988 citations · 2k across the 45 of their papers we have counts for
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On the classification of smooth toric surfaces with exactly one exceptional curve
Victor Batyrev
We classify all smooth projective toric surfaces containing exactly one exceptional curve. We show that every such surface is isomorphic to either or a surfa…
Spherical amoebae and a spherical logarithm map
Victor Batyrev, Megumi Harada, Johannes Hofscheier +1
Let be a connected reductive algebraic group over with a maximal compact subgroup . Let be a (quasi-affine) spherical homogeneous space. In the first part…
Hodge Theory of Hypersurfaces in Toric Varieties and Recent Developments in Quantum Physics
Victor V. Batyrev
This is the author's Habilitation which took place at University of Essen on July 11, 1993. The manuscript contains two parts. The first one is devoted to the author's combinatoria…
Projecting lattice polytopes according to the Minimal Model Program
Victor V. Batyrev
The Fine interior of a -dimensional lattice polytope is the set of all points having integral distance at least to any integral suppo…
Canonical models of toric hypersurfaces
Victor V. Batyrev
Let be a nondegenerate hypersurface in -dimensional torus defined by a Laurent polynomial with a -dimensional Newton polytope . The subset $F(P)…
On the stringy Hodge numbers of mirrors of quasi-smooth Calabi-Yau hypersurfaces
Victor V. Batyrev
Mirrors of quasi-smooth Calabi-Yau hypersurfaces in weighted projective spaces can be obtained as Calabi-Yau compactifications of non-de…
Mirror symmetry for quasi-smooth Calabi-Yau hypersurfaces in weighted projective spaces
Victor Batyrev, Karin Schaller
We consider a -dimensional well-formed weighted projective space as a toric variety associated with a fan in $N_{\overline{w}} \otim…
On the Fine Interior of Three-dimensional Canonical Fano Polytopes
Victor Batyrev, Alexander Kasprzyk, Karin Schaller
The Fine interior of a -dimensional lattice polytope is a rational subpolytope of which is important for constructing minimal birational models of non-de…
The stringy Euler number of Calabi-Yau hypersurfaces in toric varieties and the Mavlyutov duality
Victor Batyrev
We show that minimal models of nondegenerated hypersufaces defined by Laurent polynomials with a -dimensional Newton polytope are Calabi-Yau varieties if and only if the…
Satellites of spherical subgroups
Victor Batyrev, Anne Moreau
Let be a complex connected reductive algebraic group. Given a spherical subgroup and a subset of the set of spherical roots of , we define, up to conjuga…
On the algebraic stringy Euler number
Victor Batyrev, Giuliano Gagliardi
We are interested in stringy invariants of singular projective algebraic varieties satisfying a strict monotonicity with respect to elementary birational modifications in the Mori…
Stringy Chern classes of singular toric varieties and their applications
Victor Batyrev, Karin Schaller
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 inters…
Conifold degenerations of Fano 3-folds as hypersurfaces in toric varieties
Victor Batyrev, Maximilian Kreuzer
There exist exactly 166 4-dimensional reflexive polytopes such that the corresponding 4-dimensional Gorenstein toric Fano varieties have at worst terminal singularities in codimens…
The arc space of horospherical varieties and motivic integration
Victor Batyrev, Anne Moreau
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…
On generalisations of Losev-Manin moduli spaces for classical root systems
Victor Batyrev, Mark Blume
Losev and Manin introduced fine moduli spaces of stable -pointed chains of projective lines. The moduli space is isomorphic to the toric variety $X(A…
The functor of toric varieties associated with Weyl chambers and Losev-Manin moduli spaces
Victor Batyrev, Mark Blume
A root system of rank defines an -dimensional smooth projective toric variety associated with its fan of Weyl chambers. We give a simple description of the functo…
Classification of Gorenstein Toric Del Pezzo Varieties in arbitrary dimension
Victor Batyrev, Dorothee Juny
A -dimensional Gorenstein toric Fano variety is called Del Pezzo variety if the anticanonical class is a -multiple of a Cartier divisor. Our purpose is to give…
On the geometry of SL(2)-equivariant flips
Victor Batyrev, Fatima Haddad
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. Moreov…
Constructing new Calabi-Yau 3-folds and their mirrors via conifold transitions
Victor Batyrev, Maximilian Kreuzer
We construct a surprisingly large class of new Calabi-Yau 3-folds with small Picard numbers and propose a construction of their mirrors using smoothings of toric hypersur…
Integral Cohomology and Mirror Symmetry for Calabi-Yau 3-folds
Victor Batyrev, Maximilian Kreuzer
In this paper, we compute the integral cohomology groups for all examples of Calabi-Yau 3-folds obtained from hypersurfaces in 4-dimensional Gorenstein toric Fano varieties. Among…
The Cox Ring of a Del Pezzo Surface Has Rational Singularities
Oleg N. Popov
For a smooth Del Pezzo surface the direct sum of global sections of all isomorphism classes of invertible sheaves on it can be almost canonically endowed with a ring structure, cal…
The Cox ring of a Del Pezzo surface
Victor V. Batyrev, Oleg N. Popov
Let be a smooth Del Pezzo surface obtained from by blowing up points in general position. It is well known that for the Picard group…
Mixed toric residues and Calabi-Yau complete intersections
Victor V. Batyrev, Evgeny N. Materov
Using Cayley trick, we define the notions of mixed toric residues and mixed Hessians associated with Laurent polynomials .We conjecture that the values of mixed to…
Toric Residues and Mirror Symmetry
Victor V. Batyrev, Evgeny N. Materov
We develop some ideas of Morrison and Plesser and formulate a precise mathematical conjecture which has close relations to toric mirror symmetry. Our conjecture, we call it Toric R…