On toric varieties of high arithmetical rank

Virtual Resolutions of Monomial Ideals on Toric Varieties

We use cellular resolutions of monomial ideals to prove an analog of Hilbert's syzygy theorem for virtual resolutions of monomial ideals on smooth toric varieties.

Vanishing Theorems on Toric Varieties

We use Cox's description for sheaves on toric varieties and results about the local cohomology with respect to monomial ideals to give a characteristic free approach to vanishing results on arbitrary toric varieties. As an application, we give a proof of a strong form of Fujita's conjecture in the case of toric varieties. We also prove that every sheaf on a toric variety corresponds to a module over the homogeneous coordinate ring, generalizing Cox's result for the simplicial case.

环面簇上向量丛的分裂

Splitting of Vector Bundles on Toric Varieties

本文在类似Eisenbud-Erman-Schreyer射影空间乘积情形的附加假设下，证明了任意光滑射影环面簇的Horrocks型分裂准则。

Cartier isomorphism for toric varieties

In this paper I show that, just as in the smooth case, the Cartier operator induces an isomorphism on the Zariski-deRham complex of any toric variety.

Arithmetically Cohen--Macaulay bundles on homogeneous varieties of Picard rank one

In this paper, we study arithmetically Cohen--Macaulay (ACM) bundles on homogeneous varieties $G/P$. Indeed we characterize the homogeneous ACM bundles on $G/P$ of Picard rank one in terms of highest weights. This is a generalization of the result on $G/P$ of classical types presented by Costa and Miró-Roig for type $A$, and Du, Fang, and Ren for types $B,C$ and $D$. As a consequence we prove that only finitely many irreducible homogeneous ACM bundles, up to twisting line bundles, exist over all such $G/P$. Moreover, we derive the list of the highest weights of the irreducible homogeneous ACM …

Tate resolutions on toric varieties

We develop an analogue of Eisenbud-Floystad-Schreyer's Tate resolutions for toric varieties. Our construction, which is given by a noncommutative analogue of a Fourier- Mukai transform, works quite generally and provides a new perspective on the relationship between Tate resolutions and Beilinson's resolution of the diagonal. We also develop a Beilinson-type resolution of the diagonal for toric varieties.

算术 Cohen-Macaulay 簇的 Cohen-Macaulay 表示型

The Cohen-Macaulay representation type of arithmetically Cohen-Macaulay varieties

本文证明射影空间中具有 Cohen-Macaulay 分次坐标环的约化闭子概形几乎都是野 Cohen-Macaulay 型，并对少数例外情形给出完全分类。

F-Signature of Affine Toric Varieties

We express the F-signature of the coordinate ring of an affine toric variety as the volume of a certain polytope, generalizing a formula of Watanabe and Yoshida. We also compute the F-signature of pairs and triples of a toric singularity.

On simplicial toric varieties of codimension 2

We describe classes of toric varieties of codimension 2 which are either minimally defined by 3 binomial equations over any algebraically closed field, or are set-theoretic complete intersections in exactly one positive characteristic.

Arithmetically Cohen-Macaulay Bundles on complete intersection varieties of sufficiently high multidegree

Recently it has been proved that any arithmetically Cohen-Macaulay (ACM) bundle of rank two on a general, smooth hypersurface of degree at least three and dimension at least four is a sum of line bundles. When the dimension of the hypersurface is three, a similar result is true provided the degree of the hypersurface is at least six. We extend these results to complete intersection subvarieties by proving that any ACM bundle of rank two on a general, smooth complete intersection subvariety of sufficiently high multi-degree and dimension at least four splits. We also obtain partial results in t…

Standard monomial theory and toric degenerations of Richardson varieties inside Grassmannians and flag varieties

We study toric degenerations of opposite Schubert and Richardson varieties inside degenerations of Grassmannians and flag varieties. These degenerations are parametrized by matching fields in the sense of Sturmfels and Zelevinsky. We construct so-called restricted matching field ideals whose generating sets are understood combinatorially through tableaux. We determine when these ideals are toric and coincide with Gröbner degenerations of Richardson varieties using the well established standard monomial theory for Grassmannians and flag varieties.

Syzygies of projective toric varieties

This paper has been subsumed by math.AG/0502240

Murphy's law for toric vector bundles on smooth projective toric varieties

We show that the moduli space of rank three toric vector bundles on smooth projective toric varieties satisfies Murphy's law, that is, every singularity type of finite type arises somewhere on this space.

Quotients of smooth projective toric varieties by $μ_p$ in positive characteristics $p$

In this paper we show that quotients of smooth projective toric varieties by $μ_p$ in positive characteristics $p$ are toric varieties.

Geometry of multigraded rings and embeddings of toric varieties

We use homogeneous spectra of multigraded rings to construct toric embeddings of a large family of projective varieties which preserve some of the birational geometry of the underlying variety, generalizing the well-known construction associated to Mori Dream Spaces.

Singular Loci of Hibi toric varieties

We first construct explicit bases for the cotangent spaces at singular points on Hibi toric varieties, i.e., toric varieties associated to distributive lattices. We then determine the singular loci of these toric varieties.

半正规环面簇

Seminormal toric varieties

本文提出半正规环面簇的组合刻画，利用带锥群结构的扇，扩展了正规环面簇类，同时简化了非正规情形的组合数据。

Derived Categories of Toric Varieties

We prove that the derived categories for toric varieties have complete exceptional collections.

On derived categories of arithmetic toric varieties

We begin a systematic investigation of derived categories of smooth projective toric varieties defined over an arbitrary base field. We show that, in many cases, toric varieties admit full exceptional collections. Examples include all toric surfaces, all toric Fano 3-folds, some toric Fano 4-folds, the generalized del Pezzo varieties of Voskresenskii and Klyachko, and toric varieties associated to Weyl fans of type $A$. Our main technical tool is a completely general Galois descent result for exceptional collections of objects on (possibly non-toric) varieties over non-closed fields.

Toric varieties whose canonical divisors are divisible by their dimensions

We totally classify the projective toric varieties whose canonical divisors are divisible by their dimensions. In Appendix, we show that Reid's toric Mori theory implies Mabuchi's characterization of the projective space for toric varieties.

On the equations and classification of toric quiver varieties

Toric quiver varieties (moduli spaces of quiver representations) are studied. Given a quiver and a weight there is an associated quasiprojective toric variety together with a canonical embedding into projective space. It is shown that for a quiver with no oriented cycles the homogeneous ideal of this embedded projective variety is generated by elements of degree at most $3$. In each fixed dimension $d$ up to isomorphism there are only finitely many $d$-dimensional toric quiver varieties. A procedure for their classification is outlined.

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.

Standard monomial theory and toric degenerations of Richardson varieties in flag varieties

We study standard monomial bases for Richardson varieties inside the flag variety. In general, writing down a standard monomial basis for a Richardson variety can be challenging, as it involves computing so-called defining chains or key tableaux. However, for a certain family of Richardson varieties, indexed by compatible permutations, we provide a very direct and straightforward combinatorial rule for writing down a standard monomial basis. We apply this result to the study of toric degenerations of Richardson varieties. In particular, we provide a new family of toric degenerations of Richard…

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.