Milena Hering
Let A be an ample line bundle on a projective toric variety X of dimension n. We show that if l>=n-1+p, then A^l satisfies the property N_p. Applying similar methods, we obtain a combinatorial theorem: For a given lattice polytope P we give a criterion for an integer m to guarantee that mP is normal.
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In a first part of this paper, we prove constancy of the canonical graded Betti table among the smooth curves in linear systems on Gorenstein weak Fano toric surfaces. In a second part, we show that Green's canonical syzygy conjecture holds for all smooth curves of genus at most 32 or Clifford index at most 6 on arbitrary toric surfaces. Conversely we use known results on Green's conjecture (due to Lelli-Chiesa) to obtain new facts about graded Betti tables of projectively embedded toric surfaces.
This paper has been subsumed by math.AG/0502240
The purpose of the paper is to give a sharp asymptotic description of the weights that appear in the syzygies of a toric variety. We prove that as the positivity of the embedding increases, in any strand of syzygies, torus weights after normalization stabilize to the same fixed shape that we explicitly specify.
Syzygies of algebraic varieties through symmetric products of algebraic curves
本文综述了利用代数曲线对称积的几何方法证明的关于代数簇合冲的若干近期结果，包括亏格猜想、割线簇与切发展曲面的合冲、格林猜想及渐近合冲。
We study syzygies of abelian varieties via the methods of Caucci and Ito based on computations of cohomological rank functions. We provide some strong evidences to a conjecture of Ito and Lozovanu.
Using multigraded Castelnuovo-Mumford regularity, we study the equations defining a projective embedding of a variety X. Given globally generated line bundles B_1, ..., B_k on X and integers m_1, ..., m_k, consider the line bundle L := B_1^m_1 \otimes ... \otimes B_k^m_k. We give conditions on the m_i which guarantee that the ideal of X in P(H^0(X,L)) is generated by quadrics and the first p syzygies are linear. This yields new results on the syzygies of toric varieties and the normality of polytopes.
Asymptotic syzygies of a normal crossing variety follow the same vanishing behavior as one of its smooth components, unless there is a cohomological obstruction arising from how the smooth components intersect each other. In that case, we compute the asymptotic syzygies in terms of the cohomology of the simplicial complex associated to the normal crossing variety. We combine our results on normal crossing varieties with knowledge of degenerations of certain smooth projective varieties to obtain some results on asymptotic syzygies of those smooth projective varieties.
We show that the linear syzygy spaces of elliptic normal curves, their secant varieties and of bielliptic canonical curves are spanned by geometric syzygies.
On the Jacobian syzygies for generic toric models
本文针对仿射环面中的一般超曲面，在射影空间关联超曲面配置，并计算了相应雅可比代数的最小分次分解。
We use the language of multiplier ideals in order to relate the syzygies of an abelian variety in a suitable embedding with the local positivity of the line bundle inducing that embedding. This extends to higher syzygies a result of Hwang and To on projective normality.
We show how a natural constant introduced by Jiang and Pareschi for a polarized abelian variety encodes information about the syzygies of the section ring of the polarization. As a particular case this gives a quick and characteristic-free proof of Lazarsfeld's conjecture on syzygies of abelian varieties, originally proved by Pareschi in characteristic zero.
We study property $N_p$ for Kummer varieties embedded by a power of an ample line bundle.
The geometric and algebraic properties of smooth projective varieties with 1-regular structure sheaf are well understood, and the complete classification of these varieties is a classical result. The aim of this paper is to study the next case: smooth projective varieties with 2-regular structure sheaf. First, we give a classification of such varieties using adjunction mappings. Next, under suitable conditions, we study the syzygies of section rings of those varieties to understand the structure of the Betti tables, and show a sharp bound for Castelnuovo-Mumford regularity.
