i_k}\mid j_1+\cdots +j_n>0\}.\)$$\begin{aligned}{\hat{V}}:=\{x\in V\mid (e_0^{i_1}\otimes \cdots \otimes e_0^{i_k})^*(x)=1\}\end{aligned}$$
 (ii)
We may, and often will, consider \({\hat{V}}\) as a complement to a hyperplane section \(\{[x]:(e_0^{i_1}\otimes \cdots \otimes e_0^{i_k})^*(x)=0\}\) of \(\mathbb {P}(V)\). Thus \({{\hat{V}}}\) may be regarded as an affine open subset of \(\mathbb {P}(V)\).
 (iii)
We denote \(A:=V_1^{\otimes s_1}\otimes \dots \otimes V_n^{\otimes s_n}\).
 (iv)We also define$$\begin{aligned}&B:= \prod _{i=1}^n\hat{V_i}^{\times s_i}\subset V_1^{s_1}\oplus \dots \oplus V_n^{s_n},\\&B':= \prod _{i=1}^n\hat{V_i}\subset V_1\oplus \dots \oplus V_n. \end{aligned}$$
 (v)
Finally, we denote \(\pi :A{\setminus }\{0\}\rightarrow \mathbb {P}(A)\) and obtain the following diagram of open subsets:
Remark 2.3
Since \(\mathbb {P}(S^{s_1}(V_1)\otimes \dots \otimes S^{s_n}(V_n))\subset \mathbb {P}(A)\), we can study the Segre–Veronese variety as a subvariety of \(X=\overline{\pi \circ \psi (B')}\subset \mathbb {P}(A)\). Note that the image of \(\tilde{\psi }_B\) does not contain the origin.
2.1 Cumulants
In this setting we may apply the (nonlinear) coordinate systems of B, called cumulants and presented in Michalek et al. [29]. For the motivations to consider them, coming from algebraic statistics, we refer the reader to Refs. [34, 41, 42]. A general mathematical setting for these methods is well presented in Ciliberto et al. [10]. Further results are obtained for other varieties, e.g. Grassmannians and spinor varieties [28]. However, in other cases we do not obtain toric coverings. Still, we believe that similar methods can be applied to a larger class of secant and tangential varieties.
Notation 2.4
Basis elements of A are of the form \(e^1_{c^1_1,\ldots ,c^1_{s_1}}\otimes \cdots \otimes e^n_{c^n_1,\ldots ,c^n_{s_n}}\) and are in natural correspondence with tuples \((c^1_1,\ldots ,c^1_{s_1},\ldots ,c^n_1,\ldots ,c^n_{s_n})\), where \( 0\le c^i_j\le d_i\) for \(1\le i\le n\) and \(1\le j\le s_i\). Let us denote the set of these tuples by C(A) and for each \(c\in C(A)\) the corresponding basis element by e(c). Finally, denote the dual basis elements by \(x(c)=e(c)^*\). Similarly, we denote \(C({\hat{A}})=C(A){\setminus }\{(0,\ldots ,0)\}\).
Definition 2.5
(degree, ordering) Given a tuple \(c\in C({\hat{A}})\), the number of its nonzero entries is called the degree of c. Given \(c_1,c_2\in C({\hat{A}})\), we say that \(c_1\le c_2\) if \(c_1\) can be obtained from \(c_2\) by setting some entries to zero.
Thus, we have a natural poset structure on \(C({\hat{A}})\), which induces a poset structure on the basis of \(\hat{A}\). Simply speaking, the ordering on the basis is defined by replacing \(e^i_j\) by \(e^i_0\) in the tensor product elements.
Example 2.6
Consider \(t_2:=e^1_1\otimes e^1_0\otimes e^2_5\), \(t_1:=e^1_0\otimes e^1_0\otimes e^2_5\) and \(t_0:=e^1_1\otimes e^1_3\otimes e^2_4\). We have \(t_1<t_2\) and \(t_1,t_2\) are not comparable with \(t_0\).
Definition 2.7
A thick interval partition of a tuple \(c\in C({\hat{A}})\) of degree at least two is an increasing sequence of indices \({1\brack 1}=b_0<\ldots <b_k={n\brack s_n}\) such that \(\deg c_{[b_i:b_{i+1}]}\ge 2\) for each i. The set of all thick interval partitions of c will be denoted by IP(c). It is always nonempty as it contains \(\left\{ {1\brack 1},{n\brack s_n}\right\} \).
Now we can recall the coordinate systems from [29, Sec. 2].
