On the vanishing of higher syzygies of curves. II
Marian Aprodu
本文证明了Green猜想对于亏格为$g$的一般$k$-曲线，对于$g\geq k(k-1)/2$ .
We verify Green's conjecture for a generic $k$-gonal curve of genus $g$, for $g\geq k(k-1)/2$.
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Effective Non-vanishing of Asymptotic Adjoint Syzygies
本文的目的是建立光滑簇上伴随型线丛的合轨的一个有效的非消灭定理，当嵌入的正性增加时。本文的目的是证明对于一个伴随型因子$B = K_X+ bA$与$b \geq n+1$，可以得到任意$X$的一个有效的陈述，它专门针对Ein和Lazarsfeld的论文“代数变种的渐近合势”中的Veronese合势的陈述.在此基础上，我们也给出了该论文中问题7.9的答案。
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。
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。
Vanishing Theorems and Syzygies for K3 Surfaces and Fano Varieties
本文证明了K3曲面上的几个强消失定理。作为它们的应用，我们得到了K3曲面和Fano簇的更高的合点结果。
Geometric syzygies of elliptic normal curves and their secant varieties
证明了椭圆正态曲线及其割线簇和双椭圆标准曲线的线性合点空间都是几何合点所张成的。
We relate the equianalytic and the equisingular deformations of a reduced complex plane curve to the Jacobian syzygies of its defining equation. Several examples and conjectures involving rational cuspidal curves are discussed.
Syzygies of 5-gonal Canonical Curves
我们证明了对于奇亏格$g\geq 13$和偶亏格$g\geq 28$的$5$-曲线，曲线的$\lceil \frac{g-1}{2}\rceil$次合模不是由特殊度束$5$扫过的涡卷的合模决定的.
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.
We analyze the vanishing and non-vanishing behavior of the graded Betti numbers for Segre embeddings of products of projective spaces. We give lower bounds for when each of the rows of the Betti table becomes non-zero, and prove that our bounds are tight for Segre embeddings of products of P^1. This generalizes results of Rubei concerning the Green-Lazarsfeld property N_p for Segre embeddings. Our methods combine the Kempf-Weyman geometric technique for computing syzygies, the Ein-Erman-Lazarsfeld approach to proving non-vanishing of Betti numbers, and the theory of algebras with straightening…
A theorem of Green says that a line bundle of degree at least $2g+1+p$ on a smooth curve $X$ of genus $g$ has property $N_p$. We prove a similar conclusion for certain singular, reducible curves $X$ under suitable degree bounds over all irreducible components of $X$.
Special Linear Series and Syzygies of Canonical Curves of Genus 9
在这篇论文中，我们给出了亏格为9的不可约非奇异标准曲线$C$的合势的完整描述。这包括$C$的所有可能Betti表的集合。并给出了这些Betti表与$C$上特殊线性级数的个数和类型的直接对应关系。特别是对于$\operatorname{Cliff}(C)=3$，曲线$C$包含在由5次因子的基点自由束构造的4维有理法向涡卷$X\subset \mathbb{P}^8$上的行列式曲面$Y$中。
Toward GIT stability of syzygies of canonical curves
我们引入了标准曲线合点的GIT稳定性问题，并着眼于稳定曲线的模空间的标准模型的GIT构造。作为这个方向的第一步，我们证明了奇亏格的一般标准曲线的第一合点的半稳定性。
Secant spaces and syzygies of special line bundles on curves
在射影曲线$C$上的特殊线丛$L$上，我们引入了一个称为$(Δ_q)$的几何条件。当$L=K_C$时，此条件意味着gon $(C) \ge q+2$。对于一个任意的特殊$L$，我们表明，$(Δ_3)$意味着$L$具有著名的属性$(M_3)$，推广了类似的结果证明了Voisin的情况下$L=K_C$。
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…
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.
