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Flexible affine cones and flexible coverings
Flexible affine cones and flexible coverings
Michałek, Mateusz; Perepechko, Alexander; Süß, Hendrik
2018-06-05 00:00:00
Math. Z. https://doi.org/10.1007/s00209-018-2069-2 Mathematische Zeitschrift Flexible afﬁne cones and ﬂexible coverings 1,2,3 4,5 Mateusz Michałek · Alexander Perepechko · Hendrik Süß Received: 3 December 2016 / Accepted: 19 February 2018 © The Author(s) 2018 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 surface Mathematics Subject Classiﬁcation 14R20 · 14J50 In 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 The research of M. Michalek was supported by IP Grant 0301/IP3/2015/73 of the Polish Ministry of Science. Mateusz Michałek mmichalek@impan.pl Alexander Perepechko perepeal@gmail.com Hendrik Süß hendrik.suess@manchester.ac.uk Mathematical Institut, Polish Academy of Sciences, Sniadeckich 8, 00-956 Warsaw, Poland Freie Universität, Berlin, Germany Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany Kharkevich Institute for Information Transmission Problems, 19 Bolshoy Karetny per., 127994 Moscow, Russia Moscow Institute of Physics and Technology, 9 Institutskiy per., Dolgoprudny, 141701 Moscow Region, Russia School of Mathematics, The University of Manchester, Alan Turing Building, Oxford Road, Manchester M13 9PL, UK 123 M. Michałek et al. 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 k of characteristic zero. We consider actions of the additive group G = (k, +) on X. The subgroup of Aut(X ) generated by all G -actions is called the special automorphism group of X and will be denoted by SAut(X ). We are interested in transitivity of the SAut(X )-actions on the regular locus X . reg Recall that an action of a group G on a set M is m-transitive if for every two m-tuples (x ,..., x ) and (x ,..., x ) of pairwise distinct elements of M there exists g ∈ G such 1 m that g · x = x for i = 1,... m. We have the following result. Theorem 0.1 ([1, Theorem 0.1]) Let X be an irreducible afﬁne variety of dimension ≥ 2. Then the following conditions are equivalent. (i) The group SAut(X ) acts transitively on the regular locus X . reg (ii) The group SAut(X ) acts m-transitively on X for every m > 0. reg (iii) The tangent space of every x ∈ X is spanned by tangent vectors to orbits of G - reg 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 ≥ 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 ≥ 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 , and H be a very ample divisor on Y . If each subset Y \U is the support of an i i effective Q-divisor D linearly equivalent to H, then the afﬁne cone X = AffCone Yis i H ﬂexible. d d 1 n Recall that a Segre–Veronese variety is an embedding of the direct product P ×···× P of projective spaces by the very ample line bundle of the form O(s ) ··· O(s ).Further, 1 n given a projective variety X ⊂ P , 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 (P(V )) × ··· × v (P(V )) be a Segre–Veronese variety. Then s 1 s n 1 n the afﬁne cone over the secant variety of X is ﬂexible. Further, if s = ··· = s = 1,then 1 n 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 ambi- ent 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 T -varieties 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 2-torus action. We use below the list of Fano threefolds in Mori–Mukai’s classiﬁcation [26]. 123 Flexible afﬁne cones and ﬂexible coverings 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 2-torus action in Mori–Mukai’s classiﬁcation are ﬂexible. The main tool to obtain these results is the combinatorial description of T -varieties 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 T -variety of com- plexity one is ﬂexible. 1 Flexibility of afﬁne cones n+1 n Lemma 1.1 Let X ⊂ A be the afﬁne cone over a projective variety Y ⊂ P of dimension ≥ 1. Consider a subgroup G ⊂ Aut X such that • the canonical G -action on X by homotheties sends G-orbits to G-orbits, • all G-orbits are locally closed, and • there is an orbit Gx ⊂ X \{0}, whose image Y under the projection π : X \{0}→ Yis an open subset in Y with complement of codimension ≥ 2. −1 ∗ Then Gx = π (Y ) and is open in X. ∗ −1 ∗ Proof Since the G-action is G -equivariant, X = π (Y ) is a union of G-orbits, whose ∗ ∗ projections coincide with Y . Hence X = λGx,where all G-orbits are closed in λ∈G X . Let us show that Gx = X . Assume the contrary. Then dim Gx = dim Y and the stabilizer S ⊂ G of the orbit Gx is ﬁnite. Indeed, two points v, v ∈ Gx lie in the same G -orbit if m m and only if v = λ · v for some λ ∈ G . The latter is equivalent to λGx = Gx, i.e., λ ∈ S. So, Gv ∩ G v = Sv for any v ∈ X . Denote by X the blow up of X at 0. This is the total space of the line bundle O (−1) over Y . Consider the quotient morphism /S × × μ : X − −→ X = X /S |S| given by t → t on every trivialization chart, where t is the coordinate of the ﬁbers. Then μ(Gx ) intersects each ﬁber at most once, so it is a meromorphic, nonvanishing section of the line bundle X → Y . Indeed, it is a graph of a rational function on trivialization charts of the line bundle. However, the subset D ⊂ Y , where our rational section of the line bundle X → Y is not deﬁned or vanishes, if non-empty, is a Cartier divisor. 