Notation 2.8
Lemma 2.9
Each one of the sets \(\{x(c)\}_{ c\in C({\hat{A}})},\) \(\{y(c)\}_{ c\in C({\hat{A}})},\) and \(\{z(c)\}_{ c\in C({\hat{A}})}\) is an algebraically independent system of functions generating \({\mathcal {O}}({\hat{A}})\). In other words, \(\{z(c)\}\) is a coordinate system on \({\hat{A}}\) as an affine space.
Proof
Since y(c) is a sum of x(c) and of terms of smaller degree, the endomorphism of \(\mathbf {k}[{\hat{A}}]\) that maps x(c) to y(c) for each c is invertible. The same holds for \(\{y(c)\}\) and \(\{z(c)\}\). So, the statement follows. \(\square \)
2.2 Secant
Lemma 2.10
Proof
2.3 Torus action
We can infer the following decomposition of \(\sec _X\).
Notation 2.11
Lemma 2.12
There is a decomposition \(\sec _X = m\circ {{\mathrm{rep}}}.\) In particular, m is a monomial parameterization of \({{\mathrm{Sec}}}X\).
Proof
Straightforward. \(\square \)
This monomial parameterization of \({{\mathrm{Sec}}}X\) already provides a structure of a toric variety on \({{\mathrm{im}}}(m)={\hat{A}}\cap {{\mathrm{Sec}}}X,\) hence provides us with a toric chart of \({{\mathrm{Sec}}}X\). Below we describe in detail its structure.
Notation 2.13
Notation 2.14
Proposition 2.15
 (i)
\({{\mathrm{Sec}}}X\cap {\hat{A}}= {\hat{A}}_1\times S_2\) via natural projections along coordinates z(c),
 (ii)
\({\hat{A}}_1=X\cap {\hat{A}}\cong \mathbb {A}^N\), where \(N={\sum _{i=1}^n\dim {\hat{V}}_i},\)
 (iii)
\(S_2\cong {{\mathrm{AffCone}}}(X_P)\), where \(X_P\) is a projective toric variety with polarization corresponding to the polytope P, see, e.g., [11, §2.3] for the construction.
Proof
The morphism m is a direct product of \(m_{\{0\}\times B'\times \{0\}}\) and \(m_{\mathbb {A}^1\times \{0\}\times B'}\), which respectively parameterize \({\hat{A}}_1\) and \(S_2\). This implies (i). The Segre–Veronese variety X is also parameterized by \(m_{\{0\}\times B'\times \{0\}}\), thus (ii) holds.
Let us consider the standard torus \(T={{\mathrm{Spec}}}\mathbf {k}[M]\subset B'\) and a projectivization \(\pi _z:{\hat{A}}{\setminus }\{0\}\rightarrow \mathbb {P}({\hat{A}})\) along z(c)coordinates. Then \(X_P=\overline{\pi _z(S_2)}\) is parameterized by a Tequivariant map \(\pi _z\circ m_{\{0\}\times \{0\}\times B'}\), which is defined by the set of monomials \(\{(e^i_{c^i_j})^{*}\mid \deg c>1\}\subset {\mathcal {O}}(B')\) corresponding exactly to lattice points \(P\cap M\). This implies (iii). \(\square \)
Proposition 2.16
Proof
These propositions imply the following relationship of the tangential and secant varieties, which, in turn, implies Zak’s theorem [40] for Segre–Veronese varieties.
Corollary 2.17
 (i)
P is not contained in the hyperplane \(\sum _{i,j}(\chi ^i_j)^*= 2\),
 (ii)
\(\dim {{\mathrm{Sec}}}X=2\dim X+1\),
 (iii)
\({{\mathrm{Sec}}}X\ne {{\mathrm{Tan}}}X,\)
 (iv)
\(\dim {{\mathrm{Tan}}}X = \dim {{\mathrm{Sec}}}X 1\).
Proof

\((i)\Rightarrow (ii)\) The assumption implies \(\dim P=d\). Hence, \(\dim S_2=d+1\) and \(\dim {{\mathrm{Sec}}}X=\dim {\hat{A}}_1+\dim S_2=2d+1\).

\((i)\Rightarrow (iv)\) As before, \(\dim \mathrm {Cone}(P)=d\). Hence, \(\dim {{\mathrm{Tan}}}X=\dim {\hat{A}}_1+d=2d\). The implications \((ii)\Rightarrow (iii)\) and \((iv)\Rightarrow (iii)\) are obvious.

