Marian Aprodu, Gianluca Pacienza
We use the Brill-Noether theory to prove the Green conjecture for exceptional curves on K3 surfaces. Such curves count among the few ones having Clifford dimension at least three. We obtain our result by adopting an infinitesimal approach due to Pareschi, and using a degenerate version of a theorem due to Hirschowitz-Ramanan-Voisin.
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We give a very simple criterion in order to ensure that the Green-Griffiths locus of a projective manifold is the whole manifold. Next, we use it to show that the Green-Griffiths locus of any projective manifold uniformized by a bounded symmetric domain of rank greater than one is the whole manifold. In particular, this clarifies an old example given by M. Green to S. Lang.
Green's Conjecture states the following : syzygies of the canonical model of a curve are simple up to the p^th stage if and only if the Clifford index of C is greater than p. We prove that the generic curve of genus g satisfies Green's conjecture.
Green's canonical syzygy conjecture asserts a simple relationship between the Clifford index of a smooth projective curve and the shape of the minimal free resolution of its homogeneous ideal in the canonical embedding. We prove the analogue of this conjecture formulated by Bayer and Eisenbud for a class of non-reduced curves called ribbons. Our proof uses the results of Voisin and Hirschowitz-Ramanan on Green's conjecture for general smooth curves.
Green's conjecture for general covers
该研究证明了高克利福德维数曲线的几类覆盖满足格林合冲猜想，还发现皮卡数最小为9的Nikulin曲面截面曲线满足经典格林猜想但不满足Prym-格林猜想，给出了模空间R_g中格林猜想成立的明确轨迹。
Based on computeralgebra experiments we formulate a refined version of Green's conjecture and a conjecture of Schicho-Schreyer-Weimann which conjecturally also holds in positive characteristic. The experiments are done by using our Macaulay2 package, which constructs random canonically embedded curves of genus $g\leq 15$ over arbitrary small finite fields.
Let X be a smooth genus g curve equipped with a simple morphism f: X -> C, where C is either the projective line or more generally any smooth curve whose gonality is computed by finitely many pencils. Here we apply a method developed by Aprodu to prove that if g is big enough then X satisfies both Green and Green-Lazarsfeld conjectures. We also partially address the case in which the gonality of C is computed by infinitely many pencils.
Green's Conjecture predicts than one can read off special linear series on an algebraic curve, by looking at the syzygies of its canonical embedding. We extend Voisin's results on syzygies of K3 sections, to the case of K3 surfaces with arbitrary Picard lattice. This, coupled with results of Voisin and Hirschowitz-Ramanan, provides a complete solution to Green's Conjecture for smooth curves on arbitrary K3 surfaces.
We show that Generic Green's conjecture holds for generic binary curves, through a detailed analysis of the family of scrolls containing fixed rational normal curves.
Green's conjecture is proved for smooth curves C lying on a rational surface S with an anticanonical pencil, under some mild hypotheses on the line bundle L defined by C. Constancy of Clifford dimension, Clifford index and gonality of curves in the linear system |L| is also obtained.
We prove the existence of a full exceptional collection for the derived category of equivariant matrix factorizations of an invertible polynomial with its maximal symmetry group. This proves a conjecture of Hirano--Ouchi. In the Gorenstein case, we also prove a stronger version of this conjecture due to Takahashi. Namely, that the full exceptional collection is strong.
The gonality conjecture predicts that the gonality of a curve can be read off Koszul cohomology of line bundles of sufficiently large degree. We verify this conjecture for generic curves of odd genus. The even-genus case was previously solved in the joint work arXiv:math.AG/0301261.
Manin's Conjecture predicts the rate of growth of rational points of a bounded height after removing those lying on an exceptional set. We study whether the exceptional set in Manin's Conjecture is a thin set.
We prove the Green conjecture for generic curves of odd genus. That is we prove the vanishing $K_{k,1}(X,K_X)=0$ for $X$ generic of genus $2k+1$. The curves we consider are smooth curves $X$ on a K3 surface whose Picard group has rank 2. This completes our previous work, where the Green conjecture for generic curves of genus $g$ with fixed gonality $d$ was proved in the range $d\geq g/3$, with the possible exception of the generic curves of odd genus.
