Marian Aprodu
We apply a degenerate version of a result due to Hirschowitz, Ramanan and Voisin to verify Green and Green-Lazarsfeld conjectures over explicit open sets inside each $d$-gonal stratum of curves $X$ with $d<[g_X/2]+2$. By the same method, we verify the Green-Lazarsfeld conjecture for any curve of odd genus and maximal gonality. The proof invokes Voisin's solution to the generic Green conjecture as a key argument.
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We show that for $5$-gonal curves of odd genus $g\geq 13$ and even genus $g\geq 28$ the $\lceil \frac{g-1}{2}\rceil$-th syzygy module of the curve is not determined by the syzygies of the scroll swept out by the special pencil of degree $5$.
Ein and Lazarsfeld have shown that one can read off the gonality of an algebraic curve from its syzygies in the embedding defined by any one line bundle of sufficiently large degree. This note extends their approach and shows that the gonality can be detected from the syzygies of an embedding by any line bundle of degree at least 4g-3.
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。
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…
We show that a small variant of the methods used by Voisin in her study of canonical curves leads to a surprisingly quick proof of the gonality conjecture of Green and the second author, asserting that one can read off the gonality of a curve C from its resolution in the embedding defined by any one line bundle of sufficiently large degree. More generally, we establish a necessary and sufficient condition for the asymptotic vanishing of the weight one syzygies of the module associated to an arbitrary line bundle on C.
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。
We verify Green's conjecture for a generic $k$-gonal curve of genus $g$, for $g\geq k(k-1)/2$.
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 develop in this article an algorithm that, given a projective curve $C$, computes a \textit{gonal map}, that is, a finite morphism from $C$ to the projective line of minimal degree. Our method is based on the computation of scrollar syzygies of canonical curves. We develop an improved version of our algorithm for curves with a unique gonal map and we discuss a characterization of such curves in terms of Betti numbers. Finally, we derive an efficient algorithm for radical parametrization of curves of gonality $\le 4$.
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.
A note on an effective bound for the gonality conjecture
本文改进了gonality猜想中的有效界，将线束次数从4g-3降至4g-4，并指出两个例外情形。
The present paper is related to a conjecture made by Green and Lazarsfeld concerning 1-linear syzygies of curves embedded by complete linear systems of sufficiently large degrees. Given a smooth, irreducible, complex, projective curve $X$, we prove that the least integer $q$ for which the property $(M_q)$ fails for a line bundle $L$ on $X$ does not depend on $L$ as soon as its degree becomes sufficiently large. Consequently, this number is an invariant of the curve, and the statement of Green-Lazarsfeld's conjecture is equivalent to saying that this invariant equals the gonality of the curve. …
We describe a systematic way of constructing effective divisors on the moduli space of stable curves of genus g having exceptionally small slope. We prove that any divisor on \bar{M}_g consisting of curves failing a certain Green-Lazarsfeld syzygy type condition, provides a counterexample to the Harris-Morrison Slope Conjecture. These divisors generalize our original isolated counterexample to the Slope Conjecture which was the divisor on M_{10} of curves lying on K3 surfaces. We also introduce a new stratification of M_g, somewhat similar to the classical stratification given by gonality, but…
In this note, we give a slight improvement of a result of A. Küronya and V. Lozovanu about higher syzygies on abelian surfaces.
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。
Syzygies and Koszul modules in geometry
本文综述了Koszul模和余式在代数几何中的最新进展，包括对Chen秩、Green猜想等猜想的探讨及开放问题。
For all $2\leq q\leq \dim(X)$ and most relevant $p$ values, the dimension of the asymptotic Koszul cohomology group $K_{p,q}(X, B;L_d)$ grows exponentially with $d$.
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…
We discuss recent progress on syzygies of curves, including proofs of Green's and Gonality Conjectures as well as applications of Koszul cycles to the study of the birational geometry of various moduli spaces of curves. We prove a number of new results, including a complete solution to Green's Conjecture for arbitrary hexagonal curves. Finally, we propose several new conjectures on syzygies, including a Prym-Green conjecture for l-roots of trivial bundles as well as a strong Maximal Rank Conjecture for generic curves. To appear in the Proceedings of Clay Mathematical Institute.
In 1967, Shioda \cite{Shi1} determined the ring of invariants of binary octavics and their syzygies using the symbolic method. We discover that the syzygies determined in \cite{Shi1} are incorrect. In this paper, we compute the correct equations among the invariants of the binary octavics and give necessary and sufficient conditions for two genus 3 hyperelliptic curves to be isomorphic over an algebraically closed field $k$, $\ch k \neq 2, 3, 5, 7$. For the first time, an explicit equation of the hyperelliptic moduli for genus 3 is computed in terms of absolute invariants.
Recently, Barraza-Rojas have described the action of the full automorphisms group on the Fermat curve of degree $p$, for $p$ a prime integer, and obtained the group algebra decomposition of the corresponding Jacobian variety. In this short note we observe that the factors in such a decomposition are given by the Jacobian varieties of certain $p$-gonal curves.
We consider the generic Green conjecture on syzygies of a canonical curve, and particularly the following reformulation thereof: {\it For a smooth projective curve $C$ of genus $g$ in characteristic 0, the condition ${\rm Cliff} C>l$ is equivalent to the fact that $K_{g-l'-2,1}(C,K_C)=0, \forall l'\leq l$.} We propose a new approach, which allows up to prove this result for generic curves $C$ of genus $g(C)$ and gonality ${\rm gon(C)}$ in the range $$\frac{g(C)}{3}+1\leq {\rm gon(C)}\leq\frac{g(C)}{2}+1.$$
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