Remarks on syzygies of $d$-gonal curves
我们应用一个退化的版本的结果，由于Hirschowitz，Ramanan和Voisin验证绿色和绿色Lazarsfeld代数内的每个$d$ -层曲线$X$与$d<[g_X/2]+2$显式开集。用同样的方法，我们对任意奇亏格和极大角性曲线证明了Green-Lazarsfeld猜想。证明调用Voisin的解决方案，通用的绿色猜想作为一个关键参数。
Curves with Jacobian syzygies of the same degree
本文研究雅可比合系分次模由极小次数分量生成的复射影平面曲线，发现该类曲线规模庞大，包含光滑曲线、极大Tjurina曲线及满足特定三重点条件的直线构形。
Inspired by the methods of Voisin, the first two authors recently proved that one could read off the gonality of a curve C from the syzygies of its ideal in any one embedding of sufficiently large degree. This was deduced from from a vanishing theorem for the asymptotic syzygies associated to an arbitrary line bundle B on C. The present paper extends this vanishing theorem to a smooth projective variety X of arbitrary dimension. Specifically, given a line bundle B on X, we prove that if B is p-jet very ample (i.e. the sections of B separate jets of total weight p+1) then the weight one Koszul …
Let $C$ be a trigonal curve of genus $g\ge 5$ and let $T$ be the unique trigonal line bundle inducing a map $π: C \stackrel{3:1}{\longrightarrow} \mathbb{P}^1$. This note provides a short and easy proof of the normal generation for the residual line bundle $K_C\otimes T^{-1}$ for curves of genus $g\ge 7$. Moreover, we compute the minimal free resolution of the embedded curve $C\subset \mathbb{P}(H^0(C,K_C\otimes T^{-n})^*)$ for the residual line bundle $K_C\otimes T^{-n}$ for $n\ge 1$ and $g \ge 3n+4$.
We survey three results on syzygies of curves beyond Green's conjecture, with a particular emphasis on drawing connections between the study of syzygies and other topics in moduli theory.
Linear syzygies of projective space
研究线性合系q=1情形下Veronese簇合系的Ein-Lazarsfeld猜想，证明在最高和次高阶线性合系上与猜想一致的消失性命题。
The Rank of Syzygies of Canonical Curves
该研究证明一般典范曲线的线性合冲空间由最小秩合冲张成，属于代数几何与交换代数领域的理论成果。
Double Rational Normal Curves with Linear Syzygies
一种是描述有理正态曲线上的二重结构，这种曲线是由二次方程从理论上定义的，具有线性合点，将平面上的二重线推广到平面上
Syzygies using vector bundles
研究了射影空间中由大次线丛嵌入的曲线的合势。证明利用了与给定的线丛相关联的向量丛的合矢与截面空间之间的关系。
Syzygies of canonical ribbons on higher genus curves
研究高亏格曲线上典范ribbon的合冲，证明一般ribbon的两种Clifford指标相等，指出分裂ribbon的指标相等与亏格猜想、Green猜想相关，并将其简化为辅助合冲模Koszul上同调群的消失问题。
On the vanishing of higher syzygies of curves
本文涉及Green和Lazarsfeld关于嵌在充分大次数的完全线性系中的曲线的1-线性合型的一个猜想。给定一个光滑的，不可约的，复杂的，射影曲线$X$，我们证明了最小整数$q$的属性$(M_q)$失败的线丛$L$ $X$不依赖于$L$，只要它的程度变得足够大。因此，这个数是曲线的不变量，而格林-拉扎斯菲尔德猜想的陈述等价于说这个不变量等于曲线的角性。我们验证了猜想平面曲线，曲线躺在Hirzebruch表面上，并为通用曲线具有足够大的相比，gonality的属。最后，我们的文件证明，格林的典型猜想持有曲线躺在Hirzebruch表面。
New evidence for Green's conjecture on syzygies of canonical curves
本文证明了当亏格$g$为奇数时，标准曲线的两种减弱形式的Green图是等价的。
Syzygies of curves and the effective cone of \bar{M}_g
我们描述了一个系统的方法来构造有效因子的模空间的稳定曲线的亏格g具有非常小的斜率。我们证明了\bar{M}_g上的任何因子都是由不满足某种Green-Lazarsfeld合冲型条件的曲线组成的，它为Harris-Morrison斜率猜想提供了一个反例.这些因子将我们原来的孤立反例推广到斜率猜想，斜率猜想是位于K3曲面上的曲线的M_{10}上的因子。我们还引入了M_g的一个新的分层，有点类似于由角性给出的经典分层，但其中的超椭圆曲线的类似物是K3曲面的截面。最后证明了g<23的模空间M_{g,n}是一般型的.