\((iii)\Rightarrow (i)\) If P is contained in the hyperplane, then all monomials corresponding to lattice points in P are of the same degree. In particular, \(X_P'\cong {{\mathrm{AffCone}}}(X_P)\).\(\square \)
Example 2.18
1. Consider the Veronese surface \(Y=\mathbb {P}^2\hookrightarrow \mathbb {P}^5\). It is a projective toric variety corresponding to the simplex \(S={{\mathrm{conv}}}(0,2\chi _1,2\chi _2)\subset \langle \chi _1,\chi _2\rangle \cong \mathbb {Z}^2\), i.e. parameterized by characters of a twodimensional torus that correspond to the lattice points of S. By Proposition 2.15, the variety \(X_P\) that defines the secant is parameterized by \(P=S\cap \{\chi _1^*+\chi _2^*\ge 2\}\), i.e., by the lattice points \(2\chi _1,\chi _1+\chi _2,2\chi _2\). Hence, both factors of \({{\mathrm{Sec}}}X\cap {\hat{A}}={\hat{A}}_1\times S_2\) are of dimension two, so \(\dim {{\mathrm{Sec}}}X=4\). Thus, the secant variety is degenerate and defined by \(z(1,2)^2=z(1,1)z(2,2)\).
2. Consider the Segre product \(\mathbb {P}^1\times \mathbb {P}^1\times \mathbb {P}^1\). The representing polytope is a cube \([0,1]^3\). The secant variety is represented by a polytope with vertices (1, 1, 0), (1, 0, 1), (0, 1, 1), (1, 1, 1). We see that the affine cone over it is the whole affine space; indeed, in this case the secant variety is nondegenerate and fills the whole ambient space. The tangential variety is a hypersurface defined by the equation \(z(1,1,0)z(1,0,1)z(0,1,1)=z(1,1,1)^2\).
Theorem 2.19
The tangential and secant varieties of a Segre–Veronese variety are covered by complements of hyperplane sections. Each complement is an affine toric variety without torus factors. In case of the secant variety, these are always normal toric varieties. In case of the tangential variety they are normal if the underlying variety is the Segre product.
Proof
By [39, Theorem 2.2] we know that \({{\mathrm{Sec}}}X\) is normal. By [29, Proposition 8.5] we know that \({{\mathrm{Tan}}}X\) is normal, when X is the Segre product. The open subsets \({{\mathrm{Sec}}}X\cap {\hat{A}}\) and \({{\mathrm{Tan}}}X\cap {\hat{A}}\) are toric varieties by Propositions 2.15 and 2.16. Moreover, they do not contain torus factors, since they are products of an affine space \({\hat{A}}_1\) with either an affine cone over a projective toric variety \(X_P\) or a nondegenerate toric variety \(X_P^\prime \). By taking such subsets for various choices of basis vectors \(e^i_0\), \(i=1,\ldots ,n\), we obtain the statement. \(\square \)
Theorem 2.20
Let \(X=v_{s_1}(\mathbb {P}(V_1))\times \dots \times v_{s_n}(\mathbb {P}(V_n))\) be a Segre–Veronese variety. Then the affine cone over the secant variety of X is flexible. Further, if \(s_1=\dots =s_n=1\), then also the affine cone over the tangential variety of X is flexible.
Proof
By [5, Theorem 0.2] we know that all the affine charts from Theorem 2.19 are flexible. Now, by Theorem 1.4 we obtain flexibility of the affine cone. \(\square \)
Example 2.21
Consider the third Veronese embedding \(X=v_3(\mathbb {P}^1)\). It is represented by characters in the interval \(S={{\mathrm{conv}}}(0,3)\). Then \({{\mathrm{Tan}}}X\cap {\hat{A}}\cong \mathbb {A}^1\times X_P'\), where the monoid of characters associated to the toric variety \(X_P'\) is generated by \(\{2,3\}\). Namely, \(X_P'\) is the curve with a cusp singularity at the origin. Thus, \({{\mathrm{Tan}}}X\) is a surface, whose singular locus is the curve X. This example can be generalized to tangential varieties of other Segre–Veronese varieties provided that at least one of the Veronese factors is of degree at least 3.
3 The combinatorial description of Tvarieties
We consider a normal variety X with an effective action of an algebraic torus \(T \cong (\mathbb {G}_m)^r\). Then X is called a Tvariety of complexity \((\dim X  \dim T)\). Here, the case of a complexityone torus is the most widely studied one, with contributions by many different authors [2, 16, 21, 24, 35, 37]. In the following we restrict ourselves to the case of rational Tvarieties of complexity one. Following [4] and generalising the classification of toric varieties by their fans, we introduce some combinatorial language to classify rational Tvarieties of complexity one.