123 M. Michałek et al. On the other hand, under our assumptions D ⊆ Y \Y is of codimension at least 2. Therefore, D is empty and μ(Gx ) is a global section of X → Y disjoint with the zero- section. Since Y is of positive dimension X → Y is non-trivial. This gives a contradiction. So, the group G acts on X transitively. Corollary 1.2 Under the setup of Lemma 1.1, if Y is smooth in codimension one and π(Gx ) = Y coincides with the regular locus Y ,then Gx = X \{0}. reg reg Lemma 1.3 Let Y ⊂ P be a linearly nondegenerate projective variety, and let H = Y ∩ {x = 0} be a hyperplane section of Y . Suppose that U = Y \H is endowed with a G - 0 a action φ : G × U → U. Let π : X \{x = 0}→ U be the natural projection, where X = a 0 n+1 AffCone Y ⊂ A is the afﬁne cone over Y . Then X admits a G -action φ : G × X → X a a normalised by the G -action of the cone, such that −1 • φ(G ×{π(x )}) = π(φ(G ×{x })) for any x ∈ π (U ). a a −1 • φ is trivial on X \π (U ). In other words, π : X \{x = 0}→ U provides a correspondence between φ-orbits and φ-orbits. Proof Let Y be deﬁned by a homogeneous ideal I ⊂ k[x ,..., x ] which does not contain 0 n x , k[X]= k[x ,..., x ]/I,and U ={x
= 0}⊂ Y . 0 0 n 0 There exists a natural embedding ρ : U → X, ρ(U ) ={x = 1}⊂ X. On the other hand, π : X \{x = 0}→ U is a trivial G -bundle. Therefore, we may extend the G -action φ on 0 m a ˜ ˜ U to a G -action φ on X \{x = 0} deﬁned by a homogeneous locally nilpotent derivation δ. a 0 ˜ ˜ There exists d ∈ N such that x δ(x ) ∈ k[X ] for i = 1,..., n.Since x ∈ ker δ,a i 0 d+1 ˆ ˜ homogeneous derivation δ = x δ on X is locally nilpotent. The corresponding G -action φ is normalised by the G -action, coincides with φ on the hypeplane section {x = 1} U m 0 of X, and is trivial on {x = 0}⊂ X. Theorem 1.4 Let Y be a normal projective variety covered by ﬂexible afﬁne open subsets U ,i ∈ I , and H be a very ample divisor on Y . If each subset Y \U is the support of an i i effective Q-divisor D linearly equivalent to H, then the afﬁne cone X = AffCone Yis i H ﬂexible. Proof The tangent space at 0 to X must contain all lines through 0 contained in X.As X is n+1 covered by such lines, the tangent space is equal to the linear span of X.If X = A the theorem is trivial. Otherwise, the origin is a singular point of X, which we assume from now on. For each subset U there exists a ﬁnite number of G -actions {φ } such that the orbit of i a ij the group generated by them is the regular locus of U ,see [1, Prop. 1.5]. Let k ∈ N be such that kD is a Z-divisor for any i ∈ I . For each action φ we can consider a lifted action φ on i ij ij AffCone Y as in Lemma 1.3. Since the Veronese map X → AffCone Y is unramiﬁed kH kH outside the vertex, Theorem 1.3 of Masuda and Miyanishi [27] implies the existence of an ˆ ˜ action φ on X,whoseorbits have thesameimage in Y as the orbits of φ . ij ij Let a subgroup G =φ ⊂ SAut X be generated by the G -actions on X which corre- ij a spond to all the open subsets U . Then the image of the orbit Gx of a regular point x ∈ X i reg under the projection X \{0}→ Y is equal to Y . Thus, the statement follows from Corol- reg lary 1.2,asthe variety Y is normal, in particular smooth in codimension one, and the G-orbits are locally closed by Arzhantsev et al. [1, Prop. 1.3]. 123 Flexible afﬁne cones and ﬂexible coverings Example 1.5 Consider P with coordinates x ,..., x and a smooth subvariety Y = 0 n V(x , f ) of codimension 2, where f is an irreducible homogeneous polynomial of degree d. Let q : X → P 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 ≤ n holds. n n −1 n Let U := P \H := P \V(x ). The preimage q (U ) is isomorphic to U = A ,since i i i 0 0 −1 Y does not intersect U .For i
= 0 the preimage q (U ) is given by 0 i f x V · u − · v ⊂ U × P . −1 Hence, q (U ) is coveredbythe afﬁnecharts 0 −1 ∞ −1 U := q (U )\[v = 0] and U := q (U )\[u = 0], (1) i i i i the ﬁrst one being an afﬁne space and the second one being isomorphic to f x v x x v 0 0 n V − · ⊂ Spec k ,..., , . x u x x u i i i n−1 In the notation of Arzhantsev et al. [5] this is a suspension over A and hence ﬂexible by Theorem 0.2 in loc. cit. Given a divisor D ⊂ P ,wedenoteby D its strict transform on the blowup. We also denote by E the exceptional divisor at Y and by H the pullback of a general hyperplane. To see that the given covering is polar ([22, Def. 3.7]) with respect to every ample divisor, note that −1 ∞ 0 X \q (U ) = E ∪ H ; X \U = H ∪ H ; X \U = H ∪ V( f ); for i > 0. (2) 0 0 i 0 i i i Now, X has Picard group Z with generators [H ] and [E ]. The effective cone is generated by [E ] and [H]−[E ] and the nef-cone by [H ] and d[H]−[E ]. Moreover, [V( f )]= d[H]−[E ] and [H ]=[H]−[E ] hold. So, for every afﬁne charts from (1) the boundary components in (2) span a cone in the Neron-Severi space containing the whole nef cone of X. Hence, every ample class can be expressed as a positive linear combination of the complement components. In other words, for every afﬁne charts U from (1) there is an effective divisor D with support X \U , which lies in the chosen ample class. By Theorem 1.4 the ﬂexibility of the corresponding afﬁne cone follows. n n Let further be q : X → P the combined blowup of P in Y as above and additionally in the point y = (1 : 0 : ··· : 0). Similarly, using the same notation and denoting by E the exceptional divisor of the blowup in the point y, we have the following ﬂexible charts on X : i ∞ 0 U := X \(H ∪ E ∪ H ), U := X \(H ∪ H ∪ E ), U := X \(H ∪ V( f ) ∪ E ). 