Cohomological rank functions and syzygies of projective bundles on abelian varieties
本文利用Jiang–Pareschi上同调秩函数及Caucci–Ito技术研究阿贝尔簇上射影丛的合冲，从而推广并细化了Chiantipalli关于性质$N_p$的结果。
Syzygies of some rational homogeneous varieties
本文推广Manivel关于旗簇上丰富线丛幂次满足性质$(N_p)$的结果至$B$、$C$、$D$、$G_2$型有理齐次簇，研究其合冲。
Geography of Landau-Ginzburg models and threefold syzygies
本文研究环面 Landau-Ginzburg 模型在极端收缩与极小模型纲领下的行为，提出并部分证明了 Fano 簇的极端收缩、极小模型纲领与其环面 Landau-Ginzburg 模型退化之间的对应关系，并应用于计算光滑 Fano 三维簇的初等合冲。
Syzygies of Kummer varieties
本文证明 Kummer 簇上丰富线丛的 $m$ 次幂满足 Green-Lazarsfeld 性质 $(N_p)$ 的条件为 $m > (p+2)/2$，其合冲行为为阿贝尔簇情形的一半。
Vanishing of weight one syzygies of projective varieties
本文探究射影簇上权1 Koszul上同调的消失条件，结合(M_q)与(N_p)性质推导相关推论，深化了对极小自由消解的认知。
Effective syzygies of secant varieties
该研究针对代数几何中割线簇的性质问题，给出了仅依赖于空间维度X和参数p的伴随线丛有效界，推进了割线簇合冲理论的研究。
In this article we prove some strong vanishing theorems on K3 surfaces. As an aplication of them, we obtain higher syzygy results for K3 surfaces and Fano varieties.
We consider the Grassmannian $\mathbb{G}r(k,n)$ of $(k+1)$-dimensional linear subspaces of $V_n=H^0({¶^1},Ø_{¶^1}(n))$. We define $\frak{X}_{k,r,d}$ as the classifying space of the $k$-dimensional linear systems of degree $n$ on $¶^1$ whose basis realize a fixed number of polynomial relations of fixed degree, say a fixed number of syzygies of a certain degree. The first result of this paper is the computation of the dimension of $\frak{X}_{k,r,d}$. In the second part we make a link between $\frak{X}_{k,r,d}$ and the Poncelet varieties. In particular, we prove that the existence of linear syzyg…
Let X be a smooth complex projective variety of dimension n and let A be an ample and basepoint free divisor. We prove $K_X+mA$ satisfies property $N_p$ for $m\geqslant n+1+p$. We also show the graded ring of sections $R(X, K_X+mA)$ is Koszul for $m\geqslant n+2$.
Asymptotic nonvanishing of syzygies of algebraic varieties
本文建立了光滑射影簇渐近合冲的精确非消没结果，细化Ein-Lazarsfeld定理，并结合先前消没结果完全确定了线丛分次截面模极小自由分解的渐近形状。
We give simple geometric proofs of Aprodu-Farkas-Papadima-Raicu-Weyman's theorem on syzygies of tangent developable surfaces of rational normal curves and Raicu-Sam's result on syzygies of K3 carpets. As a consequence, we obtain a quick proof of Green's conjecture for general curves of genus $g$ over an algebraically closed field $\mathbf{k}$ with $\operatorname{char}(\mathbf{k}) = 0$ or $\operatorname{char}(\mathbf{k}) \geq \lfloor (g-1)/2 \rfloor$. We also show the arithmetic normality of tangent developable surfaces of arbitrary smooth projective curves of large degree.
A structure theorem for syzygies of del Pezzo varieties
本文利用Buchsbaum-Eisenbud结构定理，证明了定义del Pezzo簇的二次型间最高线性合子可表示为元素为线性形式楔积的斜对称矩阵的列。
In this paper, we show that a general polarized abelian variety $(X,L)$ of type $(1,\dots,1,d)$ and dimension $g$ satisfies property $(N_p)$ if $ d \geq \sum_{i=0}^{g} (p+2)^i$. In particular, the case $p=0$ affirmatively solves a conjecture by L. Fuentes Garc\'ıa on projective normality.