Let us denote the character lattice of the torus T by M and the dual lattice by N. For the associated \(\mathbb {Q}\)vector spaces we write \(M_\mathbb {Q}\) and \(N_\mathbb {Q}\).
Definition 3.1
Remark 3.2
Affine charts correspond to fdivisors \(\mathcal {S}\) such that \(\mathcal {S}_P\) consists of a single polyhedron (and its faces) and \(\deg \mathcal {S}= \sum _{P\in \mathbb {P}^1} \mathcal {S}_P\). These objects are called pdivisors in [4, 19].
Example 3.3
In Fig. 1 we sketched the nontrivial slices of an fdivisor as well as its degree. It describes the blowup of the quadric threefold in one point, see [33].
Lemma 3.4
[20, Remark 1.8.] An fdivisor describes a subtorus action on a toric variety if and only if \(\mathcal {S}_P\) equals a lattice translate of the tail fan for all but at most two \(P \in \mathbb {P}^1\).
Definition 3.5
 (i)
all \(h_P\) have the same linear part, which will be denoted by \({{\mathrm{lin}}}h\),
 (ii)
only finitely many of them differ from \({{\mathrm{lin}}}h\).
 (i)
if \(h_P\le 0\) for all \(P\in \mathbb {P}^1\), then \(D_h\) is effective.
 (ii)in this case \(X(\mathcal {S}) {\setminus }{{\mathrm{supp}}}D_h\) is given by the fdivisorwhere \([h_P=0]\) denotes the polyhedral subcomplex of \(\mathcal {S}_P\) consisting of those polyhedra on which \(h_P\) vanishes.$$\begin{aligned}{}[h=0]:= \left( \sum _P [h_P=0] \otimes P,\quad \deg \mathcal {S}\cap [{{\mathrm{lin}}}h = 0]\right) , \end{aligned}$$(5)
Theorem 3.6
If \(D_h\) is ample, then \(h_P\) is strongly concave for every \(P \in \mathbb {P}^1\) and \(h_P^*(u) \ge 0\) for every \(u \in \Box _h\).
Proof
By Petersen and Süß [31, Theorem 3.28], \(h_P\) has to be strongly concave and by [19, Prop. 3.1(i)] we get that \(h_P^*(u) \ge 0\). \(\square \)
Remark 3.7
On a Tvariety every divisor is linearly equivalent to some torus invariant divisor. This follows for example from Fulton et al. [15, Theorem 1].
4 Affine cones over projective Tvarieties
From now on we assume that the Tvarieties which we consider are proper over the base, i.e. the corresponding fdivisors \(\mathcal {S}\) satisfy the condition that all its slices \(\mathcal {S}_P\) are subdivisions of \(N_\mathbb {Q}\).
Definition 4.1
(Equivariant covering by toric charts) A Tvariety is called equivariantly covered by toric charts, if there is an open covering by toric varieties \(U_i\) such that the torus T acts as a subtorus of the embedded torus of \(U_i\).
Lemma 4.2
The Tvariety \(X(\mathcal {S})\) is equivariantly covered by toric charts if and only if for every maximal polyhedron \(\Delta \) in \(\mathcal {S}_P\), \(P\in \mathbb {P}^1\), all but at most two slices contain a lattice translate of \({{\mathrm{tail}}}(\Delta )\). In particular, either \(X(\mathcal {S})\) itself is toric or there is at most one \(P \in \mathbb {P}^1\) such that \(\mathcal {S}_P\) does not contain a lattice vertex.
Proof
The first part is a corollary of Lemma 3.4. To prove the last statement, we consider two points P, Q such that \(\mathcal {S}_P,\mathcal {S}_Q\) contain only nonlattice vertices. Now, consider a third point R and a maximal polyhedron \(\Delta \subset \mathcal {S}_R\). Since there is no lattice translate of \({{\mathrm{tail}}}(\Delta )\) in \(\mathcal {S}_P\) and \(\mathcal {S}_Q\), \(\Delta \) itself must be a translated cone. Notice that all maximal cones in \(\mathcal {S}_R\) must be translated by the same lattice point. Indeed, otherwise they would not cover \(N_\mathbb {Q}\) and there would exist a different maximal dimensional polyhedron that could not be a lattice translate of its tail cone. Hence, for any \(R\notin \{P,Q\}\) the slice \(\mathcal {S}_R\) is just a translated tail fan. By Lemma 3.4, \(X(\mathcal {S})\) is a toric variety. \(\square \)
Remark 4.3
By [6, Appendix] this criterion is fulfilled for all smooth complete rational Tvarieties of complexity one. Hence, they are covered by affine charts isomorphic to affine spaces.