0 i i 0 i 0 i i The ﬁrst two are afﬁne spaces and the last one is a suspension as before. We see that the com- plements of the afﬁne charts always consist of three components with classes corresponding to one of the triples ([H]−[E ], [E ], [H]−[E ]), ([H]−[E ], d[H]−[E ], [E ]), ([H]−[E ], [H]−[E ], [E ]). Further, each triple spans a cone containing the nef cone of X , which is spanned by d[H]−[E ], [H]−[E ],and [H ]. Hence, every ample class can be expressed as a posi- tive linear combination of complement components. As above this implies the ﬂexibility of the corresponding afﬁne cone by Theorem 1.4. 123 M. Michałek et al. 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 Given for each i = 1,..., n a ﬁnite-dimensional vector space V and its symmetric power S (V ), s ∈ Z ,the Segre–Veronese variety i i >0 s s 1 n X = v (P(V )) × ··· × v (P(V )) ⊂ P(S (V ) ⊗ ··· ⊗ S (V )) s 1 s n 1 n is deﬁned as the embedding of the product P(V )×···× P(V ) by the very ample line bundle 1 n O(s ) ··· O(s ). 1 n 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. i i For each V ,1 ≤ i ≤ n, we denote d = dim V − 1and ﬁx abasis e ,..., e of V .We i i i i 0 d ⊗s also denote elements of the basis of V by i i i e = e ⊗ ··· ⊗ e . i ,...,i i i 1 s 1 s Thus, the symmetric power is S (V ) = ⊗s i ∗ i ∗ {v ∈ V | (e ) (v) = (e ) (v) for any permutation σ ∈ S }. i i ,...,i i ,...,i 1 s σ(1) σ(s) This allows us to embed the Veronese cone into the Segre cone and obtain the following diagram: e s 1 s V ⊕ ··· ⊕ V V ⊕· · · ⊕ V 1 n s s ⊗s ⊗s 1 n 1 n S (V ) ⊗ ··· ⊗ S (V ) ⊂ V ⊗· · · ⊗ V , 1 n 1 n s ⊗···⊗s V V 1 n where • e : (v ,...,v ) → (v ,...,v ,v ,...,v ,v ,...,v ) is the diagonal embedding, 1 n 1 1 2 2 3 n s s s s 1 1 1 2 1 n 1 n • ψ : (v ,...,v ,v ,...,v ,v ,...,v ) → v ⊗ ··· ⊗ v is the parameterization of n n 1 1 2 2 3 1 the Segre cone, which is a nonlinear map, • ψ = ψ | ◦ e is the parameterization of the Segre–Veronese cone, and im(e) ⊗s i s • s : V → S (V ) are the natural symmetrizing projections. V i i i Notation 2.2 (i) For a vector space V = V ⊗ ··· ⊗ V we denote i i 1 k i i k ∗ V := {x ∈ V | (e ⊗ ··· ⊗ e ) (x ) = 1} 0 0 i i and regard it as a vector space with basis {e ⊗ ··· ⊗ e | j + ··· + j > 0}. 1 n j j 1 k (ii) We may, and often will, consider V as a complement to a hyperplane section {[x]: 1 k ∗ (e ⊗ ··· ⊗ e ) (x ) = 0} of P(V ). Thus V may be regarded as an afﬁne open subset 0 0 of P(V ). 123 Flexible afﬁne cones and ﬂexible coverings ⊗s ⊗s 1 n (iii) We denote A := V ⊗ ··· ⊗ V . (iv) We also deﬁne ×s 1 s B := V ⊂ V ⊕ ··· ⊕ V , i =1 B := V ⊂ V ⊕ ··· ⊕ V . i 1 n i =1 (v) Finally, we denote π : A\{0}→ P(A) and obtain the following diagram of open subsets: s1 s B ⊂ V ⊕ ··· ⊕ V V ⊕· · · ⊕ V ⊃ B 1 n 1 n ⊗s s1 sn 1 ⊗sn S (V ) ⊗ ··· ⊗ S (V ) ⊂ V ⊗· · · ⊗ V ⊃ A 1 n 1 n P(A) ⊃ A. s s 1 n Remark 2.3 Since P(S (V ) ⊗ ··· ⊗ S (V )) ⊂ P(A), we can study the Segre–Veronese 1 n variety as a subvariety of X = π ◦ ψ(B ) ⊂ P(A). Note that the image of ψ | does not contain the origin. 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. 1 n Notation 2.4 Basiselementsof A are of the form e ⊗···⊗ e and are in natural 1 n c ,...,c c ,...,c 1 sn 1 s 1 1 n n i correspondence with tuples (c ,..., c ,..., c ,..., c ),where 0 ≤ c ≤ d for 1 ≤ i ≤ n s s 1 1 1 n j and 1 ≤ j ≤ s . Let us denote the set of these tuples by C (A) and for each c ∈ C (A) the corresponding basis element by e(c). Finally, denote the dual basis elements by x (c) = e(c) . Similarly, we denote C (A) = C (A)\{(0,..., 0)}. Deﬁnition 2.5 (degree, ordering) Given a tuple c ∈ C (A), the number of its nonzero entries is called the degree of c.Given c , c ∈ C (A),wesay that c ≤ c if c can be obtained 1 2 1 2 1 from c by setting some entries to zero. Thus, we have a natural poset structure on C (A), which induces a poset structure on the basis i i of A. Simply speaking, the ordering on the basis is deﬁned by replacing e by e in the tensor j 0 product elements. 1 1 2 1 1 2 1 1 2 Example 2.6 Consider t := e ⊗ e ⊗ e , t := e ⊗ e ⊗ e and t := e ⊗ e ⊗ e .We 2 1 0 1 0 5 0 0 5 1 3 4 have t < t and t , t are not comparable with t . 1 2 1 2 0 123 M. Michałek et al. Denote the set of indices 1 1 n n Ind(A) = ,..., ,..., ,..., 1 s 1 s 1 n and endow it with a natural lexicographic order, namely, the order of appearance above. ˆ ˆ Given c ∈ C (A) and a subset of indices I ⊂ Ind(A), we introduce the index tuple c , ∈ I, 1 n i j c = (b ,..., b ), where b = 1 s j i 0, ∈ / I. ˆ ˆ Note that {b ∈ C (A) | b ≤ c}={c | I ⊂ Ind(A)}. We will use either one-element subsets I = or subsets of the following form, where i , i ∈ Ind(A): 1 2 I =[i : i ]:={i | i ≤ i < i }⊂ Ind(A). 1 2 1 2 Deﬁnition 2.7 A thick interval partition of a tuple c ∈ C (A) of degree at least two is an 1 n increasing sequence of indices = b < ... < b = such that deg c ≥ 2for 0 k [b :b ] 1 s i i +1 each i. The set of all thick interval partitions of c will be denoted by IP(c).Itisalways 1 n nonempty as it contains , . 1 s Now we can recall the coordinate systems from [29, Sec. 2]. Notation 2.8 For each c ∈ C (A) we denote x (c), deg c = 1, y(c) := deg(c)−deg(b) (−1) x (b) x (c ), deg c > 1. i i i 0 0 0 (b)≤(c) c