Let $C$ be a plane rational curve of degree $d$ and $p:\tilde C \rightarrow C$ its normalization. We are interested in the splitting type $(a,b)$ of $C$, where $\mathcal{O}_{\mathbb{P}^1}(-a-d)\oplus \mathcal{O}_{\mathbb{P}^1}(-b-d)$ gives the syzigies of the ideal $(f_0,f_1,f_2)\subset K[s,t]$, and $(f_0,f_1,f_2)$ is a parameterization of $C$. We want to describe in which cases $(a,b)=(k,d-k)$ ($2k\leq d)$, via a geometric description; namely we show that $(a,b)=(k,d-k)$ if and only if $C$ is the projection of a rational curve on a rational normal surface in $\mathbb{P}^{k+1}$.
A key result for syzygies of curves is Voisin's proof of Green's conjecture for the canonical embedding of a general curve of any genus. Her primary tools were the Lazarsfeld Mukai bundle on a K3 surface and a representation of Koszul cohomology on the Hilbert scheme of points on the surface. In this note we construct representations of the Koszul complex on Grassmann varieties; Voisin's setup arises as the inverse image of one of the maps. Using a different map, we give a substantially shorter proof of Voisin's result for K3 surfaces of even sectional genus.
We explore the concept of projections of syzygies and prove two new technical results; we firstly give a precise characterization of syzygy schemes in terms of their projections, secondly, we prove a converse to Aprodu's Projection Theorem. Applying these results, we prove that extremal syzygies of general curves of non-maximal gonality embedded by a linear system of sufficiently high degree arise from scrolls. Lastly, we prove Green's Conjecture for general covers of elliptic curves (of arbitrary degree) as well as proving a new result for curves of even genus and maximal gonality.
Secant rank and syzygies of projections of elliptic normal curves
研究椭圆正则曲线投影的syzygies，通过切线秩确定投影曲线的Green-Lazarsfeld指数，证明一般点的指数为ceil(d/2)-3。
Geometric Syzygies of Canonical Curves of even Genus lying on a K3-Surface
基于Voisin [2001]的最新结果，我们将位于K3曲面上的偶亏格g=2k的标准曲线C的线性链中的最后一个非零合合空间描述为k-2重嵌入P^{k+1}的环境空间. Green和Lazarsfeld [1984]从g^1_{k+1}构造的几何合点在这个P^{k+1}上形成了一个由100多条有理正态曲线组成的非退化构形。这证明了格林猜想[1984]的一个自然推广，即在这种情况下，几何合点应该跨越所有合点的空间。
Effective gonality theorem on weight-one syzygies of algebraic curves
本文针对Green-Lazarsfeld冈奇性猜想的有效版本未解决的问题，建立权一合冲的有效零化定理，证明特定线丛次数下猜想成立，且因平面曲线反例达到最优，给出猜想的完整解答。
Syzygies of secant varieties of curves of genus 2
本文利用Boij-Söderberg理论，通过计算Betti图的特殊位置确定贡献的纯图，完整计算了亏格2曲线的割线簇的Betti图，解决了权重k+1线性链长度未知的问题。
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 the secant variety of a curve
证明了2 g +3次以上线性正态平滑曲线的正割簇是算术Cohen-Macaulay数，并利用这一信息研究了正割簇的分次贝蒂数.
Geometric Syzygies of Mukai Varieties and General Canonical Curves with Genus at most 8
本文描述了截面亏格6，7，8的Mukai簇的最小秩最后合型空间。在此基础上，我们显示：1。亏格为6的一般标准曲线的第一几何合点在P^4中形成一个非退化的5条线的构型。2.亏格为7的一般标准曲线的第一几何合点在P^15中的旋量簇S上形成一个非退化的、线性法向的、84次直纹曲面。3.亏格为8的一般标准曲线的第二几何合点在P^20中的2重嵌入P^5上形成14条二次曲线的非退化构型。这证明了格林猜想[1984]的一个自然推广，即在这些情况下，几何合点应该跨越所有合点的空间。本文将结果1和3推广到数学AG/0108078中偶数亏格的一般曲线。结果2是本文的主要新结果。
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。
Syzygies and logarithmic vector fields along plane curves
研究了平面曲线$C$定义方程的雅可比理想的合矢与沿$C$的对数向量场层的稳定性、因子$C$的自由度以及$C$的Torelli性质（Dolgachev-Kapranov意义下）之间的关系.我们特别表明，曲线与少量的节点和尖点是Torelli在这个意义上。
These lectures discuss recent advances on syzygies on algebraic curves, especially concerning the Green, the Prym-Green and the Green-Lazarsfeld Secant Conjectures. The methods used are largely geometric and variational, with a special emphasis on examples and explicit calculations. The notes are based on series of lectures given in Daejeon (March 2013), Rome (November-December 2015) and Guanajuato (February 2016).
