Gonality and Clifford Index of curves on $K3$ Surfaces

光滑aCM曲线在$\mathbb{P}^3$中四次曲面的第二亏约数

Second gonality of smooth aCM curves on quartic surfaces in $\mathbb{P}^3$

研究光滑aCM曲线在$\mathbb{P}^3$中四次曲面的第二亏约数，通过曲线上的网计算其亏约数，确定最小的线性系参数。

Green's Conjecture for curves on rational surfaces with an anticanonical pencil

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.

Max Noether's Theorem for Integral Curves

We generalize, for integral curves, a celebrated result of Max Noether on global sections of the n-dualizing sheaf of a smooth nonhyperelliptic curve. This is our main result. We also obtain an embedding of a non-Gorenstein curve in a way that we can express the dimensions of the components of the ideal in terms of the main invariants of the curve. Afterwards we focus on gonality, Clifford index and Koszul cohomology of non-Gorenstein curves by allowing torsion free sheaves of rank 1 in their definitions. We find an upper bound for the gonality, which agrees with Brill-Noether's one for a rati…

Very special divisors on 4-gonal real algebraic curves

Given a real curve, we study special linear systems called "very special" for which the dimension does not satisfy a Clifford type inequality. We classify all these very special linear systems when the gonality of the curve is small.

Exceptional linear systems on curves on Enriques surfaces

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.

A note on Harris Morrison sweeping families of maximal gonality

Harris and Morrison constructed semistable families f:F \to Y of k-gonal curves of genus g such that for every k the corresponding modular curves give a sweeping family in the k-gonal locus in the moduli space. Their construction depends on the choice of a smooth curve X. We show that if the genus g(X) is sufficiently high with respect to g, then the ratio K_F^2 / χ(O_F) is 8 asymptotically with respect to g(X). We show also that if the gonality is maximal and some conjectured estimates of Harris and Morrison hold, the slope of the fibration f: F\to Y is 12 asymptotically with respect to g and…

The Mercat Conjecture for stable rank 2 vector bundles on generic curves

It has been a long-standing problem to find an adequate definition of a Clifford index for higher rank vector bundles on curves, which should capture the complexity of the curve in its moduli space. An interesting proposal in rank 2 has been put forward by Mercat, who conjectured that the second Clifford index of a curve should be equal to its classical Clifford index, defined in terms of gonality. Using moduli of sheaves on generic K3 surfaces, we prove Mercat's conjecture for generic curves of every genus. Furthermore, for odd g we identify an effective divisor in the moduli space M_g along …

Counter-examples of high Clifford index to Prym-Torelli

We give examples of curves of arbitrarily high Clifford index such that the Prym map is not injective at any of their étale double covers.

Prym曲线的Prym-canonical Clifford指数与维度

The Prym-canonical Clifford index

本文提出了Prym曲线的两个新不变量，通过分类和计算展示了其在双曲曲线中的应用。

Clifford index for reduced curves

We extend the notion of Clifford index to reduced curves with planar singularities by considering rank 1 torsion free sheaves. We investigate the behaviour of the Clifford index with respect to the combinatorial properties of the curve and we show that Green's conjecture holds for certain classes of curves given by the union of two irreducible components.

Bundles of rank 2 with small Clifford index on algebraic curves

In this paper, we remove a technical hypothesis from a previous paper. This allows us to construct stable bundles of rank 2 with Clifford index significantly smaller than the classical Clifford index on suitably chosen curves of any genus $\ge13$.

Brill-Noether theory of curves on Enriques surfaces, II. The Clifford index

We complete our study of linear series on curves lying on an Enriques surface by showing that, with the exception of smooth plane quintics, there are no exceptional curves on Enriques surfaces, that is, curves for which the Clifford index is not computed by a pencil.

Bundles of rank 3 on curves of Clifford index 3

Two definitions of the Clifford index for vector bundles on a smooth projective curve $C$ of genus $g\ge4$ were introduced in a previous paper by the authors. In another paper the authors obtained results on one of these indices for bundles of rank 3. In this paper we extend these results in the case where $C$ has classical Clifford index 3. In particular we prove Mercat's conjecture for bundles of rank 3 for $g \leq 8$ and $g \geq 13$ when $C$ has classical Clifford index 3. We obtain complete results in the case of genus 7.

The slope of fibred surfaces: unitary rank and Clifford index

We prove new slope inequalities for relatively minimal fibred surfaces, showing an influence of the relative irregularity, of the unitary rank and of the Clifford index on the slope. The argument uses Xiao's method and a new Clifford-type inequality for subcanonical systems on non-hyperelliptic curves.

On bundles of rank 3 computing Clifford indices

Let $C$ be a smooth irreducible projective algebraic curve defined over the complex numbers. The notion of the Clifford index of $C$ was extended a few years ago to semistable bundles of any rank. Recent work has been focussed mainly on the rank-2 Clifford index, although interesting results have also been obtained for the case of rank 3. In this paper we extend this work, obtaining improved lower bounds for the rank-3 Clifford index. This allows the first computations of the rank-3 index in non-trivial cases and examples for which the rank-3 index is greater than the rank-2 index.

