Margherita Lelli-Chiesa
We show that the Brill-Noether locus M^3_{18,16} is an irreducible component of the Gieseker-Petri locus in genus 18 having codimension 2 in the moduli space of curves. This result disproves a conjecture predicting that the Gieseker-Petri locus is always divisorial.
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We use tropical techniques to prove a case of the Gieseker-Petri Theorem. Specifically, we show that the general curve of arbitrary genus does not admit a Gieseker-Petri special pencil.
The paper has been withdrawn by change of content and some errors in the examples.
The Gieseker-Petri locus GP_g is defined as the locus inside M_g consisting of curves which violate the Gieseker-Petri Theorem. It is known that GP_g has always some divisorial components and it has been conjectured that it is of pure codimension 1 inside M_g. We prove that this holds true for genus up to 13.
We show that certain divisors of Brill-Noether and Gieseker-Petri type span extremal rays of the effective cone in the moduli space of stable genus one curves with $n$ ordered marked points. In particular, they are different from the infinitely many extremal rays found in \cite{ChenCoskun}.
We develop a framework to apply tropical and nonarchimedean analytic techniques to multiplication maps on linear series and study degenerations of these multiplications maps when the special fiber is not of compact type. As an application, we give a tropical criterion for a curve over a valued field to be Gieseker-Petri general.
We prove a smoothness result for spaces of linear series with prescribed ramification on twice-marked elliptic curves. In characteristic 0, we then apply the Eisenbud-Harris theory of limit linear series to deduce a new proof of the Gieseker-Petri theorem, along with a generalization to spaces of linear series with prescribed ramification at up to two points. Our main calculation involves the intersection of two Schubert cycles in a Grassmannian associated to almost-transverse flags.
We analyze the stratification of the moduli space S_g of spin curves of genus g given by the dimension of the theta-characteristic. Using the relation between gaussian maps and the strata S_g^r, we construct "regular" components of S_g^r having expected codimension r(r+1)/2 inside S_g. We also relate moduli spaces of pointed curves with a moving spin structure to the classical Gieseker-Petri loci in M_g. We show that the locus of curves for which the Gieseker-Petri theorem fails for a pencil is always a divisor on M_g. Finally, we give a sufficient criterion for the injectivity of Gaussian map…
We perform an intersection theoretic study of the rational map between two different moduli spaces of stable curves which associates to a curve its corresponding Brill-Noether locus (in the case this locus has virtual dimension 1). We then use these results to describe the cone of moving divisors on M_g. Several other applications to moduli spaces of Prym varieties are presented. In a different direction, we prove that the locus in M_g of curves failing to satisfy the Gieseker-Petri theorem is supported in codimension 1 for every possible type of linear series.
In this note the geography of minimal surfaces of general type admitting $\mathbb{Z}_2^2$-actions is studied. More precisely, it is shown that Gieseker's moduli space $\mathfrak{M}_{K^2,χ}$ contains surfaces admitting a $\mathbb{Z}_2^2$-action for every admissible pair $(K^2, χ)$ such that $2χ-6\leq K^2\leq 8χ-8$ or $K^2=8χ$. The examples considered allow to prove that the locus of Gorenstein stable surfaces is not closed in the KSBA-compactification $\overline{\mathfrak{M}}_{K^2,χ}$ of Gieseker's moduli space $\mathfrak{M}_{K^2,χ}$ for every admissible pair $(K^2, χ)$ such that $2χ-6\leq K^2\…
We show that on a generic curve and under some conditions on the degree and genus, there exists a component B of the locus of stable vector bundles of rank r and degree d with at least k sections of the expected dimension such that for a generic vector bundle E of B, the Petri map is injective.
Let $\M_g$ be the course moduli space of complex projective nonsingular curves of genus $g$. We prove that when the Brill-Noether number $ρ(g,1,n)$ is non-negative the Petri locus $P^1_{g,n}\subset \M_g$ has a divisorial component whose closure has a non-empty intersection with $Δ_0$. In order to prove the result we show that the scheme $G^1_n(Γ)$ that parametrizes degree $n$ pencils on a curve $Γ$ is isomorphic to a component of the Hilbert scheme parametrizing certain curves on the symmetric square $Γ_2$ of $Γ$ and we study the properties of such a family of curves.
Let C be a smooth projective complex curve of genus $g\geq2$. We investigate the Brill-Noether locus consisting of stable bundles of rank 2 and canonical determinant having at least $k$ independent sections. Using the Hecke correpondence we construct a fundamental class, which determines the non-emptiness of this locus at least when $C$ is a Petri curve. We prove that in many expected cases the Brill-Noether locus is non-empty. For some values of $k$ the result is best possible.
Let $X \subset \mathbb P^3$ be a very general sextic surface over complex numbers. In this paper we study certain Brill-Noether problems for moduli of rank $2$ stable bundles on $X$ and its relation with rank $2$ weakly Ulrich and Ulrich bundles. In particular, we show the non-emptiness of certain Brill-Noether loci and using the geometry of the moduli and the notion of the Petri map on higher dimensional varieties, we prove the existence of components of expected dimension. We also give sufficient conditions for the existence of rank $2$ weakly Ulrich bundles $\mathcal E$ on $X$ with $c_1(\ma…
The symplectic Brill--Noether locus ${\mathcal S}_{2n, K}^k$ associated to a curve $C$ parametrises stable rank $2n$ bundles over $C$ with at least $k$ sections and which carry a nondegenerate skewsymmetric bilinear form with values in the canonical bundle. This is a symmetric determinantal variety whose tangent spaces are defined by a symmetrised Petri map. We obtain upper bounds on the dimensions of various components of ${\mathcal S}_{2n, K}^k$. We show the nonemptiness of several ${\mathcal S}_{2n, K}^k$, and in most of these cases also the existence of a component which is generically smo…
On the local structure of the Brill-Noether locus of locally free sheaves on a smooth variety
本文研究光滑簇上局部自由层的Brill-Noether轨迹的局部结构，通过分析无穷小形变函子Def_E^k及对(E,U)的形变，刻画了切锥、切空间，并将凝聚系统模空间的经典结果推广到任意秩和任意维数。
Local structure of theta divisors and related loci of generic curves
本文研究一般曲线 theta 除子及 Brill-Noether 轨迹的局部结构，证明在 Petri 映射单射条件下轨迹局部 étale 同构于切锥，并由此解决单值性猜想等问题。
Brill-Noether loci
本文通过研究椭圆曲线链邻域中的非奇异曲线，重新证明了 Brill-Noether 轨迹在特定余维数范围内的非空性及 Petri 映射的满射性，并推广了同余维数轨迹互异性的已知结果。
Smoothness results for the schemes of special divisors on general k-gonal curves
本文研究一般k-边曲线C上的特殊除子方案W^r_d(C)的光滑性，通过Brill-Noether退化方案的细化，证明其在特定开子集上光滑，并探讨invertible sheaves的Petri映射性质。
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