We study the local cohomology modules H^i_B(R) for a reduced monomial ideal B in a polynomial ring R=k[X_1,...,X_n]. We consider a grading on R which is coarser than the Z^n-grading such that each component of H^i_B(R) is finite dimensional and we give an effective way to compute these components. Using Cox's description for sheaves on toric varieties, we apply these results to compute the cohomology of coherent sheaves on toric varieties. We give algorithms for this computation which have been implemented in the Macaulay 2 system. We obtain also a topological description for the cohomology of rank one torsionfree sheaves 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.
We introduce the notion of Betti category for graded modules over suitably graded polynomial rings, and more generally for modules over certain small categories. Our categorical approach allows us to treat simultaneously many important cases, such as monomial ideals and toric rings. We prove that in these cases the Betti category is a finite combinatorial object that completely determines the structure of the minimal free resolution. For monomial ideals, the Betti category is the same as the Betti poset that we studied in a previous article. We describe in detail and with examples how the theo…
In this paper we study some algebraic and combinatorial behaviors of expansion functor. We show that on monomial ideals some properties like polymatroidalness, weakly polymatroidalness and having linear quotients are preserved under taking the expansion functor. The main part of the paper is devoted to study of toric ideals associated to the expansion of subsets of monomials which are minimal with respect to divisibility. It is shown that, for a given discrete polymatroid $P$, if toric ideal of $P$ is generated by double swaps then toric ideal of any expansion of $P$ has such a property. This …
This article concerns monomial ideals fixed by differential operators of affine semi-group rings over $\mathbb{C}$. We give a complete characterization of when this happens. Perhaps surprisingly, every monomial ideal is fixed by an infinite set of homogeneous differential operators and is in fact determined by them. This opens up a new tool for studying monomial ideals. We explore applications of this to (mixed) multiplier ideals and other variants as well as give examples of detecting ideal membership in integrally closed powers and symbolic powers of squarefree monomial ideals.
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.
Given a birational map in the three dimensional projective space defined by monomials of degree $d$, we prove that its inverse is defined by monomials of degree at most $d^2-d+1$.
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…
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.
The goal of this paper is to produce a formula for the multiplier ideals of monomial space curves in the spirit of Howald's formula for the multiplier ideals of monomial ideals. This is achieved by constructing a toric blowup of affine space in such a way that a log resolution of the monomial curve may be constructed from this toric variety in a well controlled manner.
We construct a 3-dimensional complete intersection toric variety on which the subadditivity formula doesn't hold, answering negatively a question by Takagi and Watanabe. A combinatorial proof of the subadditivity formula on 2-dimensional normal toric varieties is also provided.
We describe a class of affine toric varieties $V$ that are set-theoretically minimally defined by codim $V+1$ binomial equations over fields of any characteristic.
Let $I_M$ and $I_N$ be defining ideals of toric varieties such that $I_M$ is a projection of $I_N$, i.e. $I_N \subseteq I_M$. We give necessary and sufficient conditions for the equality $I_M=rad(I_N+(f_1,...,f_s))$, where $f_1,...,f_s$ belong to $I_M$. Also a method for finding toric varieties which are set-theoretic complete intersection is given. Finally we apply our method in the computation of the arithmetical rank of certain toric varieties and provide the defining equations of the above 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.
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.
Mirror symmetry for a toric variety involves Laurent polynomials whose symplectic topology is related to the algebraic geometry of the toric variety. We show that there is a monodromy action on the monomially admissible Fukaya-Seidel categories of these Laurent polynomials as the arguments of their coefficients vary that corresponds under homological mirror symmetry to tensoring by a line bundle naturally associated to the monomials whose coefficients are rotated. In the process, we introduce the monomially admissible Fukaya-Seidel category as a new interpretation of the Fukaya-Seidel category…
In this article, we study monomial curves, toric ideals and monomial algebras associated to $4$-generated pseudo symmetric numerical semigroups. Namely, we determine indispensable binomials of these toric ideals, give a characterization for these monomial algebras to have strongly indispensable minimal graded free resolutions. We also characterize when the tangent cones of these monomial curves at the origin are Cohen-Macaulay.
We construct the RR varieties as the fiber products of Bott-Samelson varieties over Richardson varieties. We study their homogeneous coordinate rings and standard monomial theory.
We introduce the notion of a \emph{resolution supported on a poset}. When the poset is a CW-poset, i.e. the face poset of a regular CW-complex, we recover the notion of cellular resolution as introduced by Bayer and Sturmfels. Work of Reiner and Welker, and of Velasco, has shown that there are monomial ideals whose minimal free resolutions are not cellular, hence cannot be supported on any CW-poset. We show that for any monomial ideal there is a \emph{homology CW-poset} that supports a minimal free resolution of the ideal. This allows one to extend to every minimal resolution, essentially verb…
The Hodge algebra structures on the homogeneous coordinate rings of Grassmann varieties provide semi-toric degenerations of these varieties. In this paper we construct these semi-toric degenerations using quasi-valuations and triangulations of Newton-Okounkov bodies.
We provide a combinatorial characterization of monomial linear systems on toric varieties whose general member is quasismooth. This is given both in terms of the Newton polytope and in terms of the matrix of exponents of a monomial basis.
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.
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.
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.
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.
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.
We describe the equivariant cobordism ring of smooth toric varieties. This equivariant description is used to compute the ordinary cobordism ring of such 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.
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.
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.
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.
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…
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.
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.
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.
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.
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.
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.
We study the pullback maps on cohomology groups for equivariant rational maps (i.e., monomial maps) on toric varieties. Our method is based on the intersection theory on toric varieties. We use the method to determine the dynamical degrees of monomial maps and compute the degrees of the Cremona involution.
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.
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.
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.
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.