To get flexibility for every affine cone we need to strengthen the condition in Lemma 4.2.
Theorem 4.4
Let \(X={\mathfrak {X}}(\mathcal {S})\) be a Tvariety such that for any maximal polyhedron \(\Delta \in \mathcal {S}_y\), \(y\in \mathbb {P}^1\), at most two slices contain a polyhedron with the same tail cone \({{\mathrm{tail}}}(\Delta )\) that is not a lattice translate of \({{\mathrm{tail}}}(\Delta )\). Then for every very ample divisor H the corresponding affine cone is flexible.
Proof
By Theorem 1.4 it is enough to show that there exists a Tinvariant Hpolar covering by toric charts. Let us first rephrase this condition in terms of fdivisors. Remember that being Hpolar means that the complement of every chart is the support of an effective divisor linearly equivalent to H. Having Tinvariant charts means that we have to choose the effective divisors above to be Tinvariant. Therefore, we are looking for a collection of support functions h, which via (4) give rise to divisors \(D_h \sim H\), where \(\sim \) denotes linear equivalence. Moreover, the open subsets \(X {\setminus } {{\mathrm{supp}}}D_h\) have to cover X. By (5) the latter is equivalent to the fact that for every maximal polyhedron \(\Delta _P \in \mathcal {S}_P\) there exists a strictly concave nonpositive support function h on \(\mathcal {S}\) corresponding to an effective divisor \(D_h\sim H\), such that \([h_P=0]=\Delta _P\) (i.e. \(h_P_{\Delta _P}\equiv 0\) and negative elsewhere). For being a toric covering additionally we have to impose that \([h = 0]\) has only two slices that are not lattice translates of the tail fan.
We now construct such a covering for some very ample divisor H. By Remark 3.7 every divisor is linearly equivalent to a torus invariant one. Hence, using [36, Section 4] we can assume that \(H \sim D_h\) for some support function h. Fix a maximal polyhedron \(\Delta \subset \mathcal {S}_Q\). Then \(h_Q_{\Delta }\) is affine linear, i.e. \(h_Q(v)=\langle u, v \rangle + a\). By concavity this implies \(u \in \Box _h\), with \(\Box _h\) defined as in (6). We now consider \(h':=hu\) with \(h_P'(v):=h_P(v)  \langle u, v \rangle \). Now, \(h'_P\) is again strongly concave and achieves its maximum at a polyhedron \(\Delta _P\) with tail cone \({{\mathrm{tail}}}(\Delta _P) = {{\mathrm{tail}}}(\Delta )\). Moreover, by construction we have \(0 \in \Box _{h'}\).
Now, by construction we have \(D_{h^\infty }\sim H\) and \([h^\infty _Q=0]=\Delta \). Moreover, \([h_P^\infty =0]\) is a lattice translate of \({{\mathrm{tail}}}(\Delta )\) for each \(P\notin \{0,\infty \}\). Then it describes a toric chart.
Taking these toric charts for every maximal polyhedron provides us with an Hpolar covering. Now, our result follows by Theorem 1.4. \(\square \)
Theorem 4.5
All the affine cones over the Fano threefolds Q, 2.29, 2.30, 2.31, 2.32, 3.8, 3.18, 3.19, 3.23, 3.24, 4.4, and certain elements of the families 2.24, 3.10 admitting a 2torus action in Mori–Mukai’s classification are flexible.
Proof
For all Fano threefolds from Theorem 4.5 the corresponding fdivisors are listed in Süß [33]. One can easily check that the precondition of Theorem 4.4 is fulfilled in every case. \(\square \)
Example 4.6
Let us illustrate the difference of assumptions in Lemma 4.2 and Theorem 4.4. In the lemma we are allowed to choose the polyhedron with the given tail cone. Hence, if we consider the variety given by the slice \(( \, \infty ,\,1],[ \, 1,1],[1,\infty )\) taken three times, then it does satisfy the assumptions. Indeed, if we take the maximal polytope \([\,1,1]\) in one slice, in other two slices we can take just the vertex \(\{1\}\), which is a lattice shift of the tail cone \(\{0\}\). On the other hand, in the theorem we ask for all polyhedra with the given tail cone. Here, we get three times \([1,1]\) which is not a lattice translate of \(\{0\}\). Such a difference is only possible for cones that are not fulldimensional.
