Derived Categories of Toric Varieties

Nash problem for stable toric varieties

We present a solution for Nash problem for stable toric varieties. We also introduce Nash problem for pairs and prove it for toric pairs.

Cobordism ring of toric varieties

We describe the equivariant cobordism ring of smooth toric varieties. This equivariant description is used to compute the ordinary cobordism ring of such varieties.

Notes on toric varieties from Mori theoretic viewpoint

The main purpose of this notes is to supplement the paper reid, which treated Minimal Model Program (also called Mori's Program) on toric varieties. We calculate lengths of negative extremal rays of toric varieties. As an application, we obtain a generalization of Fujita's conjecture for singular toric varieties. We also prove that every toric variety has a small projective toric $\bQ$-factorialization.

Equivariant Burnside groups and toric varieties

We study $G$-equivariant birational geometry of toric varieties, where $G$ is a finite group.

Toric Sarkisov links of toric Fano varieties

We explain a web of Sarkisov links that overlies the classification of Fano weighted projective spaces in dimensions 3 and 4, extending results of Prokhorov.

Comments on toric varieties

Here are few notes on not necessarily normal toric varieties and resolution by toric blow-up. These notes are independent of, but in the same spirit as the earlier preprint arXiv:math.AG/0306221. That is, they focus on the fact that toric varieties are locally given by monoid algebras.

Deformations of Toric Varieties via Minkowski Sum Decompositions of Polyhedral Complexes

We generalized the construction of deformations of affine toric varieties of K. Altmann and our previous construction of deformations of weak Fano toric varieties to the case of arbitrary toric varieties by introducing the notion of Minkowski sum decompositions of polyhedral complexes. Our construction embeds the original toric variety into a higher dimensional toric variety where the image is given by a prime binomial complete intersection ideal in Cox homogeneous coordinates. The deformations are realized by families of complete intersections. For compact simplicial toric varieties with at w…

A note on the Frobenius morphism on toric varieties

We give a new, shorter computation of Frobenius push-forwards of line bundles on toric varieties.

Notes on toric varieties

These notes survey some basic results in toric varieties over a field with examples and applications. A computer algebra package (written by the second author) is described which deals with both affine and projective toric varieties in any number of dimensions (written in both the software packages MAGMA and GAP). Among other things, the package implements a desingularization procedure for affine toric varieties, constructs some error-correcting codes associated with toric varieties, and computes the Riemann-Roch space of a divisor on a toric variety.

Formal Group Rings of Toric Varieties

In this paper we use formal group rings to construct an algebraic model of the $T$-equivariant oriented cohomology of smooth toric varieties. Then we compare our model with known results of equivariant cohomology of toric varieties to justify our construction.

On the Hodge structure of degenerating hypersurfaces in toric varieties

We introduce an algebraic method for describing the Hodge filtration of degenerating hypersurfaces in projective toric varieties. For this purpose, we show some fundamental properties of logarithmic differential forms on proper equivariant morphisms of toric varieties.

The Zariski-Lipman conjecture for toric varieties

We give a short proof of the Zariski-Lipman conjecture for toric varieties: any complex toric variety with locally free tangent sheaf is smooth.

Automorphisms of nonnormal toric varieties

In this paper we prove criteria for a nonnormal toric variety to be flexible, to be rigid and to be almost rigid. For rigid and almost rigid toric varieties we describe the automorphism group explicitly.

环面簇上补的显式界

An explicit bound for complements on toric varieties

该研究针对环面法诺簇上的klt补问题，推导得出其指标的有效上界，为代数几何中环面簇相关研究提供了显式的量化结果。

Wild hypersurface bundles over toric varieties

In this paper, we investigate when there exists a wild hypersurface bundle over a smooth proper toric variety in positive characteristic. In particular, we determine the possibilities for toric varieties with Picard number at most three or toric Fano varieties of dimension at most four. Moreover, we can construct wild hypersurface bundles over them.