It was recently established by the first two authors that multiplier ideals on a smooth variety satisfy some special syzygetic properties. The purpose of this note is to show how some of these can be extended to the singular setting.
In recent years, the equations defining secant varieties and their syzygies have attracted considerable attention. The purpose of the present paper is to conduct a thorough study on secant varieties of curves by settling several conjectures and revealing interaction between singularities and syzygies. The main results assert that if the degree of the embedding line bundle of a nonsingular curve of genus $g$ is greater than $2g+2k+p$ for nonnegative integers $k$ and $p$, then the $k$-th secant variety of the curve has normal Du Bois singularities, is arithmetically Cohen--Macaulay, and satisfie…
Singularities and syzygies of secant varieties of smooth projective varieties
本文研究光滑射影簇的高阶割线簇，证明曲面情形在正线丛下具有Du Bois奇点及线性合冲，并揭示其奇点类型由曲面结构层上同调决定，同时给出高维低阶情形的类似结果及高阶情形的否定性结论。
We prove that the minimal free resolution of the secant variety of a curve is asymptotically pure. As a corollary, we show that the Betti numbers of converge to a normal distribution.
Syzygies of secant varieties of smooth projective curves and gonality sequences
本文证明光滑射影曲线的gonality序列可由其充分大度数线丛嵌入下割线簇的合冲读出，并将Ein-Lazarsfeld与Rathmann的gonality猜想推广到割线簇情形。
Suppose X is a projective variety, which needs not be smooth, and L an ample divisor on X. We show that there are integers c and b such that for any nonnegative integer p, L^d is normally generated and embeds X as a variety who defining ideal has linear syzygies upto the p-th step (i.e. L^d has property N_p) for all d >= cp + b.
The purpose of this paper is to prove Ein--Lazarsfeld's conjecture on asymptotic vanishing of syzygies of algebraic varieties. This result, together with Ein--Lazarsfeld's asymptotic nonvanishing theorem, describes the overall picture of asymptotic behaviors of the minimal free resolutions of the graded section rings of line bundles on a projective variety as the positivity of the line bundles grows. Previously, Raicu reduced the problem to the case of products of three projective spaces, and we resolve this case here.
This paper studies the asymptotic behavior of the syzygies of a smooth projective variety X as the positivity of the embedding line bundle grows. We prove that as least as far as grading is concerned, the minimal resolution of the ideal of X has a surprisingly uniform asymptotic shape: roughly speaking, generators eventually appear in almost all degrees permitted by Castelnuovo-Mumford regularity. This suggests in particular that a widely-accepted intuition derived from the case of curves -- namely that syzygies become simpler as the degree of the embedding increases -- may have been misleadin…
In this thesis we looked into three different problems which share, as a common factor, the exstensive use of the Fourier--Mukai transform as research tool. In the first Part we investigated the syzygies of Kummer varieties (i.e. quotients of abelian varieties by the $\mathbb{Z}/2\Z$-action induced by the group operation), extending to higher syzygies results on projective normality and degree of equations of Sasaki, Kempf, and Khaled. The second Part of this Thesis (partially written in collaboration with Z.Jiang and M. Lahoz) is dedicated to the study of pluricanonical linear systems on vari…
The rational homology groups of the packing complexes are important in algebraic geometry since they control the syzygies of line bundles on projective embeddings of products of projective spaces (Segre--Veronese varieties). These complexes are a common generalization of the multidimensional chessboard complexes and of the matching complexes of complete uniform hypergraphs, whose study has been a topic of interest in combinatorial topology. We prove that the multivariate version of representation stability, a notion recently introduced and studied by Church and Farb, holds for the homology gro…
This paper is a survey of recent work on the asymptotic behavior of the syzygies of a smooth complex projective variety as the positivity of the embedding line bundle grows. After a quick overview of results from the 1980s and 1990s concerning the linearity of the first few terms of a resolution, we discuss a non-vanishing theorem to the effect that from an asymptotic viewpoint, essentially all of the syzygy modules that could be non-zero are in fact non-zero. We explain the quick new proof of this result in the case of Veronese varieties due to Erman and authors, and we explore some results a…
Fibered toric varieties
本文引入纤维化环面簇，用以阐明一般环面簇丛的拓扑与相交理论结果，并据此计算光滑完备环面簇的等变上同调环。
In the present paper, we consider upper bounds of higher linear syzygies i.e. graded Betti numbers in the first linear strand of the minimal free resolutions of projective varieties in arbitrary characteristic. For this purpose, we first remind `Partial Elimination Ideals (PEIs)' theory and introduce a new framework in which one can study the syzygies of embedded projective schemes well using PEIs theory and the reduction method via inner projections. Next we establish fundamental inequalities which govern the relations between the graded Betti numbers in the first linear strand of an algebrai…
We investigate toric varieties defined by arrangements of hyperplanes and call them strongly symmetric. The smoothness of such a toric variety translates to the fact that the arrangement is crystallographic. As a result, we obtain a complete classification of this class of toric varieties. Further, we show that these varieties are projective and describe associated toric arrangements in these varieties.