Example 4.7
We are coming back to the blowup of the quadric threefold from Example 3.3. We may check that the corresponding fdivisor in Fig. 1 fulfills the condition of Theorem 4.4. Hence, all the affine cones over the blowup of the quadric threefold are flexible.
Example 4.8
The hypersurface \(\mathbb {V}(x_0y_0^2+x_1y_1^2+x_2y_2^2) \subset \mathbb {P}^2 \times \mathbb {P}^2\) is 2.24 from our list in Theorem 4.5. Hence, every affine cone over this variety is flexible. In particular, this is true for the affine cone over the Segre embedding.
5 Total coordinate spaces
We recall the definition of Cox rings.
Definition 5.1
The relative spectrum \({\hat{X}} = {{\mathrm{Spec}}}_X({{\mathrm{\mathcal {R}}}})\) is called the universal torsor of X. It is an open subset of the absolute spectrum \({\overline{X}} = {{\mathrm{Spec}}}({{\mathrm{\mathcal {R}}}}(X))\), which is called the total coordinate space of X. By construction, the Cox ring is graded by the class group of X inducing an action of the torus \({{\mathrm{Spec}}}k[{{\mathrm{Cl}}}(X)]\) on the total coordinate space.
In the following we are studying flexibility of total coordinate spaces for several classes of varieties.
5.1 Del Pezzo surfaces
Since the smooth del Pezzo surfaces of degrees 6, 7, 8, and 9 are toric, their total coordinate spaces are just affine spaces and hence flexible. The remaining del Pezzo surfaces are blowups \(X_r\) of \(\mathbb {P}^2\) in r points of general position, where \(4 \le r \le 8\). Their Cox rings are described for example in Batyrev and Popov [8], Derenthal [13], and Testa et al. [38].
An exceptional curve on X is a curve of selfintersection \(\,1\) and anticanonical degree 1. On every del Pezzo surface there are only finitely many of them, we denote their number by N(r). Seen as an effective divisor every such curve C corresponds to a section in the degree[C] part of the Cox ring. This section is uniquely determined up to scaling by a nonzero constant.
We will use the following facts from Batyrev and Popov [8].
Theorem 5.2
 (i)
\({{\mathrm{\mathcal {R}}}}(X_r)=\mathbf {k}[e_1,\ldots ,e_{N(r)}]/I\) for \(4\le r\le 7\);
 (ii)
\({{\mathrm{\mathcal {R}}}}(X_8)=\mathbf {k}[e_1,\ldots ,e_{N(r)}]/I\oplus \langle f_1,f_2\rangle _\mathbf {k}\) as a vector space, where \(f_1, f_2 \in H^0(X_8,{\mathcal {O}}(K_{X_8}))\) are elements of degree one with respect to the \(\mathbb {Z}\)grading by the anticanonical degree of a divisor class.
Theorem 5.2 shows that the Cox ring \({{\mathrm{\mathcal {R}}}}(X)\) is generated by elements of degree 1 and \(Y_r:={{\mathrm{Proj}}}({{\mathrm{\mathcal {R}}}}(X_r))\) comes with an embedding into \(\mathbb {P}^{N1}\), where N (resp. \(N2\)) is the number of exceptional curves in the case \(4\le r \le 7\) (resp. \(r=8\)).
In this situation the total coordinate space \({\overline{X}}_r\) is the affine cone over this embedding.
Proposition 5.3
Let e be a section corresponding to an exceptional curve. Then the principal open subset \((Y_r)_e\) is isomorphic to \(X_{r1}\).
Proof
This can be found for example in the proof of Proposition 3.4 in Batyrev and Popov [8]. \(\square \)
Theorem 5.4
The total coordinate spaces of smooth del Pezzo surfaces are flexible.
Proof
As said above, it is enough to check the statement for \(X_r\) with \(4 \le r \le 8\). We will go by induction. The del Pezzo surface \(X_3\) is toric. Therefore, it has a flexible total coordinate space. Now, consider \(X_r\) with \(4 \le r \le 8\). Then we have seen that \({\overline{X}}_r\) is the affine cone over \(Y_r\). Moreover, the principal open subsets corresponding to sections of exceptional curves are isomorphic to \({\overline{X}}_{r1}\) and hence flexible by induction hypothesis. It remains to check that these principal open subsets cover \(Y_r\) to conclude flexibility of \({\overline{X}}_r\) from Theorem 1.4. For \(4 \le r \le 7\) this follows directly from Theorem 5.2(i). For the case \(r=8\) we have to take care for the remaining generators. By Theorem 5.2(ii) their squares are contained in the ideal \((e_1 \ldots , e_N)\) generated by the sections corresponding to exceptional curves, but then the common vanishing of \(e_1 \ldots , e_N\) implies the vanishing of the remaining generators and hence \(Y_r = \bigcup _{i=1}^N (Y_r)_{e_i}\). \(\square \)
5.2 Smooth complexityone Tvarieties
In Hausen and Süß [18] the Cox rings of Tvarieties are studied. For the case of a complexityone action they have a very particular form.