The Green-Griffiths-Lang conjecture says that for every complex projective algebraic variety $X$ of general type there exists a proper algebraic subvariety of $X$ containing all nonconstant entire holomorphic curves $f:\mathbb{C} \to X$. Using equivariant localisation on the Demailly-Semple jet differentials bundle we give an affirmative answer to this conjecture for generic projective hypersurfaces $X \subset \mathbb{P}^{n+1}$ of degree $\mathrm{deg}(X) \ge n^{9n}$.
The Green-Lazarsfeld secant conjecture predicts that the syzygies of a curve of sufficiently high degree are controlled by its special secants. We prove this conjecture for all curves of Clifford index at least two and not bielliptic and for all line bundles of a certain degree. Our proof is based on a classic result of Martens and Mumford on Brill-Noether varieties and on a simple vanishing criterion that comes from the interpretation of syzygies through symmetric products of curves.
In this short note we discuss the exceptional locus for the Lang-Vojta's conjecture in the case of the complement of two completely reducible hyperplane sections in a cubic surface. Using elementary methods, we show that generically the exceptional set is the union of the remaining 21 lines in the surface. We also describe examples in which the exceptional set is strictly larger.
A proof of generic Green's conjecture in odd genus
本文通过避免复杂计算，给出了Voisin关于奇数属一般曲线格林猜想的定理的新证明。
The Green-Griffiths-Lang conjecture says that for every complex projective algebraic variety $X$ of general type there exists a proper algebraic subvariety of $X$ containing all nonconstant entire holomorphic curves $f:\mathbb{C} \to X$. We construct a compactification of the invariant jet differentials bundle over complex manifolds motivated by an algebraic model of Morin singularities and we develop an iterated residue formula using equivariant localisation for tautological integrals over it. We show that the polynomial GGL conjecture for a generic projective hypersurface of degree $\mathrm{…
The Prym-Green conjecture in even genera up to 30
本文证明了20至30所有偶数亏格的Prym-Green猜想，并利用有理结点曲线与块Wiedemann算法完成证明。
This survey paper is based on my IMPANGA lectures given in the Banach Center, Warsaw in January 2011. We study the moduli of holomorphic map germs from the complex line into complex compact manifolds with applications in global singularity theory and the theory of hyperbolic algebraic varieties.
We prove new cases of Vojta's conjectures for surfaces in the context of function fields, with truncation equal to one and providing an effective explicit description of the exceptional set. We also prove a general and explicit result towards Campana's conjecture over complex function fields of curves. Our methods rely on a local study of $ω$-integral varieties.
Structure theorems for exceptional objects and exceptional collections of the bounded derived category of coherent sheaves on del Pezzo surfaces are established by Kuleshov and Orlov. In this paper we propose conjectures which generalize these results to weak del Pezzo surfaces. Unlike del Pezzo surfaces, an exceptional object on a weak del Pezzo surface is not necessarily a shift of a sheaf and is not determined by its class in the Grothendieck group. Our conjectures explain how these complications are taken care of by spherical twists, the categorification of $(-2)$-reflections acting on the…
The goal of this work is to study the existence and properties of non constant entire curves f drawn in a complex irreducible n-dimensional variety X, and more specifically to show that they must satisfy certain global algebraic or differential equations as soon as X is projective of general type. By means of holomorphic Morse inequalities and a probabilistic analysis of the cohomology of jet spaces, we are able to reach a significant step towards a generalized version of the Green-Griffiths-Lang conjecture.