Linear syzygies of curves in weighted projective space
本文将Green的$N_p$条件推广到加权射影空间，证明曲线高次嵌入满足这些条件，并在广泛情形下建立了正性与低次虚拟合冲之间的联系。
The gonality conjecture on syzygies of algebraic curves of large degree
我们表明，一个小的变化所使用的方法Voisin在她的研究规范曲线导致一个令人惊讶的快速证明的gonality猜想的绿色和第二作者，声称可以读出的gonality曲线C从其决议中的嵌入定义的任何一个线丛足够大的程度。更一般地，我们建立了C上任意线丛所对应的模的权1合轨渐近为零的充要条件。
We prove two statements concerning the linear strand of the minimal free resolution of a curve of fixed gonality. Firstly, we show that a general curve C of genus g of non-maximal gonality k\leq (g+1)/2 satisfies Schreyer's Conjecture, that is, b_{g-k,1}(C,K_C)=g-k, and all its highest order linear syzygies are of Eagon-Northcott type. This is a statement going beyond Green's Conjecture and predicts that all highest order linear syzygies in the canonical embedding of C are determined by the syzygies of the (k-1)-dimensional scroll containing C. Secondly, we formulate an optimal effective versi…
On Higher Syzygies of ruled varieties over a curve
对于亏格为$g$的光滑投影曲线$C$上秩为$n+1$的向量丛$\mathcal{E}$，设$X=¶_C (\mathcal{E})$具有投影映射$π:X\to C$ .本文研究了$X$中齐次坐标环的最小自由分解。本文首先阐明了$X$上非常丰富的线丛的高合势与$π$的纤维被同一线丛Veronese嵌入的高合势之间的关系。更确切地说，让$H = \mathcal{O}_{¶_C (\mathcal{E})} (1)$重言式线丛，我们证明，如果$(¶^n,\mathcal{O}_{¶^n} (a))$满足性质$N_p$，那么$(X,aH+π^*B)$满足性质$N_p$的所有$B \in {Pic}C$有足够大的程度（定理\ref{thm:positive}）。并得到了性质$N_p$的${deg}(B)$的有效界（定理\ref{thm:1}，\ref{thm:2}，\ref{thm:3}和\ref{thm:4}）.对于相反的情况，我们得到了一些部分的答案（推论\ref{cor:negative}）。其次，利用这些结果，我们证明了一些Mukai型陈述。特别地，当${rank}(\mathcal{E}) \geq g$和$μ^- (\mathcal{E})$是整数时，Mukai的猜想对于$X$成立（推论\ref{cor:Mukai}）。最后对所有的$n$，构造了一个$n$维规则簇$X$和一个充…
Equations and syzygies of the first secant variety to a smooth curve
我们证明了如果C是一条线性正规平滑曲线，嵌入一个度至少为2g+3+p的线丛，则该曲线的割线簇满足N_{3,p} .
We apply the theory of Groebner bases to the computation of free resolutions over a polynomial ring, the defining equations of a canonically embedded curve, and the unirationality of the moduli space of curves of a fixed genus.
We introduce a relativization of the secant sheaves used by Ein, Green and Lazarsfeld and apply this construction to the study of syzygies of canonical curves. As a first application, we give a simpler proof of Voisin's Theorem for general canonical curves. This completely determines the terms of the minimal free resolution of the coordinate ring of such curves. Secondly, in the case of curves of even genus, we enhance Voisin's Theorem by providing a structure theorem for the last syzygy space, resolving the Geometric Syzygy Conjecture in even genus.
