arXiv subjects

Victor Batyrev

Publications and source records attributed to Victor Batyrev.

At least 19 recordsLinked to original sources

arXiv2024

On the classification of smooth toric surfaces with exactly one exceptional curve

We classify all smooth projective toric surfaces $S$ containing exactly one exceptional curve. We show that every such surface $S$ is isomorphic to either $\mathbb{F}_1$ or a surface $S_r$ defined by a rational number $r \in \mathbb{Q} \setminus \mathbb{Z}$ $(r >1)$. If $a:= [ r]$ then $S_r$ is obtained from the minimal desingularization of the weighted projective plane $\mathbb{P}(1, 2, 2a+1)$ by toric blow-ups whose quantity equals the level of the rational number $\{ r \} \in (0,1)$ in the classical Farey tree. Moreover, we show that if $r = b/c$ with coprime $b$ and $c$, then $S_r$ is the minimal desingularization of the weighted projective plane $\mathbb{P}(1, c, b)$. We apply $2$-dimensional regular fans $\Sigma_r$ of toric surfaces $S_r$ for constructing $2$-dimensional colored fans $\Sigma^c$ of minimal horospherical $3$-folds having a regular $SL(2) \times \mathbb{G}_m$-action. The latter are minimal toric $3$-folds $V_r$ classified by Z. Guan. We establish a direct combinatorial connection between the $3$-dimensional fans $\widetilde{\Sigma}^c_r$ of $3$-folds $V_r$ and the $2$-dimensional fans $\Sigma_r$ of surfaces $S_r$.

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arXiv2024

Spherical amoebae and a spherical logarithm map

Let $G$ be a connected reductive algebraic group over $\mathbb{C}$ with a maximal compact subgroup $K$. Let $G/H$ be a (quasi-affine) spherical homogeneous space. In the first part of the paper, following Akhiezer's definition of spherical functions, we introduce a $K$-invariant map $sLog_{\Gamma, t}: G/H \to \mathbb{R}^s$ which depends on a choice of a finite set $\Gamma$ of dominant weights and $s = |\Gamma|$. We call $sLog_{\Gamma, t}$ a spherical logarithm map. We show that when $\Gamma$ generates the highest weight monoid of $G/H$, the image of the spherical logarithm map parametrizes $K$-orbits in $G/H$. This idea of using the spherical functions to understand the geometry of the space $K \backslash G/H$ of $K$-orbits in $G/H$ can be viewed as a generalization of the classical Cartan decomposition. In the second part of the paper, we define the spherical amoeba (depending on $\Gamma$ and $t$) of a subvariety $Y$ of $G/H$ as $sLog_{\Gamma, t}(Y)$, and we ask for conditions under which the image of a subvariety $Y \subset G/H$ under $sLog_{\Gamma, t}$ converges, as $t \to 0$, in the sense of Kuratowski to its spherical tropicalization as defined by Tevelev and Vogiannou. We prove a partial result toward answering this question, which shows in particular that the valuation cone is always contained in the Kuratowski limit of the spherical amoebae of $G/H$. We also show that the limit of the spherical amoebae of $G/H$ is equal to its valuation cone in a number of interesting examples, including when $G/H$ is horospherical, and in the case when $G/H$ is the space of hyperbolic triangles.

math.AG↗

arXiv2020

Mirror symmetry for quasi-smooth Calabi-Yau hypersurfaces in weighted projective spaces