Brill-Noether theory and Green's conjecture for general curves on simple abelian surfaces
本文改进Lazarsfeld式向量丛技术，计算简单阿贝尔曲面非本原线性系统中曲线的角亏与Brill-Noether轨迹维数，推导出阿贝尔曲面上曲线的一般Green猜想。
Bernardi and Tirabassi show the existence of full strong exceptional collections consisting of line bundles on smooth toric Fano $3$-folds under assuming Bondal's conjecture, which states that the Frobenius push-forward of the structure sheaf $\mc O_X$ generates the derived category $D^b(X)$ for smooth projective toric varieties $X$. In this article, we show Bondal's conjecture for smooth toric Fano $3$-folds and also improve their result, using birational geometry.
In this survey paper we study Manin's Conjecture from a geometric perspective. The focus of the paper is the recent conjectural description of the exceptional set in Manin's Conjecture due to Lehmann-Sengupta-Tanimoto. After giving an extensive background, we give a precise description of this set and compute it in many examples.
Consequences of the existence of exceptional collections in arithmetic and rationality
本文证明了Orlov猜想对一般域上的算术toric varieties成立，并通过构造有理数域上的反例否定了该猜想的推广版本。
Computing Jet Differentials and the Green-Griffiths-Lang Conjecture for Complements of Smooth Plane Curves
针对光滑平面曲线补集的Green-Griffiths-Lang猜想，本文开发了计算负扭曲不变对数2-喷流微分族的有效方法，给出该猜想的计算准则并对若干曲线族验证，明确了相关例外轨迹。
In this paper we deal with algebro-geometrical problems connected with testing S-duality conjecture for super-symmetric Yang-Mills quantum field theories in four dimensions. We describe all field configurations such that beta function coefficient for gauge groups of rank 1 and 2 is zero or negative. We also give special series of such field configurations for gauge groups of an arbitrary rank. Realization of one of the series discovers a connection with exceptional bundles. That points again at relation between S-duality and string duality.
We prove that two weakened forms of Green's conjectures for canonical curves are equivalent when the genus $g$ is odd.
We use Green's canonical syzygy conjecture for generic curves to prove that the Green-Lazarsfeld gonality conjecture holds for generic curves of genus g, and gonality d, if $g/3<d<[g/2]+2$.
A Zariski dense exceptional set in Manin's conjecture: dimension 2
本文构造了一个次数为1的 del Pezzo 曲面，其几何例外集是 Zariski 稠密的，从而提供了特征0下维度2的原始 Manin 猜想的第一个反例，并在特定假设下证明了存在无限多个此类反例。
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.
In this paper, we apply liaison theory to the Eisenbud-Green-Harris conjecture and prove that the conjecture holds for a certain subclass of homogeneous ideals in the linkage class of a complete intersection ideal. In the case of three variables, we prove that the conjecture holds for Gorenstein ideals.
Generalizing the well-known Green Conjecture on syzygies of canonical curves, Green and Lazarsfeld formulated in 1986 the Secant Conjecture predicting that a line bundle L of sufficiently high degree on a curve has a non-linear p-syzygy if and only if L fails to be (p+1)-very ample. Via lattice theory for special K3 surfaces, Voisin's solution of the classical Green Conjecture and calculations on moduli stacks of pointed curves, we prove: (1) The Green-Lazarsfeld Secant Conjecture in various degree of generality, including its strongest possible form in the divisorial case in the universal Jac…
We prove that the gonality among the smooth curves in a complete linear system on a $K3$ surface is constant except for the Donagi-Morrison example. This was proved by Ciliberto and Pareschi under the additional condition that the linear system is ample. As a consequence we prove that exceptional curves on $K3$ surfaces satisfy the Eisenbud-Lange-Martens-Schreyer conjecture and explicitly describe such curves. They turn out to be natural extensions of the Eisenbud-Lange-Martens-Schreyer examples of exceptional curves on $K3$ surfaces.
The geometric exceptional set in Manin's conjecture for Batyrev and Tschinkel's example
针对Batyrev和Tschinkel提出的Fano 5-维Fermat三次曲面丛X，在特征0域上显式构造有限薄映射，证明其生成的薄集与Lehmann-Sengupta-Tanimoto猜想的Manin猜想例外集一致。
The main purpose in this paper is to study the gonality, the Clifford index and the Clifford dimension on linearly equivalent smooth curves on Enriques surfaces. The method is similar to techniques of M.Green $\&$ R.Lazarsfeld and G.Pareschi.