Families of toric varieties

The goal of this paper is to define families of toric varieties and to study their properties. These families are locally trivial fibrations over some base, whose fibres are isomorphic to a fixed complete toric variety. The study is motivated by the occurence of such families in the context of gauged sigma models for toric varieties.

Gromov-Witten invariants of a class of toric varieties

We describe the quantum cohomology rings of a class of toric varieties. The description includes, in addition to the (already known) ring presentations, the (new) analogues for toric varieties of the sorts of quantum Giambelli formulas which exist already for Grassmannian varieties, flag varieties, etc.

Derived categories of toric varieties III

We prove derived McKay correspondence in special cases and the decomposition of toric K-equivalence into flops.

A geometric characterisation of toric varieties

We prove a conjecture of Shokurov which characterises toric varieties using log pairs.

Toric Fano varieties with divisorial contractions to curves

In this paper, we obtain a complete classification of smooth toric Fano varieties equipped with extremal contractions which contract divisors to curves for any dimension. As an application, we obtain a complete classification of smooth projective toric varieties which can be equivariantly blown-up to Fano along curves.

Derived categories of toric varieties II

We prove two theorems on the derived categories of toric varieties, the existence of an exceptional collection consisting of sheaves for a divisorial extraction and the finiteness of Fourier-Mukai partners.

Jumping deformations of complete toric varieties

We construct one-parameter complex analytic families whose special fibers are complete toric varieties. Under some assumptions, the general fibers of these families also become toric varieties and we can explicitly describe the corresponding fans from the data of the fans corresponding to the special fibers. Using these families, we give a deformation family for a certain toric weakened Fano 3-fold. Moreover, we get some examples of toric weakened Fano 4-folds.

Toric Degeneration of Bott-Samelson-Demazure-Hansen varieties

In this paper we construct a degeneration of Bott-Samelson-Demazure-Hansen varieties to toric varieties in an algebraic family and study the geometry of the resulting toric varieties. We give a natural set of torus invariant curves that generate the Chow group of $1$-cycles of the limiting toric variety and express the ample cone of this toric variety as a sub-cone of the ample cone of the corresponding Bott-Samelson-Demazure-Hansen variety. We also give a description of Extremal and Mori rays and determine when this toric variety is Fano.

Mirror Symmetry for Calabi-Yau complete intersections in Fano toric varieties

Generalizing the notions of reflexive polytopes and nef-partitions of Batyrev and Borisov, we propose a mirror symmetry construction for Calabi-Yau complete intersections in Fano toric varieties.

Toric Varieties as Spectra of Homogeneous Prime Ideals

We describe the construction of a class of toric varieties as spectra of homogeneous prime ideals.

Notes on toric varieties from Mori theoretic viewpoint, II

We give new estimates of lengths of extremal rays of birational type for toric varieties. We can see that our new estimates are the best by constructing some examples explicitly. As applications, we discuss the nefness and pseudo-effectivity of adjoint bundles of projective toric varieties. We also treat some generalizations of Fujita's freeness and very ampleness for toric varieties.

Vanishing Theorems on Toric Varieties in Positive Characteristic

We use the liftability of the relative Frobenius morphism of toric varieties and the strong liftability of toric varieties to prove the Bott vanishing theorem, the degeneration of the Hodge to de Rham spectral sequence and the Kawamata-Viehweg vanishing theorem for log pairs on toric varieties in positive characteristic. These results generalize those results of Danilov, Buch-Thomsen-Lauritzen-Mehta, Mustata and Fujino to the case where concerned Weil divisors are not necessarily torus invariant.

Makar-Limanov invariants of nonnormal affine toric varieties

We prove the equality of the Makar-Limanov invariant and the modified Makar-Limanov invariant in the case of not necessary normal affine toric varieties. Also we give a combinatorial description of these invariants.