=b [ ] j j 0 0 0 ˆ ˆ Then for each c ∈ C (A) we introduce a function in k[A] y(c), deg c = 1, z(c) := (−1) y(c ), deg c > 1. [b :b ] (b ,...,b )∈IP(c) m=1 m−1 m 0 k Lemma 2.9 Each one of the sets {x (c)} , {y(c)} , and {z(c)} is an alge- ˆ ˆ ˆ c∈C (A) c∈C (A) c∈C (A) braically independent system of functions generating O(A).Inother words, {z(c)} is a coordinate system on A as an afﬁne space. Proof Since y(c) is a sum of x (c) and of terms of smaller degree, the endomorphism of k[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. Secant The secant variety Sec X ⊂ P(A) of the Segre–Veronese variety X is parameterized by a map sec : A × B × B → A,(t,v,w) → π(t · ψ(v) + (1 − t ) · ψ(w)). (3) Hereinafter, given a tuple of degree one, we denote the index of its only non-zero entry by . Generalizing [29, Lemma 3.1], we obtain the following result. 123 Flexible afﬁne cones and ﬂexible coverings i ∗ Lemma 2.10 Let {(e ) | 1 ≤ i ≤ n, 0 ≤ j ≤ d } be the set of coordinate functions on the i ∗ 1 ﬁrst copy of B in (3) and {(e ) } the respective set on the second one. Then ∗ ∗ i i i ◦ ◦ ◦ t e + (1 − t ) e , deg c = 1 with c
= 0, i i ◦ ◦ j ⎢ ◦ c c j j ⎢ ◦ ◦ sec : z(c) → ∗ ∗ deg(z(c))−2 i i t (1 − t )(1 − 2t ) i e − e , deg c > 1, i i c
=0 c c j j for each c ∈ C (A). s s 1 n Proof Let Y be the afﬁne cone over the Segre product P(V ) × ··· × P(V ) . Then the 1 n secant of Y is parameterized by ˆ ˜ ˜ sec : A × B × B → A,(t,v,w) → π(tψ(v) + (1 − t )ψ (w)). Thus, sec = sec ◦(id ×e × e). The statement follows after applying [29, Lemma 3.1] to X Y sec . Torus action We can infer the following decomposition of sec . 1 1 Notation 2.11 Let rep : A × B × B → A × B × B , be a reparameterization: t (1 − t ) rep : (t,v,w) → , t v + (1 − t )w, (1 − 2t )(w − v) (1 − 2t ) and let m : A × B × B → A, be a monomial map: i i ◦ ◦ e , deg c = 1 with c
= 0, ◦ j ⎢ ◦ ∗ j ⎢ ◦ m : z(c) → t e , deg c > 1. c
=0 Lemma 2.12 There is a decomposition sec = m ◦ rep . In particular, m is a monomial parameterization of Sec X. Proof Straightforward. This monomial parameterization of Sec X already provides a structure of a toric variety on im(m) = A ∩ Sec X, hence provides us with a toric chart of Sec X. Below we describe in detail its structure. Notation 2.13 Let us introduce the following closed subsets of A: A ={z(c) = 0 | deg c > 1}, A ={z(c) = 0 | deg c = 1}, S = Sec X ∩ A . 2 2 1 i ∗ i i ∗ i That is, (e ) (v) = v and (e ) (w) = w respectively. j j j j 123 M. Michałek et al. Notation 2.14 (Lattice Polytope P) Consider the lattice M = Zχ ,where 1≤i ≤n1≤ j ≤d i j i i ∗ χ = (e ) .Let P ⊂ M ⊗ Q be the lattice polytope deﬁned by inequalities j j χ ≥ 0, 1 ≤ i ≤ n, 1 ≤ j ≤ d , ⎪ j (χ ) ≤ s , 1 ≤ i ≤ n, j =1 ⎩ (χ ) ≥ 2. 1≤i ≤n 1≤ j ≤d Proposition 2.15 In the terminology above, ˆ ˆ (i) Sec X ∩ A = A × S via natural projections along coordinates z(c), 1 2 ˆ ˆ ∼ ˆ (ii) A = X ∩ A A ,where N = dim V , 1 i i =1 (iii) S = AffCone(X ),where X is a projective toric variety with polarization corre- 2 P P sponding to the polytope P, see, e.g., [11, §2.3] for the construction. Proof The morphism m is a direct product of m| and m| 1 , which respec- {0}×B ×{0} A ×{0}×B tively parameterize A and S . This implies (i). The Segre–Veronese variety X is also 1 2 parameterized by m| , thus (ii) holds. {0}×B ×{0} Let us consider the standard torus T = Spec k[M]⊂ B and a projectivization ˆ ˆ π : A\{0}→ P(A) along z(c)-coordinates. Then X = π (S ) is parameterized by a z P z 2 i ∗ T -equivariant map π ◦ m| , which is deﬁned by the set of monomials {(e ) | z {0}×{0}×B deg c > 1}⊂ O(B ) corresponding exactly to lattice points P ∩ M. This implies (iii). Proposition 2.16 Let Tan X ⊂ P(A) be the tangential variety of X. Then ˆ ˆ Tan X ∩ A = A × X , where X ⊂ A is a nondegenerate toric variety parameterized by m| 1 . P {− }×{0}×B Proof We present here a sketch of a proof for k = C,see [29]and [28,Lem.3.3]for a −1 1 −1 complete proof. Consider ( ,v,v + w) ∈ A × B × B .If → 0, then sec ( ,v,v + w) tends to an element of T X → Tan X corresponding to w. On the other hand, by π ◦ψ(v) Lemma 2.10, i i i ◦ ◦ ◦ v − w , deg c = 1 with c
= 0, i i ◦ ◦ j c c j j −1 ◦ ◦ lim z(c)(sec ( ,v,v + w)) = ⎣ 1 i →0 − i 2w , deg c > 1. 4 c
=0 Thus, the decomposition follows. It remains to check that X is nondegenerate. Indeed, O(X )→ O(B ) does not contain invertible elements. 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 The following conditions are equivalent: i ∗ (i) P is not contained in the hyperplane (χ ) = 2, i, j j (ii) dim Sec X = 2dim X + 1, (iii) Sec X