This is the text of my lecture (in french) at the Bourbaki Seminar (november 2003) on the proof by Claire Voisin of the Green conjecture for a generic curve. This conjecture predicts the structure of the minimal resolution of the ideal of a canonically embedded curve in terms of the existence of certain special linear systems on the curve.
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.
We extend the theory of Koszul modules to the bi-graded case, and prove a vanishing theorem that allows us to show that the Canonical Ribbon Conjecture of Bayer and Eisenbud holds over a field of characteristic zero or at least equal to the Clifford index. Our results confirm a conjecture of Eisenbud and Schreyer regarding the characteristics where the generic statement of Green's conjecture holds. They also recover and extend to positive characteristics results due to Aprodu and Voisin asserting that Green's Conjecture holds for generic curves of each gonality.
Difference varieties and the Green-Lazarsfeld Secant Conjecture
通过引入差异簇的方法，证明了任意亏格g曲线在除子情形下的Green-Lazarsfeld割线猜想。
Given a family of smooth complex projective varieties, the Hodge conjecture predicts the algebraicity of the locus of Hodge classes. This was proven unconditionnally by Cattani, Deligne and Kaplan in 1995. In a similar way, conjectures on algebraic cycles have led Green and Griffiths to conjecture the algebraicity of the zero locus of normal functions. This corresponds to a mixed version of the theorem of Cattani, Deligne and Kaplan. This result has been proven recently by Brosnan-Pearlstein, Kato-Nakayama-Usui, and Schnell building on work of M. Saito. We will present some of the ideas around…
We prove new instances of Simpson's rigidity conjecture which states that quasi-unipotent rigid local systems should be motivic. We construct new relative motives over the fourfold punctured Riemann sphere which give rise to $G_2$-rigid local systems which are not rigid in the group $\GL_7.$
We exhibit approximately fifty Betti diagrams of free resolutions of rings of smooth, connected canonical curves of genera $9$-$14$ in prime characteristics between $2$ and $11$. Generic Green's conjecture is verified for genera $9$ and $10$ for characteristics $2$, $5$, $7$, and $11$.
Proof of a Conjecture on the Sequence of Exceptional Numbers, Classifying Cyclic Codes and APN Functions
本文证明了一个分类例外数的猜想，指出每个例外数具有 $2^i+1$ 或 $4^i-2^i+1$ 的形式，该猜想同时关联密码学中的APN函数与编码理论中的二元循环码。
The relative Green-Griffiths-Lang conjecture for families of varieties of maximal Albanese dimension
本文将Green-Griffiths-Lang猜想推广至相对情形，并利用Nevanlinna理论中的截断第二主定理型估计，证明了该猜想的强形式对极大Albanese维数簇族成立。
Classification of full exceptional collections of line bundles on three blow-ups of $\mathbb{P}^{3}$
本文针对三维射影空间在点、直线或扭曲三次曲线处爆破所得簇，证明了 Kuznetsov 全性猜想，并给出长度为 6 的线丛全例外收集的显式分类。
Generic Green's Conjecture and Generic Geometric Syzygy Conjecture in Positive Characteristic
本文研究正特征中典范曲线的合冲，在 p≥(g+4)/2 时重新证明一般 Green 猜想，并在 p>g 时确立一般偶亏格曲线最后合冲空间的几何合冲猜想关键情形。
The Prym-Green Conjecture predicts that the resolution of a generic n-torsion paracanonical curve of every genus is natural. The conjecture has mostly been studied so far for level 2, that is, for Prym-canonical curves. Using a construction of Barth and Verra that realizes torsion bundles on sections of special K3 surfaces, we prove the Prym-Green Conjecture for curves of odd genus g and torsion bundles of sufficiently high order with respect to g. We also give partial results in even genus. In the process, we confirm the expectation of Barth and Verra concerning the number of curves in a fixe…