Proposition 5.5
If we impose the additional condition that the Tvariety is equivariantly covered by toric charts (which is fulfilled in the smooth case), then we can conclude the following.
Proposition 5.6
 (i)
\(N, m, n_0,\ldots ,n_m\in \mathbb {Z}_{>0}\);
 (ii)
\(z_2,\ldots , z_m\) are distinct elements of \(\mathbf {k}^*\);
 (iii)
for \(\ell =0,\ldots ,m\), \(A_\ell \) is a monomial in \(\mathbf {k}[T_{\ell ,1},\ldots ,T_{\ell ,n_\ell }]\);
 (iv)
for \(\ell =1,\ldots ,m\) the monomial \(A_\ell \) is linear in at least one variable.
Proof
The first statements follow directly from Proposition 5.5 and Lemma 4.2. For the Fano case note that the Cox ring of a Fano variety is logterminal by Brown [9] and Gongyo et al. [17] and factorial by [7]. Hence, by Remark 6.4 in [25] we obtain the last statement. \(\square \)
Proposition 5.7
Let X be a complexityone Tvariety equivariantly covered by toric charts and \(\overline{X}\) be the total coordinate space of X. Then \(\overline{X}\) is flexible.
Proof
Let \(\mathbf {k}[\overline{X}]\) be as in Proposition 5.6. We assume that \(\overline{X}\) is naturally embedded into an affine space \(\mathbb {A}^N={{\mathrm{Spec}}}\mathbf {k}[S_1,\ldots ,S_N;T_{\ell ,j}\mid 0\le \ell \le m, 1\le j\le n_\ell ].\)
The images of monomials \(A_0,\ldots ,A_m\) in \(\mathbf {k}[\overline{X}]\) span a twodimensional subspace, and no two of them are collinear. Therefore, we may permute \(A_0,\ldots ,A_m\) along with a proper change of their coefficients, indices of variables, and numbers \(z_i\). \(\square \)
Lemma 5.8
The point \(x\in \overline{X}\) is singular if and only if there are at least three monomials \(A_i\) such that all their partial derivatives are vanishing at x.
Proof
Let x be singular and denote \(L_\ell =z_\ell \cdot A_0 + A_1 + A_\ell \) for \(\ell =2,\ldots ,m\). Then there is a nontrivial linear combination \(L\in \langle L_2,\ldots ,L_m\rangle _\mathbf {k}\), whose partial derivatives vanish at x. Since L is a sum of at least three monomials \(A_i\), whose partial derivatives also vanish, the statement follows.
Conversely, given three monomials with partial derivatives vanishing at x, we assume that they are \(A_0,A_1,A_2\). Then we take \(L=L_2\). \(\square \)
So, for a smooth point \(x\in X\), up to permutation of monomials and variables we assume that each monomial \(A_i,\,i=2,\ldots ,m,\) has a nonzero partial derivative, say, \(\frac{\partial A_i}{\partial T_{i, 1}}(x)\ne 0\). Moreover, we may choose \(T_{i,1}\) to be linear in \(A_i\). Indeed, if \(T_{i,1}\) is not linear, then \(A_i(x)\ne 0\), and we take \(T_{i,1}\) to be a linear variable by 5.6 (iv). We denote \(B_i=\frac{\partial A_i}{\partial T_{i,1}}\), which is nonzero at x.
As said before, for a chosen smooth point x we have \(B_i(x)\ne 0\). Let us take another smooth point \(y\in \overline{X}\) with nonzero coordinates and move x to y by \(\mathbb {G}_a\)actions, denoting images of x by same letter. By translations along coordinates \(S_1,\ldots ,S_{n_S}\) we ‘equalize’ them, i.e., obtain \(S_i(x)=S_i(y)\), \(i=1,\ldots ,n_S.\)
Since \(c_{0,1},\ldots ,c_{0,n_0},c_{1,1},\ldots , c_{1,n_1}\in \mathbf {k}\) are arbitrary, with \(\phi \) we also equalize coordinates \(T_{0,j},\,j=1,\ldots ,n_0,\) and \(T_{1,j},\,j=1,\ldots ,n_1,\) at x and y. Now, let \(T_{1,1}\) be a linear variable in \(A_1\), then for each \(\ell =2,\ldots ,m\) we construct a \(\mathbb {G}_a\)action \(\phi _\ell \) by permuting monomials \(A_1\) and \(A_\ell \) and applying the procedure above. Since \(\frac{\partial A_1}{\partial T_{1,1}}(x)=\frac{\partial A_1}{\partial T_{1,1}}(y)\ne 0\), with \(\phi _\ell \) we may equalize coordinates \(T_{\ell ,2},\ldots ,T_{\ell ,m}\), but break the equality of coordinate \(T_{1,1}\), which we restore with \(\phi \).