Syzygies of Curves in Products of Projective Spaces
本文将射影空间中曲线合冲的研究机制推广到射影空间乘积，证明了 Gruson–Peskine–Lazarsfeld 正则性界的一个类比，并建立了多分次预解式形状与正则性区域之间的新联系。
Representations on canonical models of generalized Fermat curves and their syzygies
本文研究了射影直线 $(\mathbb{Z}/k\mathbb{Z})^n$-覆盖的典范模型，显式确定了其齐次坐标环作为分次代数、群表示及分次模的结构，并给出一阶合冲模的生成元与不可约表示分解。
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.
Green's Conjecture for curves on arbitrary K3 surfaces
格林猜想预言，人们可以通过观察正则嵌入的合势，读出代数曲线上的特殊线性级数。我们将Voisin关于K3截面合zygies的结果推广到具有任意Picard晶格的K3曲面的情况。再加上Voisin和Hirschowitz-Ramanan的结果，为任意K3曲面上的平滑曲线的Green猜想提供了一个完整的解决方案。
We study the asymptotic non-vanishing of syzygies for products of projective spaces. Generalizing the monomial methods of Ein, Erman, and Lazarsfeld \cite{einErmanLazarsfeld16} we give an explicit range in which the graded Betti numbers of $\mathbb{P}^{n_1}\times \mathbb{P}^{n_2}$ embedded by $\mathcal{O}_{\mathbb{P}^{n_1}\times\mathbb{P}^{n_2}}(d_1,d_2)$ are non-zero. These bounds provide the first example of how the asymptotic syzygies of a smooth projective variety whose embedding line bundle grows in a semi-ample fashion behave in nuanced and previously unseen ways.
Syzygies of general projections of canonical and paracanonical curves
本文研究一般典范与副典范曲线沿一般点投影后的合冲消没性质，重点考察其是否由二次超曲面截出，并基于 Macaulay2 实验提出相关猜想。
We start the study of reduced complex projective plane curves, whose Jacobian syzygy module has 3 generators. Among these curves one finds the nearly free curves introduced by the authors, and the plus-one generated line arrangements introduced by Takuro Abe. All the Thom-Sebastiani type plane curves, and more generally, any curve whose global Tjurina number is equal to a lower bound given by A. du Plessis and C.T.C. Wall, are 3-syzygy curves. Rational plane curves which are nearly cuspidal, i.e. which have only cusps except one singularity with two branches, are also related to this class of …
We present a construction explaining the existence of (unexpected) curves of degree $d+k$, passing through a set $Z$ of points on $\mathbb{P}^2$, and having a generic point $P$ of multiplicity $d$. The construction is based on the syzygies of the $k$-th powers of Jacobian of the product of lines dual to the points of $Z$. We prove also a result characterizing the unexpectedness of the curves via splitting type of the bundle of these syzygies retricted to the line dual to $P$.
Let C \in P2 be a reduced, singular curve of degree d and equation f = 0. Let Σdenote the jacobian subscheme of C. We have 0 -> E -> 3.O -> I_Σ(d-1) -> 0 (the surjection is given by the partials of f). We study the relationships between the Betti numbers of the module H^0_*(E) and the integers, d; τ, where τ= deg(Σ). We observe that our results apply to any quasi-complete intersection of type (s; s; s).
First we give a sharp upper bound for the cardinal $m$ of a minimal set of generators for the module of Jacobian syzygies of a complex projective reduced plane curve $C$. Next we discuss the sharpness of an upper bound, given by A. du Plessis and C.T.C. Wall, for the global Tjurina number of such a curve $C$, in terms of its degree $d$ and of the minimal degree $r\leq d-1$ of a Jacobian syzygy. We give a homological characterization of the curves whose global Tjurina number equals the du Plessis-Wall upper bound, which implies in particular that for such curves the upper bound for $m$ is also …
Syzygies of Prym and paracanonical curves of genus 8
本文通过三个几何证明，揭示了Prym-Green猜想在亏格8的level 2情形下失败，并期望方法能推广至其他高2可除性亏格的例外情况。
We show that vanishing of asymptotic p-th syzygies implies p-very ampleness for line bundles on arbitrary projective schemes. For smooth surfaces we prove that the converse holds when p is small, by studying the Bridgeland-King-Reid-Haiman correspondence for tautological bundles on the Hilbert scheme of points. This extends previous results of Ein-Lazarsfeld, Ein-Lazarsfeld-Yang and gives a partial answer to some of their questions. As an application of our results, we show how to use syzygies to bound the irrationality of a variety.
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