We consider a $d$-dimensional well-formed weighted projective space $\mathbb{P}(\overline{w})$ as a toric variety associated with a fan $Σ(\overline{w})$ in $N_{\overline{w}} \otimes \mathbb{N}$ whose $1$-dimensional cones are spanned by primitive vectors $v_0, v_1, \ldots, v_d \in N_{\overline{w}}$ generating a lattice $N_{\overline{w}}$ and satisfying the linear relation $\sum_i w_i v_i =0$. For any fixed dimension $d$, there exist only finitely many weight vectors $\overline{w} = (w_0, \ldots, w_d)$ such that $\mathbb{P}(\overline{w})$ contains a quasi-smooth Calabi-Yau hypersurface $X_w$ defined by a transverse weighted homogeneous polynomial $W$ of degree $w = \sum_{i=0}^d w_i$. Using a formula of Vafa for the orbifold Euler number $χ_{\rm orb}(X_w)$, we show that for any quasi-smooth Calabi-Yau hypersurface $X_w$ the number $(-1)^{d-1}χ_{\rm orb}(X_w)$ equals the stringy Euler number $χ_{\rm str}(X_{\overline{w}}^*)$ of Calabi-Yau compactifications $X_{\overline{w}}^*$ of affine toric hypersurfaces $Z_{\overline{w}}$ defined by non-degenerate Laurent polynomials $f_{\overline{w}} \in \mathbb{C}[N_{\overline{w}}]$ with Newton polytope $\text{conv}(\{v_0, \ldots, v_d\})$. In the moduli space of Laurent polynomials $f_{\overline{w}}$ there always exists a special point $f_{\overline{w}}^0$ defining a mirror $X_{\overline{w}}^*$ with a $\mathbb{Z}/w\mathbb{Z}$-symmetry group such that $X_{\overline{w}}^*$ is birational to a quotient of a Fermat hypersurface via a Shioda map.

math.AG↗

arXiv2019

On the Fine Interior of Three-dimensional Canonical Fano Polytopes

The Fine interior $\Delta^{\text{FI}}$ of a $d$-dimensional lattice polytope $\Delta$ is a rational subpolytope of $\Delta$ which is important for constructing minimal birational models of non-degenerate hypersurfaces defined by Laurent polynomials with Newton polytope $\Delta$. This paper presents some computational results on the Fine interior of all $674,\!688$ three-dimensional canonical Fano polytopes.

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arXiv2018

Satellites of spherical subgroups

Let $G$ be a complex connected reductive algebraic group. Given a spherical subgroup $H \subset G$ and a subset $I$ of the set of spherical roots of $G/H$, we define, up to conjugation, a spherical subgroup $H_I \subset G$ of the same dimension of $H$, called a satellite. We investigate various interpretations of the satellites. We also show a close relation between the Poincaré polynomials of the two spherical homogeneous spaces $G/H$ and $G/H_I$.

math.AG↗

arXiv2018

Stringy $E$-functions of canonical toric Fano threefolds and their applications

Let $Δ$ be a $3$-dimensional lattice polytope containing exactly one interior lattice point. We give a simple combinatorial formula for computing the stringy $E$-function of the $3$-dimensional canonical toric Fano variety $X_Δ$ associated with the polytope $Δ$. Using the stringy Libgober-Wood identity and our formula, we generalize the well-known combinatorial identity $\sum_{θ\preceq Δ\atop \dim (θ) =1} v(θ) \cdot v(θ^*) = 24$ holding in the case of $3$-dimensional reflexive polytopes $Δ$.

math.AG↗

arXiv2017

The stringy Euler number of Calabi-Yau hypersurfaces in toric varieties and the Mavlyutov duality

We show that minimal models of nondegenerated hypersufaces defined by Laurent polynomials with a $d$-dimensional Newton polytope $Δ$ are Calabi-Yau varieties $X$ if and only if the Fine interior of $Δ$ consists of a single lattice point. We give a combinatorial formula for computing the stringy Euler number of $X$. This formula allows to test mirror symmetry in cases when $Δ$ is not a reflexive polytope. In particular we apply this formula to pairs of lattice polytopes $(Δ, Δ^{\vee})$ that appear in the Mavlyutov's generalization of the polar duality for reflexive polytopes. Some examples of Mavlyutov's dual pairs $(Δ, Δ^{\vee})$ show that the stringy Euler numbers of the corresponding Calabi-Yau varieties $X$ and $X^{\vee}$ may not satisfy the expected topological mirror symmetry test: $e_{\rm st}(X) = (-1)^{d-1} e_{\rm st}(X^{\vee})$. This shows the necessity of an additional condition on Mavlyutov's pairs $(Δ, Δ^\vee)$.