Proceeding in this way for each \(\ell =2,\ldots ,m\), we equalize all coordinates except \(T_{2,1},\ldots ,T_{m,1}\). But in this case the equation \(z_\ell \cdot A_0+A_1+A_\ell =0\) with condition \(B_\ell (x),B_\ell (y)\ne 0\) implies \(T_{\ell ,1}(x)=T_{\ell ,1}(y)\) for each \(\ell \). So, we may send any smooth point to any point with nonzero coordinates. Hence the action of \({{\mathrm{SAut}}}\overline{X}\) is transitive on the regular locus of \(\overline{X}\). \(\square \)
Theorem 5.9
The total coordinate space of a complete smooth rational T variety of complexity one is flexible.
5.3 Flexibility of total coordinate spaces vs. flexible coverings
In [6] it was proved that for a variety with an open covering by affine spaces one obtains flexibility of the universal torsor. However, it is not clear whether the flexibility property extends to the total coordinate space. This motivates the following even more general question.
Question 5.10
Provided a variety admits an open covering by flexible affine subsets, does this imply flexibility of the total coordinate space?
It is also tempting to try to connect flexibility of the total coordinate space of the Cox ring of X with that of affine cones over X. The following example shows that flexibility of the total coordinate space does not imply flexibility of all the affine cones.
Example 5.11
(Del Pezzo surfaces) We have seen in Sect. 5.1 that all total coordinate spaces of del Pezzo surfaces are flexible. On the other hand, del Pezzo surfaces are covered by affine spaces, which are flexible. Concerning flexibilty of affine cones it was shown in Perepechko [30] and Park and Won [32] that for degree 4 and 5 all the affine cones are flexible, but by Cheltsov et al. [23] and Kishimoto et al. [12] the anticanonical affine cones over del Pezzo surfaces of degree 3, 2, and 1 are not flexible.
One may still ask if flexibility of all the affine cones implies flexibility of the total coordinate space or for a more subtle relation, e.g. involving the grading of the Cox ring.
Question 5.12
Is there a relation between flexibility of the total coordinate space of X and the fact that all affine cones over X are flexible?
Let us give some illustrating examples for these questions.
Example 5.13
(Toric varieties) The Cox ring of a complete toric variety is a polynomial ring. Hence, the total coordinate space is flexible. On the other hand, the torus invariant affine charts and also the affine cones of a toric variety are again toric and hence flexible by Arzhantsev et al. [5, Theorem 0.2].
Example 5.14
(Blowups of a projective space in cubic hypersurfaces inside hyperplanes) The blowup constructions from Example 1.5 give varieties for which all the affine cones are flexible, as we have seen. On the other hand, the total coordinate space is flexible, as we see in the following.
Example 5.15
(Tvarieties of complexity one) The proof of Theorem 5.9 implies that for Tvarieties of complexity one the condition of being covered by toric (and hence flexible) charts is enough to deduce the flexibility of the total coordinate space. On the other hand, to conclude flexibility of all the affine cones we had to impose the stronger (technical) condition of Theorem 4.4.
Footnotes
 1.
That is, \((e_j^i)^{\prime *}(v)=v_j^i\) and \((e_j^i)^{\prime \prime *}(w)=w_j^i\) respectively.
Notes
Acknowledgements
Open access funding provided by Max Planck Society. We would like to thank Mikhail Zaidenberg for motivating questions and inspiring results and Ivan Arzhantsev for many useful remarks and suggestions. The first author started the project under Mobilnosc+ Polish Ministry of Science program, finished under DAAD PRIME program and was supported by the Foundation for Polish Science (FNP). The formulation and proof of Lemma 1.1 (A.Perepechko) were supported by a Grant from the Dynasty Foundation. The research of A. Perepechko, which lead to the results of Sect. 5, was carried out at the IITP RAS at the expense of the Russian Foundation for Sciences (Project no. 145000150).
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