Lazarsfeld-Mukai bundles on K3 surfaces associated to a pencil computing Clifford index

Let $X$ be a smooth projective K3 surface over complex numbers and $C$ be an ample curve on $X$. In this paper we will study the semistability of the Lazarsfeld-Mukai bundle $E_{C, A}$ associated to a line bundle $A$ ion $C$ such that $|A|$ is a pencil on $C$ and computes the Clifford index of $C$. We give a necessary and sufficient condition for $E_{C, A}$ being semistable.

Gonality of algebraic curves and graphs

We study the interplay between the classical theory of linear series on curves, and the recent theory of linear series on graphs. We prove that every d-gonal (weighted) graph of Hurwitz type is the dual graph of a d-gonal curve. Conversely the dual graph of a d-gonal curve is equivalent to a d-gonal graph. We define d-gonal graphs by what we call harmonic indexed morphisms. Generalizations to higher dimensional linear series, and applications to tropical curves and hyperelliptic graphs are given.

高亏格曲线上典范ribbon的合冲

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。

关于 du Val 线性系统的一个注记

A remark on du Val linear systems

本文证明了在 Halphen 曲面上的 du Val 线性系统中，光滑曲线的 Clifford 指数为常数，并在相关情形下确定了其度数由 Weierstrass 线性系实现。

A generalization of the Clifford index and determinantal equations for curves and their secant varieties

This is the author's 2008 thesis from the University of Chicago. We generalize the notion of the Clifford index to an arbitrary very ample line bundle and show how it determines when a curve and its various secant varieties have determinantal equations. In particular we establish the scheme theoretic version of a conjecture of Eisenbud-Koh-Stillman. If $L_1$ and $L_2$ are line bundles of degree at least $2g+1+k$, then ${\rm Sec}^j(C)$ is determinantally presented for $j \leq k$ and for $j \leq k+1$ if $L_1 \neq L_2$. We also give a geometric characterization of the standard Clifford index in t…

关于奇异曲线的Clifford维数

On Clifford dimension for singular curves

本文通过允许无挠层参与计算来扩展Clifford指标，研究积分曲线的Clifford维数，并在单项式情形下给出组合公式，应用于Clifford维数为2的情况。

曲线上的线性系：稳定性与 Clifford 指数

Linear series on curves: stability and Clifford index

本文研究光滑复曲线上线性系的稳定性概念，揭示对偶跨度丛稳定性与线性稳定性的联系，得到与 Clifford 指数相关的稳定性条件，并构造出积分斜率 3 的稳定向量丛及其 theta 因子。

Normal generation and Clifford index

Let $C$ be a smooth curve of genus $g\ge 4$ and Clifford index $c$. In this paper, we prove that if $C$ is neither hyperelliptic nor bielliptic with $g\ge 2c+5$ and $\mathcal M$ computes the Clifford index of $C$, then either $°\mathcal M\le \frac{3c}{2}+3$ or $|\mathcal M|=|g^1_{c+2}+h^1_{c+2}|$ and $g=2c+5$. This strengthens the Coppens and Martens' theorem (\cite{CM}, Corollary 3.2.5). Furthermore, for the latter case (1) $\mathcal M$ is half-canonical unless $C$ is a $\frac{c+2}{2}$-fold covering of an elliptic curve, (2) $\mathcal M(F)$ fails to be normally generated with $\cli(\mathcal M…

Clifford indices for vector bundles on curves

For smooth projective curves the Clifford index is an important invariant which provides a bound for the dimension of the space of sections of a line bundle. This is the first step in distinguishing curves of the same genus. In this paper we generalise this to introduce Clifford indices for semistable vector bundles on curves. We study these invariants, giving some basic properties and carrying out some computations for small ranks and for general and some special curves. For curves whose classical Clifford index is two, we compute all values of our new Clifford indices.

Higher rank Clifford indices of curves on a K3 surface

Let $(X,H)$ be a polarized K3 surface with $\mathrm{Pic}(X) = \mathbb Z H$, and let $C\in |H|$ be a smooth curve of genus $g$. We give an upper bound on the dimension of global sections of a semistable vector bundle on $C$. This allows us to compute the higher rank Clifford indices of $C$ with high genus. In particular, when $g\geq r^2\geq 4$, the rank $r$ Clifford index of $C$ can be computed by the restriction of Lazarsfeld-Mukai bundles on $X$ corresponding to line bundles on the curve $C$. This is a generalization of the result by Green and Lazarsfeld for curves on K3 surfaces to higher ra…

Xiao's conjecture for general fibred surfaces

We prove that the genus $g$, the relative irregularity $q_f$ and the Clifford index $c_f$ of a non-isotrivial fibration $f$ satisfy the inequality $q_f \leq g - c_f$. This gives in particular a proof of Xiao's conjecture for fibrations whose general fibres have maximal Clifford index.