Vanishing for Hodge ideals on toric varieties

In this article we construct a Koszul-type resolution of the p-th exterior power of the sheaf of holomorphic differential forms on smooth toric varieties and use this to prove a Nadel-type vanishing theorem for Hodge ideals associated to effective \mathbb{Q}-divisors on smooth projective toric varieties. This extends earlier results of Mustaţă and Popa.

Equivariant Todd Classes for Toric Varieties

For a complete toric variety, we obtain an explicit formula for the localized equivariant Todd class in terms of the combinatorial data -- the fan. This is based on the equivariant Riemann-Roch theorem and the computation of the equivariant cohomology and equivariant homology of toric varieties.

Toric varieties with ample tangent bundle

We give a simple combinatorial proof of the toric version of Mori's theorem that the only $n$-dimensional smooth projective varieties with ample tangent bundle are the projective spaces $\mathbb{P}^n$.

Special linear Systems on Toric Varieties

We consider linear systems on toric varieties of any dimension, with invariant base points, giving a characterization of special linear systems. We then make a new conjecture for linear systems on rational surfaces.

环面簇的等变单有理性质

Equivariant unirationality of toric varieties

本文研究环面簇在有限群作用下的等变单有理性，提出挠子理论障碍并证明其对环面自同构诱导作用的充分性。

Strong full exceptional collections on certain toric varieties with Picard number three via mutations

In this paper, we study derived categories of certain toric varieties with Picard number three that are blowing-up another toric varieties along their torus invariant loci of codimension at most three. We construct strong full exceptional collections by using Orlov's blow-up formula and mutations.

Universal Rational Parametrizations and Toric Varieties

This note proves the existence of universal rational parametrizations. The description involves homogeneous coordinates on a toric variety coming from a lattice polytope. We first describe how smooth toric varieties lead to universal rational parametrizations of certain projective varieties. We give numerous examples and then discuss what happens in the singular case. We also describe rational maps to smooth toric varieties.

Tropical Intersection Theory from Toric Varieties

We apply ideas from intersection theory on toric varieties to tropical intersection theory. We introduce mixed Minkowski weights on toric varieties which interpolate between equivariant and ordinary Chow cohomology classes on complete toric varieties. These objects fit into the framework of tropical intersection theory developed by Allermann and Rau. Standard facts about intersection theory on toric varieties are applied to show that the definitions of tropical intersection product on tropical cycles in $\R^n$ given by Allermann-Rau and Mikhalkin are equivalent. We introduce an induced tropica…

Secant varieties of toric varieties arising from simplicial complexes

Motivated by the study of the secant variety of the Segre-Veronese variety we propose a general framework to analyze properties of the secant varieties of toric embeddings of affine spaces defined by simplicial complexes. We prove that every such secant is toric, which gives a way to use combinatorial tools to study singularities. We focus on the Segre-Veronese variety for which we completely classify their secants that give Gorenstein or $\mathbb Q$-Gorenstein varieties. We conclude providing the explicit description of the singular locus.

Homogeneous Coordinates and Quotient Presentations for Toric Varieties

Generalizing cones over projective toric varieties, we present arbitrary toric varieties as quotients of quasiaffine toric varieties. Such quotient presentations correspond to groups of Weil divisors generating the topology. Groups comprising Cartier divisors define free quotients, whereas $\QQ$-Cartier divisors define geometric quotients. Each quotient presentation yields homogeneous coordinates. Using homogeneous coordinates, we express quasicoherent sheaves in terms of multigraded modules and describe the set of morphisms into a toric variety.

Toric arc schemes and quantum cohomology of toric varieties

We describe the quantum cohomology ring of a toric Fano variety $X$ in terms of the usual topological cohomology ring for an auxiliary infinite-dimensional scheme. This scheme is a part of an algebro-geometric model for the universal cover of the space of free loops with values in $X$.