= Tan X, (iv) dim Tan X = dim Sec X − 1. 123 Flexible afﬁne cones and ﬂexible coverings Then Sec X is called non-degenerate. Proof Let d = dim X. As a toric variety, X is represented by a d-dimensional polytope S, which is a product of simplices. Then P is the intersection of S with the halfspace i ∗ (χ ) ≥ 2. i, j (i ) ⇒ (ii ) The assumption implies dim P = d. Hence, dim S = d + 1 and dim Sec X = dim A + dim S = 2d + 1. 1 2 (i ) ⇒ (i v) As before, dim Cone(P) = d. Hence, dim Tan X = dim A + d = 2d.The implications (ii ) ⇒ (iii ) and (i v) ⇒ (iii ) are obvious. (iii ) ⇒ (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 = AffCone(X ). 2 5 Example 2.18 1. Consider the Veronese surface Y = P → P . It is a projective toric variety corresponding to the simplex S = conv(0, 2χ , 2χ ) ⊂χ ,χ = Z , i.e. parameterized by 1 2 1 2 characters of a two-dimensional torus that correspond to the lattice points of S. By Proposi- ∗ ∗ tion 2.15,the variety X that deﬁnes the secant is parameterized by P = S ∩{χ + χ ≥ 2}, 1 2 ˆ ˆ i.e., by the lattice points 2χ ,χ + χ , 2χ . Hence, both factors of Sec X ∩ A = A × S are 1 1 2 2 1 2 of dimension two, so dim Sec X = 4. Thus, the secant variety is degenerate and deﬁned by z(1, 2) = z(1, 1)z(2, 2). 1 1 1 3 2. Consider the Segre product P × P × P . The representing polytope is a cube [0, 1] . 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 non-degenerate 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) . 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 Sec X is normal. By [29, Proposition 8.5] we know that Tan X is normal, when X is the Segre product. The open subsets Sec X ∩ A and Tan X ∩ 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 A with either an afﬁne cone over a projective toric variety X or a non-degenerate toric variety X . By taking such subsets for various choices of basis vectors e , i = 1,..., n, we obtain the statement. Theorem 2.20 Let X = v (P(V )) × ··· × v (P(V )) be a Segre–Veronese variety. Then s 1 s n 1 n the afﬁne cone over the secant variety of X is ﬂexible. Further, if s = ··· = s = 1,then 1 n 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. Example 2.21 Consider the third Veronese embedding X = v (P ). It is represented by characters in the interval S = conv(0, 3).ThenTan X ∩ A = A × X , where the monoid of characters associated to the toric variety X is generated by {2, 3}. Namely, X is the curve P P with a cusp singularity at the origin. Thus, 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. 123 M. Michałek et al. 3 The combinatorial description of T -varieties We consider a normal variety X with an effective action of an algebraic torus T (G ) . Then X is called a T -variety of complexity (dim X − dim T ). Here, the case of a complexity- one 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 T -varieties of complexity one. Following [4] and generalising the classiﬁcation of toric varieties by their fans, we introduce some combinatorial language to classify rational T -varieties of complexity one. Let us denote the character lattice of the torus T by M and the dual lattice by N.For the associated Q-vector spaces we write M and N . Q Q For a polyhedron ⊂ N we deﬁne its tail cone as follows tail( ) := {v ∈ N | + Q · v = }. Q ≥0 Now, we consider polyhedral complexes
in N . Here, by polyhedral complex we mean a set of convex polyhedra, which is closed under the face relation and every pair of polyhedra intersect in a common face. Moreover, we assume that the set of tail cones has the structure of a polyhedral complex itself, which is called the tail fan of
and will be denoted by tail(
). Consider a pair S = ( S ⊗ P, deg S) where S are polyhedral complexes in N P P Q P∈P with some common tail fan and deg S ⊂||. Here, S ⊗ P is just a formal sum. The complexes S are called slices of S. We assume that there are only ﬁnitely many slices that differ from the tail fan tail(S) := . The set of the points P ∈ P such that S
= is called the support of S and will be denoted by supp S. Note that for every full-dimensional σ σ σ ∈ there is a unique polyhedron in S with tail( ) = σ . P P Deﬁnition 3.1 (f-divisor)Apair S as above is called an f-divisor if for any full-dimensional σ ∈ tail(S) we have either deg S ∩ σ =∅ or = deg S ∩ σ σ. An f-divisor as above corresponds to a rational T -variety of complexity one, see [19, Section 1]. Moreover, this correspondence is even functorial. In particular, invariant open subvarieties correspond to f-divisors S , such that S ⊂ S as sets of polyhedra and deg S = | tail(S )|∩ deg S. For simplicity we write S ⊂ S in this situation. As a consequence of Proposition 1.6 in [19], f-divisors S ,..., S ⊂ S give rise to an open covering if and only if their slices cover the slices of S, i.e. |S|= |S |. Remark 3.2 Afﬁne charts correspond to f-divisors S such that S consists of a single poly- hedron (and its faces) and deg S = 1 S . These objects are called p-divisors in [4,19]. P∈P Example 3.3 In Fig. 1 we sketched the non-trivial slices of an f-divisor 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 f-divisor describes a subtorus action on a toric variety if and only if S equals a lattice translate of the tail fan for all but at most two P ∈ P . 123 Flexible afﬁne cones and ﬂexible coverings Fig. 1 An f-divisor In the language of f-divisors we also may describe torus invariant Cartier divisors by support functions. A support function h on a polyhedral subdivision
is a continuous function that is afﬁne linear on every polyhedra in
. We denote by lin h the linear part of h.Thisisa piecewise linear function on the tail fan deﬁned as follows: (lin h)(v) := h(w + v) − h(w) for some w ∈ ∈