We describe the spaces of minimal rank last syzygies for the Mukai Varieties of sectional genus 6,7 and 8. Based on this we show: 1. The first geometric syzygies of a general canonical curve of genus 6 form a non degenerate configuration of 5 lines in P^4. 2. The first geometric syzygies of a general canonical curve of genus 7 form a non degenerate, linearly normal, ruled surface of degree 84 on a spinor variety S in P^15. 3. The second geometric syzygies of a general canonical curve of genus 8 form a non degenerate configuration of 14 conics on a 2-uple embedded P^5 in P^20. This proves a nat…
Let $X \subset ¶^r$ be a nondegenerate projective variety and let $ν_{\ell} : ¶^r \to ¶^N$ be the $\ell$-th Veronese embedding. In this paper we study the higher normality, defining equations and syzygies among them for the projective embedding $ν_{\ell} (X) \subset ¶^N$. We obtain that for a very ample line bundle $L \in {Pic}X$ such that $X \subset ¶H^0 (X,L)$ is $m$-regular in the sense of Castelnuovo-Mumford, $(X,L^{\ell})$ satisfies property $N_{\ell}$ for all $\ell \geq m$ (Theorem \ref{thm:main1}). This is a generalization of M. Green's work that $(¶^r,\mathcal{O}_{¶^r} (\ell))$ satisfi…
This article gives an overview of toric Fano and toric weak Fano varieties associated to graphs and building sets. We also study some properties of such toric Fano varieties and discuss related topics.
We characterize building sets whose associated nonsingular projective toric varieties are Fano. Furthermore, we show that all such toric Fano varieties are obtained from smooth Fano polytopes associated to finite directed graphs.
This paper studies two related subjects. One is some combinatorics arising from linear projections of polytopes and fans of cones. The other is quotient varieties of toric varieties. The relation is that projections of polytopes are related to quotients of projective toric varieties and projection of fans are related to quotients of general toric varieties. Despite its relation to geometry the first part is purely combinatorial and should be of interest in its own right.
The goal of this article is to study the equations and syzygies of embeddings of rational surfaces and certain Fano varieties. Given a rational surface X and an ample and base-point-free line bundle L on X, we give an optimal numerical criterion for L to satisfy property Np. This criterion turns out to be a characterization of property Np if X is anticanonical. We also prove syzygy results for adjunction bundles and a Reider type theorem for higher syzygies. For certain Fano varieties we also prove results on very ampleness and higher syzygies.
We give a structure theorem for n-dimensional smooth toric Fano varieties whose associated polytope has "many" pairs of centrally symmetric vertices.