Toward the classification of higher-dimensional toric Fano varieties

The purpose of this paper is to give basic tools for the classification of nonsingular toric Fano varieties by means of the notions of primitive collections and primitive relations due to Batyrev. By using them we can easily deal with equivariant blow-ups and blow-downs, and get an easy criterion to determine whether a given nonsingular toric variety is a Fano variety or not. As applications of these results, we get a toric version of a theorem of Mori, and can classify, in principle, all nonsingular toric Fano varieties obtained from a given nonsingular toric Fano variety by finite succession…

On deformations of toric Fano varieties

In this note we collect some results on the deformation theory of toric Fano varieties.

Mather-Jacobian multiplier ideals on toric varieties

This paper provides a formula for the Mather-Jacobian multiplier ideals of torus invariant ideals on (not necessarily normal) toric varieties that generalizes Howald's formula for the multiplier ideal of monomial ideals in a polynomial ring.

Automorphisms of Certain Projective Bundles over Toric Varieties

The automorphism group of a projective bundle P(E) over a simplicial toric variety is described when the bundle E is a direct sum of line bundles. Applications to study of moduli of complete intersections on toric varieties, including Calabi Yau m-folds and Del Pezzo surfaces, are indicated.

Morphic cohomology of toric varieties

In this paper we construct a spectral sequence computing a modified version of morphic cohomology of a toric variety (even when it is singular) in terms of combinatorial data coming from the fan of the toric variety.

Koszul duality for toric varieties

We show that certain categories of perverse sheaves on a pair of affine toric varieties defined by dual cones are Koszul dual in the sense of Beilinson, Ginzburg and Soergel. The functor expressing this duality is constructed explicitly using a combinatorial model for mixed sheaves on toric varieties.

The chow cohomology of affine toric varieties

We study the Fulton-Macpherson Chow cohomology of affine toric varieties. In particular, we prove that the Chow cohomology vanishes in positive degree. We prove an analogous result for the operational $K$-theory defined by Anderson and Payne.

容许有限轨道数幂零群作用的环面簇

Toric varieties admitting an action of a unipotent group with a finite number of orbits

本研究刻画了容许有限轨道数幂零群作用的完备单纯环面簇，并给出了二维情形下这类环面簇的完整列表，为相关代数几何研究提供了分类结果。

On some explicit semi-stable degenerations of toric varieties

We study semi-stable degenerations of toric varieties determined by certain partitions of their moment polytopes. Analyzing their defining equations we prove a property of uniqueness.

The moduli stack of affine stable toric varieties

Let X be an irreducible affine T-variety. We consider families of affine stable toric T-varieties over X and give a description of the corresponding moduli space as the quotient stack of an open subscheme in a certain toric Hilbert scheme under the action of a torus.

Pulling subdivisions of cones and blowups of monomial ideals on affine toric varieties

This short note solves the following problem: Given a map of normal toric varieties corresponding to a coherent subdivision of a cone, find an ideal such that the given map is the blowup of that ideal.

A counterexample to the maximality of toric varieties

We present a counterexample to the conjecture of Bihan, Franz, McCrory, and van Hamel concerning the maximality of toric varieties. There exists a six dimensional projective toric variety X with the sum of the mod 2 Betti numbers of X(R) strictly less than the sum of the mod 2 Betti numbers of X(C).

Smooth Toric Quiver Varieties

We study smoothness of toric quiver varieties. When a quiver $Q$ is defined with the identity dimension vector, the corresponding quiver variety is also a toric variety. So it has both fan representation and quiver representation. We work only on quivers with canonical weight and we classify smooth such toric quiver varieties. We show that a variety corresponding to a quiver with the identity dimension vector and the canonical weight is smooth if and only if it is a product of projective spaces or their blowups.

Multiplication maps and vanishing theorems for toric varieties

We use multiplication maps to give a characteristic-free approach to vanishing theorems on toric varieties. Our approach is very elementary but is enough powerful to prove vanishing theorems.

Smooth projective toric varieties whose nontrivial nef line bundles are big

For any $n\geq 3$, we explicitly construct smooth projective toric $n$-folds of Picard number $\geq 5$, where any nontrivial nef line bundles are big.