with v ∈ tail( ). Deﬁnition 3.5 (Support function on S)A support function h on an f-divisor S is a collection {h } of support functions on S such that P P P∈P (i) all h have the same linear part, which will be denoted by lin h, (ii) only ﬁnitely many of them differ from lin h. We have two kinds of torus invariant prime divisors on X (S). Horizontal prime divisors (1) correspond to rays ρ ∈ tail(S) that do not intersect deg S and are denoted by D . Vertical prime divisors correspond to vertices v in the subdivisions S and are denoted by D .Now, P P,v the divisor corresponding to the support function h is given by D =− (lin h)(ρ) · D − μ(v) · h (v)D , (4) h ρ P P,v P,v where we identify the ray with the ray generator and μ(v) denotes the minimal positive integer such that μ(v) · v is a lattice element. In particular, (i) if h ≤ 0for all P ∈ P ,then D is effective. P h (ii) in this case X (S)\ supp D is given by the f-divisor [h = 0]:= [h = 0]⊗ P, deg S ∩[lin h = 0] , (5) where [h = 0] denotes the polyhedral subcomplex of S consisting of those polyhedra P P on which h vanishes. By [36, Section 4] (or [31]), every invariant Cartier divisor arises in this way. We have D ∼ D if and only if h − h is afﬁne linear for every P, i.e. h − h =u, · + a , h h P P P P P and a = 0. Moreover, we have a criterion for ampleness expressed in the following notation. We denote by the polytope given by ={u ∈ M |u, · ≥ (lin h)} (6) h Q and consider concave piecewise afﬁne function h on as “dual” of h : h P h (u) := inf(u,v− h (v)). The deﬁnition implies that h (u) is ﬁnite for u ∈ . 123 M. Michałek et al. 1 ∗ Theorem 3.6 If D is ample, then h is strongly concave for every P ∈ P and h (u) ≥ 0 h P for every u ∈ . Proof By Petersen and Süß [31, Theorem 3.28], h has to be strongly concave and by [19, Prop. 3.1(i)] we get that h (u) ≥ 0. Remark 3.7 On a T -variety 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 T -varieties From now on we assume that the T -varieties which we consider are proper over the base, i.e. the corresponding f-divisors S satisfy the condition that all its slices S are subdivisions of N . Deﬁnition 4.1 (Equivariant covering by toric charts)A T -variety is called equivariantly covered by toric charts, if there is an open covering by toric varieties U such that the torus T acts as a subtorus of the embedded torus of U . Lemma 4.2 The T -variety X (S) is equivariantly covered by toric charts if and only if for every maximal polyhedron in S ,P ∈ P , all but at most two slices contain a lattice translate of tail( ). In particular, either X (S) itself is toric or there is at most one P ∈ P such that S 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 S , S contain only non-lattice vertices. Now, consider a third point P Q R and a maximal polyhedron ⊂ S . Since there is no lattice translate of tail( ) in S and R P S , itself must be a translated cone. Notice that all maximal cones in S must be translated Q R by the same lattice point. Indeed, otherwise they would not cover N and there would exist a different maximal dimensional polyhedron that could not be a lattice translate of its tail cone. Hence, for any R ∈{ / P, Q} the slice S is just a translated tail fan. By Lemma 3.4, X (S) is a toric variety. Remark 4.3 By [6, Appendix] this criterion is fulﬁlled for all smooth complete rational T - varieties 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 = X(S) be a T -variety such that for any maximal polyhedron ∈ S , y ∈ P , at most two slices contain a polyhedron with the same tail cone tail( ) that is not a lattice translate of tail( ). 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 T -invariant H-polar covering by toric charts. Let us ﬁrst rephrase this condition in terms of f-divisors. Remember that being H-polar means that the complement of every chart is the support of an effective divisor linearly equivalent to H.Having T -invariant charts means that we have to choose the effective divisors above to be T -invariant. Therefore, we are looking for a collection of support functions h, which via (4)giverisetodivisors D ∼ H,where ∼ denotes linear 123 Flexible afﬁne cones and ﬂexible coverings equivalence. Moreover, the open subsets X \ supp D have to cover X.By(5) the latter is equivalent to the fact that for every maximal polyhedron ∈ S there exists a strictly P P concave non-positive support function h on S corresponding to an effective divisor D ∼ H, such that [h = 0]= (i.e. h | ≡ 0 and negative elsewhere). For being a toric covering P P P 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 ∼ D for some support function h. Fix a maximal polyhedron ⊂ S . h Q Then h | is afﬁne linear, i.e. h (v) =u,v+ a. By concavity this implies u ∈ , with Q Q h deﬁned as in (6). We now consider h := h − u with h (v) := h (v) −u,v.Now, h P h is again strongly concave and achieves its maximum at a polyhedron with tail cone tail( ) = tail( ). Moreover, by construction we have 0 ∈ . P h By our precondition, we may assume without loss of generality that for every point R ∈ P\{0, ∞} the polyhedron is a lattice translate of tail( ). Assume further Q
=∞ and introduce h by h (v) := h (v) − max Im h for P
=∞, P P P h (v) := h (v) + max Im h . ∞ ∞ P
=∞ It remains to check that h (v) + max Im h ≤ 0 to see that D is indeed effective. ∞ P