On an example of Mukai

In this note we use an example of Mukai to construct semistable bundles of rank 3 with 6 independent sections on a curve of genus 9 or 11 with Clifford index strictly less than the Clifford index of the curve. The example also allows us to show the non-emptiness of some Brill-Noether loci with negative expected dimension.

Generalized Lazarsfeld-Mukai bundles and a conjecture of Donagi and Morrison

Let S be a K3 surface and assume for simplicity that it does not contain any (-2)-curve. Using coherent systems, we express every non-simple Lazarsfeld-Mukai bundle on S as an extension of two sheaves of some special type, that we refer to as generalized Lazarsfeld-Mukai bundles. This has interesting consequences concerning the Brill-Noether theory of curves C lying on S. From now on, let g denote the genus of C and A be a complete linear series of type g^r_d on C such that d<= g-1 and the corresponding Brill-Noether number is negative. First, we focus on the cases where A computes the Cliffor…

The topological period-index problem over 6-complexes

By comparing the Postnikov towers of the classifying spaces of projective unitary groups and the differentials in a twisted Atiyah-Hirzebruch spectral sequence, we deduce a lower bound on the topological index in terms of the period, and solve the topological version of the period-index problem in full for finite CW complexes of dimension at most 6. Conditions are established that, if they were met in the cohomology of a smooth complex 3-fold variety, would disprove the ordinary period-index conjecture. Examples of higher-dimensional varieties meeting these conditions are provided. We use our …

Linear series on semistable curves

The dimension of spaces of global sections for line bundles on semistable curves parametrized by the compactified Picard scheme is studied. The theorem of Riemann is shown to hold. The theorem of Clifford is shown to hold in the following cases: the curve has two components; the curve is any semistable curve and the degree is either 0 or 2g-2; the curve is stable, free from separating nodes, and the degree is at most 4. These results are all shown to be sharp. Applications to the Clifford index, to the combinatorial description of hyperelliptic curves, and to plane quintics are given.

代数曲线的线性稳定性与秩二 Clifford 指标及其应用

Linear stability and rank two Clifford indices of algebraic curves with applications

本文证明计算秩二 Clifford 指标的向量丛线性半稳定，给出线性稳定条件，并应用于相干系统，验证 Butler 猜想在特定范围内成立。

K3 projective models in scrolls

We study the projective models of complex K3 surfaces polarized by a line bundle L such that all smooth curves in |L| have non-general Clifford index. Such models are in a natural way contained in rational normal scrolls. We use this study to classify and describe all projective models of K3 surfaces of genus at most 10.

On the Xiao conjecture for plane curves

Let $f: S\longrightarrow B$ be a non-trivial fibration from a complex projective smooth surface $S$ to a smooth curve $B$ of genus $b$. Let $c_f$ the Clifford index of the generic fibre $F$ of $f$. In [arXiv:1401.7502v4] it is proved that the relative irregularity of $f$, $q_f=h^{1,0}(S)-b$ is less than or equal to $g(F)-c_f$. In particular this proves the (modified) Xiao's conjecture: $q_f\le 1+g(F)/2$ for fibrations of general Clifford index. In this short note we assume that the generic fiber of $f$ is a plane curve of degree $d\ge 5$ and we prove that $q_f\le g(F)-c_f-1$. In particular we …

Linear stability of projected canonical curves with applications to the slope of fibred surfaces

Let f :S\to B be a non locally trivial fibred surface. We prove a lower bound for the slope of f depending increasingly from the relative irregularity of f and the Clifford index of the general fibres.

Brill-Noether special cubic fourfolds of discriminant 14

We study the Brill-Noether theory of curves on K3 surfaces that are Hodge theoretically associated to cubic fourfolds of discriminant 14. We prove that any smooth curve in the polarization class has maximal Clifford index and deduce that a cubic fourfold contains disjoint planes if and only if it admits a Brill-Noether special associated K3 surface of degree 14. As an application, the complement of the pfaffian locus, inside the Noether-Lefschetz divisor of discriminant 14 in the moduli space of cubic fourfolds, is contained in the irreducible locus of cubic fourfolds containing two disjoint p…

Canonical syzygies of smooth curves on toric surfaces

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.

On hyperplane sections of K3 surfaces

Let C be a Brill-Noether-Petri curve of genus g\geq 12. We prove that C lies on a polarized K3 surface, or on a limit thereof, if and only if the Gauss-Wahl map for C is not surjective. The proof is obtained by studying the validity of two conjectures by J. Wahl. Let I_C be the ideal sheaf of a non-hyperelliptic, genus g, canonical curve. The first conjecture states that, if g\geq 8, and if the Clifford index of C is greater than 2, then H^1(P^{g-1}, I_C^2(k))=0, for k\geq 3. We prove this conjecture for g\geq 11. The second conjecture states that a Brill-Noether-Petri curve of genus g\geq 12 …