The Green-Griffiths-Lang conjecture stipulates that for every projective variety X of general type over C, there exists a proper algebraic subvariety of X containing all non constant entire curves f : C $\rightarrow$ X. Using the formalism of directed varieties, we prove here that this assertion holds true in case X satisfies a strong general type condition that is related to a certain jet-semistability property of the tangent bundle TX . We then give a sufficient criterion for the Kobayashi hyperbolicity of an arbitrary directed variety (X,V). This work is dedicated to the memory of Professor…
The algebraic Green-Griffiths-Lang conjecture for complements of very general pairs of divisors
本文证明了 $\mathbb{P}^n$ 中总次数为 $2n$ 的非常一般除子对之补集在模去真闭子簇后代数双曲，从而为 Lang-Vojta 与 Green-Griffiths 猜想提供新证据，并部分推广了已有结果。
We prove a strong vanishing result for finite length Koszul modules, and use it to derive Green's conjecture for every g-cuspidal rational curve over an algebraically closed field k with char(k) = 0 or char(k) >= (g+2)/2. As a consequence, we deduce that the general canonical curve of genus g satisfies Green's conjecture in this range. Our results are new in positive characteristic, whereas in characteristic zero they provide a different proof for theorems first obtained in two landmark papers by Voisin. Our strategy involves establishing two key results of independent interest: (1) we describ…
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 $$\frac{g(C)}{3}+1\leq {\rm gon(C)}\leq\frac{g(C)}{2}+1.$$
Quasi-étale covers of Du Val del Pezzo surfaces and Zariski dense exceptional sets in Manin's conjecture
本文构造了 Manin 猜想中具有 Zariski 稠密例外集的奇异 del Pezzo 曲面首批例子（次数为1、2和3），通过分类 Du Val del Pezzo 曲面的拟平展覆盖并研究群作用排除了高次情况，提供了系统构造方法。
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。
King's conjecture states that on every smooth complete toric variety $X$ there exists a strongly exceptional collection which generates the bounded derived category of $X$ and which consists of line bundles. We give a counterexample to this conjecture. This example is just the Hirzebruch surface $\mathbb{F}_2$ iteratively blown up three times, and we show by explicit computation of cohomology vanishing that there exist no strongly exceptional sequences of length 7.
We construct indecomposable cycles in the motivic cohomology group $H^3_{\mathcal M}(A,{\mathbb Q}(2))$ where $A$ is an Abelian surface over a number field or the function field of a base. When $A$ is the self product of the universal elliptic curve over a modular curve, these cycles can be used to prove algebraicity results for values of higher Green's functions, similar to a conjecture of Gross, Kohnen and Zagier. We formulate a conjecture which relates our work with the recent work of Bruinier-Ehlen-Yang on the conjecture of Gross-Kohnen-Zagier.
An almost perfect nonlinear (APN) function (necessarily a polynomial function) on a finite field $\mathbb{F}$ is called exceptional APN, if it is also APN on infinitely many extensions of $\mathbb{F}$. In this article we consider the most studied case of $\mathbb{F}=\mathbb{F}_{2^n}$. A conjecture of Janwa-Wilson and McGuire-Janwa-Wilson (1993/1996), settled in 2011, was that the only exceptional monomial APN functions are the monomials $x^n$, where $n=2^i+1$ or $n={2^{2i}-2^i+1}$ (the Gold or the Kasami exponents respectively). A subsequent conjecture states that any exceptional APN function …
Let X be a geometrically smooth n-dimensional projective algebraic complex hypersurface in P^{n+1}(C). Using Green-Griffiths jets, we establish the existence of nonzero global algebraic differential equations that must be satisfied by every nonconstant entire holomorphic curve C -> X if X is of general type, namely if its degree d satisfies the optimal possible lower bound: d >= n + 3. The case n = 2 dates back to Green-Griffiths 1979, while according to very recent advances (Invent. Math. 180, pp. 161-223, February 2010), the best (and only) lower degree bound known previously in arbitrary di…
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