=∞ Recall that we have 0 ∈ . The claim follows from the ampleness of D and Theorem 3.6. h h ∞ ∞ Now, by construction we have D ∼ H and [h = 0]= . Moreover, [h = 0] is a Q P lattice translate of tail( ) for each P ∈{ / 0, ∞}. Then it describes a toric chart. Taking these toric charts for every maximal polyhedron provides us with an H-polar covering. Now, our result follows by Theorem 1.4. 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 2-torus action in Mori–Mukai’s classiﬁcation are ﬂexible. Proof For all Fano threefolds from Theorem 4.5 the corresponding f-divisors are listed in Süß [33]. One can easily check that the precondition of Theorem 4.4 is fulﬁlled in every case. 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 (−∞, − 1], [− 1, 1], [1, ∞) 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 full-dimensional. Example 4.7 We are coming back to the blowup of the quadric threefold from Example 3.3. We may check that the corresponding f-divisor in Fig. 1 fulﬁlls the condition of Theorem 4.4. Hence, all the afﬁne cones over the blowup of the quadric threefold are ﬂexible. 2 2 2 2 2 Example 4.8 The hypersurface V(x y + x y + x y ) ⊂ P × P is 2.24 from our list in 0 1 2 0 1 2 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. 123 M. Michałek et al. 5 Total coordinate spaces We recall the deﬁnition of Cox rings. Deﬁnition 5.1 (Cox sheaf, Cox ring, universal torsor, total coordinate space)Let X be a complete normal variety, whose class group is a free abelian group. Assume that the classes of divisors D ,..., D form a basis of this class group. The Cox sheaf of X is deﬁned by 1 r R = O a D . i i i =1 a∈Z ≥0 It becomes a sheaf of O -algebras via the usual multiplication of sections. The algebra R(X ) of global sections of R is called the Cox ring of X. The relative spectrum X = Spec (R) is called the universal torsor of X. It is an open subset of the absolute spectrum X = Spec(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 Spec k[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 of P in r points of general position, where 4 ≤ r ≤ 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 self-intersection − 1 and anti-canonical 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 non-zero constant. We will use the following facts from Batyrev and Popov [8]. Theorem 5.2 [8, Thm 3.2 and Prop. 3.4] Let N (r ) be the number of exceptional curves on a del Pezzo surface X .Denotebye ..., e the sections corresponding to the exceptional r 1 N (r ) curves and by I the ideal of their relations. Then (i) R(X ) = k[e ,..., e ]/Ifor 4 ≤ r ≤ 7; r 1 N (r ) (ii) R(X ) = k[e ,..., e ]/I ⊕ f , f as a vector space, where f , f ∈ 8 1 N (r ) 1 2 k 1 2 H (X , O(−K )) are elements of degree one with respect to the Z-grading by the 8 X anti-canonical degree of a divisor class. Theorem 5.2 shows that the Cox ring R(X ) is generated by elements of degree 1 and N −1 Y := Proj(R(X )) comes with an embedding into P ,where N (resp. N − 2) is the r r number of exceptional curves in the case 4 ≤ r ≤ 7 (resp. r = 8). In this situation the total coordinate space X 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 ) is isomorphic to X . r e r −1 Proof This can be found for example in the proof of Proposition 3.4 in Batyrev and Popov [8]. 123 Flexible afﬁne cones and ﬂexible coverings 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 with 4 ≤ r ≤ 8. We will go by induction. The del Pezzo surface X is toric. Therefore, it has a ﬂexible total coordinate space. Now, consider X with 4 ≤ r ≤ 8. Then we have seen that X is the afﬁne cone over r r Y . Moreover, the principal open subsets corresponding to sections of exceptional curves areisomorphicto X and hence ﬂexible by induction hypothesis. It remains to check that r −1 these principal open subsets cover Y to conclude ﬂexibility of X from Theorem 1.4.For r r 4 ≤ r ≤ 7 this follows directly from Theorem 5.2(i). For the case r = 8wehavetotake care for the remaining generators. By Theorem 5.2(ii) their squares are contained in the ideal (e ..., e ) generated by the sections corresponding to exceptional curves, but then 1 N the common vanishing of e ..., e implies the vanishing of the remaining generators and 1 N hence Y = (Y ) . r r e i =1 5.2 Smooth complexity-one T -varieties In Hausen and Süß [18] the Cox rings of T -varieties are studied. For the case of a complexity- one action they have a very particular form. Proposition 5.5 [18, Corollary 4.9] Let S be an f-divisor and let us denote by S the subset (1) of rays in tail(S) that do not intersect deg S. Then the Cox ring of X (S) is given by (0) k[S , T | ρ ∈ S , P ∈ supp S,v ∈ S ] ρ P,v μ(0) μ(∞) μ(z) ∗ z · T + T + T | z ∈ supp S ∩ k μ(v) μ(P) where T := (0) T and μ(v) denotes the minimal positive integer such that P,v v∈S μ(v) · v is a lattice element. If we impose the additional condition that the T -variety is equivariantly covered by toric charts (which is fulﬁlled in the smooth case), then we can conclude the following. Proposition 5.6 The Cox ring of a complexity-one T -variety equivariantly covered by toric charts is isomorphic to k[S ,..., S ; T | 0 ≤ ≤ m, 1 ≤ j ≤ n ]/z · A + A + A | 2 ≤ ≤ m, 1 n , j 0 1 where (i) N , m, n ,..., n ∈ Z ; 0 m >0 (ii) z ,..., z are distinct elements of k ; 2 m (iii) for = 0,..., m, A is a monomial in k[T ,..., T ]; ,1 ,n (iv) for = 1,..., m the monomial A is linear in at least one variable. Moreover, if X is Fano, then we may assume that A for ≥ 3 is linear in each variable. Proof The ﬁrst statements follow directly from Proposition 5.5 and Lemma 4.2.For theFano case note that the Cox ring of a Fano variety is log-terminal by Brown [9] and Gongyo et al. [17] and factorial by [7]. Hence, by Remark 6.4 in [25] we obtain the last statement. Proposition 5.7 Let X be a complexity-one T -variety equivariantly covered by toric charts and X be the total coordinate space of X. Then X is ﬂexible. 