math.AG↗

arXiv2017

On the algebraic stringy Euler number

We are interested in stringy invariants of singular projective algebraic varieties satisfying a strict monotonicity with respect to elementary birational modifications in the Mori program. We conjecture that the algebraic stringy Euler number is one of such invariants. In the present paper, we prove this conjecture for varieties having an action of a connected algebraic group G and admitting equivariant desingularizations with only finitely many G-orbits. In particular, we prove our conjecture for arbitrary projective spherical varieties.

math.AG↗

arXiv2016

Stringy Chern classes of singular toric varieties and their applications

Let X be a normal projective Q-Gorenstein variety with at worst log-terminal singularities. We prove a formula expressing the total stringy Chern class of a generic complete intersection in X via the total stringy Chern class of X. This formula is motivated by its applications to mirror symmetry for Calabi-Yau complete intersections in toric varieties. We compute stringy Chern classes and give a combinatorial interpretation of the stringy Libgober-Wood identity for arbitrary projective Q-Gorenstein toric varieties. As an application we derive a new combinatorial identity relating d-dimensional reflexive polytopes to the number 12 in dimension d>3.

math.AG↗

arXiv2013

Lattice polytopes, finite abelian subgroups in $\SL(n,\C)$ and coding theory

We consider $d$-dimensional lattice polytopes $Δ$ with $h^*$-polynomial $h^*_Δ=1+h_k^*t^k$ for $1 2$, the main technical tool in the classification of these linear codes is the non-vanishing theorem for generalized Bernoulli numbers $B_{1,χ}^{(r)}$ associated with odd characters $χ:\F_q^*\to\C^*$ where $q=p^r$. Our result implies a complete classification of all lattice polytopes whose $h^*$-polynomial is a binomial.

math.CO↗

arXiv2012

The arc space of horospherical varieties and motivic integration

For arbitrary connected reductive group G we consider the motivic integral over the arc space of an arbitrary Q-Gorenstein horospherical G-variety associated with a colored fan and prove a formula for the stringy E-function of a horospherical variety X which generalizes the one for toric varieties. We remark that in contrast to toric varieties the stringy E-function of a Gorenstein horospherical variety X may be not a polynomial if some cones in the fan of X have nonempty sets of colors. Using the stringy E-function, we can formulate and prove a new smoothness criterion for locally factorial horospherical varieties. We expect that this smoothness criterion holds for arbitrary spherical varieties.

math.AG↗

arXiv2012

Conifold degenerations of Fano 3-folds as hypersurfaces in toric varieties

There exist exactly 166 4-dimensional reflexive polytopes such that the corresponding 4-dimensional Gorenstein toric Fano varieties have at worst terminal singularities in codimension 3 and their anticanonical divisor is divisible by 2. For every such a polytope, one naturally obtains a family of Fano hypersurfaces X with at worst conifold singularities. A generic 3-dimensional Fano hypersurface X can be interpreted as a flat conifold degeneration of some smooth Fano 3-folds Y whose classification up to deformation was obtained by Iskovskikh, Mori and Mukai. In this case, both Fano varieties X and Y have the same Picard number r. Using toric mirror symmetry, we define a r-dimensional generalized hypergeometric power series associated to the dual reflexive polytope. We show that if r =1 then this series is a normalized regular solution of a modular D3-equation that appears in the Golyshev correspondence. We expect that the multidimensional power series can be used to compute the small quantum cohomology ring of all Fano 3-folds Y with the Picard number r >1 if Y admit a conifold degeneration X.

math.AG↗

arXiv2011

The functor of toric varieties associated with Weyl chambers and Losev-Manin moduli spaces

A root system $R$ of rank $n$ defines an $n$-dimensional smooth projective toric variety $X(R)$ associated with its fan of Weyl chambers. We give a simple description of the functor of $X(R)$ in terms of the root system $R$ and apply this result in the case of root systems of type $A$ to give a new proof of the fact that the toric variety $X(A_n)$ is the fine moduli space $\bar{L}_{n+1}$ of stable $(n+1)$-pointed chains of projective lines investigated by Losev and Manin.