We investigate Gauss maps of (not necessarily normal) projective toric varieties over an algebraically closed field of arbitrary characteristic. The main results are as follows: (1) The structure of the Gauss map of a toric variety is described in terms of combinatorics in any characteristic. (2) We give a developability criterion in the toric case. In particular, we show that any toric variety whose Gauss map is degenerate must be the join of some toric varieties in characteristic zero. (3) As applications, we provide two constructions of toric varieties whose Gauss maps have some given data …
This article is based on a series of lectures on toric varieties given at RIMS, Kyoto. We start by introducing toric varieties, their basic properties and later pass to more advanced topics relating mostly to combinatorics.
Gorenstein toric Schubert varieties in Grassmannians
本文研究格拉斯曼流形中的环面Schubert簇，给出了Gorenstein环面Schubert簇的扇的显式描述，并证明其均为Fano簇。
We give a bound of $k$ for a very ample lattice polytope to be $k$-normal. Equivalently, we give a new combinatorial bound for the Castelnuovo-Mumford regularity of normal projective toric varieties.
We discuss a characteristic free version of Frobenius splittings for toric varieties and give a polyhedral criterion for a toric variety to be diagonally split. We apply this criterion to show that section rings of nef line bundles on diagonally split toric varieties are normally presented and Koszul, and that Schubert varieties are not diagonally split in general.
Toric Richardson Varieties
本文从偏序集、根系和几何三个角度刻画了所有类型的 toric Richardson 簇。
We associate a complete non-singular fan with a polygon triangulation. Such a fan appears from a certain toric Richardson variety, called of Catalan type introduced in this paper. A toric Richardson variety of Catalan type is a Fano Bott manifold. We show that toric Richardson varieties of Catalan type are classified up to isomorphism in terms of unordered binary trees. In particular, the number of isomorphism classes of $n$-dimensional toric Richardson varieties of Catalan type is the $(n+1)$th Wedderburn--Etherington number.
We investigate Gorenstein toric Fano varieties by combinatorial methods using the notion of a reflexive polytope which appeared in connection to mirror symmetry. The paper contains generalisations of tools and previously known results for nonsingular toric Fano varieties. As applications we obtain new classification results, bounds of invariants and formulate conjectures concerning combinatorial and geometrical properties of reflexive polytopes.
We give a necessary and sufficient condition for the nonsingular projective toric variety associated to a building set to be weak Fano in terms of the building set.
Given a subspace L of a vector space V, the Kalman variety consists of all matrices of V that have a nonzero eigenvector in L. Ottaviani and Sturmfels described minimal equations in the case that dim L = 2 and conjectured minimal equations for dim L = 3. We prove their conjecture and describe the minimal free resolution in the case that dim L = 2, as well as some related results. The main tool is an exact sequence which involves the coordinate rings of these Kalman varieties and the normalizations of some related varieties. We conjecture that this exact sequence exists for all values of dim L.
On Mukai's conjecture for hyperelliptic varieties
本文研究数值平凡典范线丛的光滑射影簇的合冲性质，证明了一般性结果，并在任意维数和正特征下验证了阿贝尔簇有限商的Mukai合冲猜想若干情形。
Let X be a smooth projective variety and let K be the canonical divisor of X. In this paper, we study embeddings of X given by adjoint line bundles of the form K+L, where L is an ample line bundle. When X is a regular surface (i.e. H^1(X, O_X) = 0), we obtain a numerical criterion for K+L to have property Np. When X is a regular variety of arbitrary dimension, under a mild condition, we give an explicit calculation for the regularity of the ideal sheaves of such embeddings.
We study the linear syzygies of a homogeneous ideal I in a polynomial ring S, focusing on the graded betti numbers b_(i,i+1) = dim_k Tor_i(S/I, k)_(i+1). For a variety X and divisor D with S = Sym(H^0(D)*), what conditions on D ensure that b_(i,i+1) is nonzero? Eisenbud has shown that a decomposition D = A + B such that A and B have at least two sections gives rise to determinantal equations (and corresponding syzygies) in I_X; and conjectured that if I_2 is generated by quadrics of rank at most 4, then the last nonvanishing b_(i,i+1) is a consequence of such equations. We describe obstruction…
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