123 M. Michałek et al. Proof Let k[X ] be as in Proposition 5.6.Weassume that X is naturally embedded into an afﬁne space A = Spec k[S ,..., S ; T | 0 ≤ ≤ m, 1 ≤ j ≤ n ]. 1 N , j The images of monomials A ,..., A in k[X ] span a two-dimensional subspace, and 0 m no two of them are collinear. Therefore, we may permute A ,..., A along with a proper 0 m change of their coefﬁcients, indices of variables, and numbers z . Lemma 5.8 The point x ∈ X is singular if and only if there are at least three monomials A such that all their partial derivatives are vanishing at x. Proof Let x be singular and denote L = z · A + A + A for = 2,..., m. Then there is 0 1 a non-trivial linear combination L ∈L ,..., L , whose partial derivatives vanish at x. 2 m k Since L is a sum of at least three monomials A , whose partial derivatives also vanish, the statement follows. Conversely, given three monomials with partial derivatives vanishing at x, we assume that they are A , A , A .Thenwetake L = L . 0 1 2 2 So, for a smooth point x ∈ X, up to permutation of monomials and variables we assume ∂ A that each monomial A , i = 2,..., m, has a non-zero partial derivative, say, (x )
= 0. ∂T i,1 Moreover, we may choose T to be linear in A .Indeed,if T is not linear, then A (x )
= 0, i,1 i i,1 i ∂ A and we take T to be a linear variable by 5.6 (iv). We denote B = , which is non-zero i,1 i ∂T i,1 at x. Given a set of arbitrary numbers c ,..., c , c ,..., c ∈ k, we construct a G - 0,1 0,n 1,1 1,n a 0 1 action φ on A in two steps. First, denoting a parameter of G by t,welet ∗ N N φ : k[A ]→ k[A ]⊗ k[t ], S → S , for i = 1,..., n , i i S T → T + tc B , for j = 1,..., n , 0, j 0, j 0, j k 0 k=2 T → T + tc B , for j = 1,..., n . 1, j 1, j 1, j k 1 k=2 Then, for some H , H ∈ k[A ]⊗ k[G ], 0 1 a φ (A ) =A + H B , 0 0 0 k k=2 φ (A ) =A + H B . 1 1 1 k k=2 Now, for each = 2,..., m we let T → T − (z · H + H ) B , ,1 ,1 0 1 k 2≤k≤m k
= T → T , for j = 2,..., n . , j , j ∗ N Then the trinomial z · A + A + A is ﬁxed by φ ,so φ preserves X ⊂ A . Thus, we have 0 1 constructed a G -action on the total coordinate space X, which we also denote by φ. 123 Flexible afﬁne cones and ﬂexible coverings As said before, for a chosen smooth point x we have B (x )
= 0. Let us take another smooth point y ∈ X with non-zero coordinates and move x to y by G -actions, denoting images of x by same letter. By translations along coordinates S ,..., S we ‘equalize’ them, 1 n i.e., obtain S (x ) = S (y), i = 1,..., n . i i S Since c ,..., c , c ,..., c ∈ k are arbitrary, with φ we also equalize coordinates 0,1 0,n 1,1 1,n 0 1 T , j = 1,..., n , and T , j = 1,..., n , at x and y.Now,let T be a linear variable 0, j 0 1, j 1 1,1 in A , then for each = 2,..., m we construct a G -action φ by permuting monomials 1 a ∂ A ∂ A 1 1 A and A and applying the procedure above. Since (x ) = (y)
= 0, with φ we ∂T ∂T 1,1 1,1 may equalize coordinates T ,..., T , but break the equality of coordinate T ,which we ,2 ,m 1,1 restore with φ. Proceeding in this way for each = 2,..., m, we equalize all coordinates except T ,..., T . But in this case the equation z · A + A + A = 0 with condition 2,1 m,1 0 1 B (x ), B (y)
= 0 implies T (x ) = T (y) for each . So, we may send any smooth ,1 ,1 point to any point with non-zero coordinates. Hence the action of SAut X is transitive on the regular locus of X. Theorem 5.9 The total coordinate space of a complete smooth rational T -variety of com- plexity one is ﬂexible. Proof By [6, Thm A.1], every rational smooth complete rational T -variety of complexity one is covered by afﬁne spaces, so the statement follows from Proposition 5.7. 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 anti-canonical 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 coor- dinate 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]. 123 M. Michałek et al. 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. ∗ n We can consider the k -action on P given by multiplication with the coordinate x .It n−1 comes with a natural quotient map to P being deﬁned outside the isolated ﬁxed point (1 : 0 : ... : 0). Then the centers of our blowups are ﬁxed points of the action and we obtain induced actions on X and X with natural quotient maps given by composition of the original quotient map with the blowup. Now, we may use Theorem 1.2 in Hausen and Süß [18]to calculate the Cox rings R(X ) = k[T ,..., T , T , S ]/(T T − f (T ,..., T , 0)) 0 n 1 n 0 n−1 n n and R(X ) k[T ,..., T , T , S , S ]/(T T − f (T ,..., T , 0))). 0 n 1 2 n 0 n−1 n n We see that they are suspensions over an afﬁne space and hence ﬂexible by Theorem 0.2 in Arzhantsev et al. [5]. Example 5.15 (T -varieties of complexity one) The proof of Theorem 5.9 implies that for T -varieties 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. 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. 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