math.AG↗

arXiv2011

On generalisations of Losev-Manin moduli spaces for classical root systems

Losev and Manin introduced fine moduli spaces $\bar{L}_n$ of stable $n$-pointed chains of projective lines. The moduli space $\bar{L}_{n+1}$ is isomorphic to the toric variety $X(A_n)$ associated with the root system $A_n$, which is part of a general construction to associate with a root system $R$ of rank $n$ an $n$-dimensional smooth projective toric variety $X(R)$. In this paper we investigate generalisations of the Losev-Manin moduli spaces for the other families of classical root systems.

math.AG↗

arXiv2010

A generalization of a theorem of G. K. White

An n-dimensional simplex Δ in \R^n is called empty lattice simplex if Δ\cap\Z^n is exactly the set of vertices of Δ. A theorem of G. K. White shows that if n=3 then any empty lattice simplex Δ\subset\R^3 is isomorphic up to an unimodular affine linear transformation to a lattice tetrahedron whose all vertices have third coordinate 0 or 1. In this paper we prove a generalization of this theorem for an arbitrary odd dimension n=2d-1 which in some form was conjectured by Sebő and Borisov. This result implies a classification of all 2d-dimensional isolated Gorenstein cyclic quotient singularities with minimal log-discrepancy at least d.

math.CO↗

arXiv2009

Classification of Gorenstein Toric Del Pezzo Varieties in arbitrary dimension

A $n$-dimensional Gorenstein toric Fano variety $X$ is called Del Pezzo variety if the anticanonical class $-K_X$ is a $(n-1)$-multiple of a Cartier divisor. Our purpose is to give a complete biregular classfication of Gorenstein toric Del Pezzo varieties in arbitrary dimension $n \geq 2$. We show that up to isomorphism there exist exactly 37 Gorenstein toric Del Pezzo varieties of dimension $n$ which are not cones over $(n-1)$-dimensional Gorenstein toric Del Pezzo varieties. Our results are closely related to the classification of all Minkowski sum decompositions of reflexive polygons due to Emiris and Tsigaridas and to the classification up to deformation of $n$-dimensional almost Del Pezzo manifolds obtained by Jahnke and Peternell.

math.AG↗

arXiv2008

On the geometry of SL(2)-equivariant flips

In this paper, we show that any 3-dimensional normal affine quasihomogeneous SL(2)-variety can be described as a categorical quotient of a 4-dimensional affine hypersurface. Moreover, we show that the Cox ring of an arbitrary 3-dimensional normal affine quasihomogeneous SL(2)-variety has a unique defining equation. This allows us to construct SL(2)-equivariant flips by different GIT-quotients of hypersurfaces. Using the theory of spherical varieties, we describe SL(2)-flips by means of 2-dimensional colored cones.

math.AG↗

arXiv2008

Constructing new Calabi-Yau 3-folds and their mirrors via conifold transitions

We construct a surprisingly large class of new Calabi-Yau 3-folds $X$ with small Picard numbers and propose a construction of their mirrors $X^*$ using smoothings of toric hypersurfaces with conifold singularities. These new examples are related to the previously known ones via conifold transitions. Our results generalize the mirror construction for Calabi-Yau complete intersections in Grassmannians and flag manifolds via toric degenerations. There exist exactly 198849 reflexive 4-polytopes whose 2-faces are only triangles or parallelograms of minimal volume. Every such polytope gives rise to a family of Calabi-Yau hypersurfaces with at worst conifold singularities. Using a criterion of Namikawa we found 30241 reflexive 4-polytopes such that the corresponding Calabi-Yau hypersurfaces are smoothable by a flat deformation. In particular, we found 210 reflexive 4-polytopes defining 68 topologically different Calabi--Yau 3-folds with $h_{11}=1$. We explain the mirror construction and compute several new Picard--Fuchs operators for the respective 1-parameter families of mirror Calabi-Yau 3-folds.

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