Abel Castorena, Alberto López Martín, Montserrat Teixidor i Bigas
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
We prove the Bertram-Feinberg-Mukai conjecture for a generic curve $C$ of genus $g$ and a semistable vector bundle $E$ of rank two and determinant $K$ on $C$, namely we prove the injectivity of the Petri-canonical map $S^2(H^0(E))\to H^0(S^2(E))$.
This paper establishes the correctness of a conjecture of Bertram-Feinberg and Mukai for a special class of globally generated rank-two bundles with canonical determinant over a generic Riemann surface of genus at least four.
Let C be a generic curve, E a generic vector bundle on C. Then, for every line bundle on C the twisted Petri map P:H^0(C,L\otimes E)\otimes H^0(C, K\otimes L^*\otimes E^{*})--> H^0(C, K) is injective.
A proof of Petri's general conjecture on the unobstructedness of linear systems on a general curve is proposed, using only the local properties of the deformation space of the pair (curve, line bundle).
We construct pointed Prym-Brill-Noether varieties parametrizing line bundles assigned to an irreducible étale double covering of a curve with a prescribed minimal vanishing at a fixed point. We realize them as degeneracy loci in type D and deduce their classes in case of expected dimension. Thus, we determine a pointed Prym-Petri map and prove a pointed version of the Prym-Petri theorem implying that the expected dimension holds in the general case. These results build on work of Welters and De Concini-Pragacz on the unpointed case. Finally, we show that Prym varieties are Prym-Tyurin varietie…
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…
Ranks of maps of vector bundles
本文将针对线丛的技巧推广至向量丛，并利用该方法证明了向量丛Petri map的 injectivity 及与Poincaré sheaves形变理论相关 map 的 surjectivity。
In this article we study Brill-Noether loci of moduli space of stable bundles over smooth surfaces. We define Petri map as an analogy with the case of curves. We show the non-emptiness of certain Brill-Noether loci over very general quintic hypersurface in $\mathbb{P}^3$, and use the Petri map to produce components of expected dimension.
We study the Saito-Ikeda infinitesimal invariant of the ccle defined by curves in their jacobians using rank (k+1) vector bundles and we give a criterion for which the higher cycle class map is not trivial. When k=2, this turns out to be strictly linked to the Petri map for vector bundles: an explicit construction on a curve of genus g>9 shows the existence of a non trivial element in the higher Griffiths group.
Recall that a smooth complex projective curve has a very ample canonical bundle when it is non-hyperelliptic, and according to a theorem of M. Noether the resulting embedding is projectively normal. A theorem of Petri further asserts that the homogeneous ideal is generated by quadrics if the curve is neither trigonal nor a smooth plane quintic. In this note, we prove an analogue of Noether's theorem for the symmetric square of the curve - namely, the canonical bundle of the symmetric square determines a projectively normal embedding exactly when the curve itself is neither hyperelliptic, trigo…
A genus-g du Val curve is a degree-3g plane curve having 8 points of multiplicity g, one point of multiplicity g-1, and no other singularity. We prove that the corank of the Gauss-Wahl map of a general du Val curve of odd genus (>11) is equal to one. This, together with the results of [1], shows that the characterisation of Brill-Noether-Petri curves with non-surjective Gauss-Wahl map as hyperplane sections of K3 surfaces and limits thereof, obtained in [3], is optimal.
This paper contains results on stable bundles of rank 2 with space of sections of dimension 4 on a smooth irreducible projective algebraic curve $C$. There is a known lower bound on the degree for the existence of such bundles; the main result of the paper is a geometric criterion for this bound to be attained. For a general curve $C$ of genus 10, we show that the bound cannot be attained, but that there exist Petri curves of this genus for which the bound is sharp. We interpret the main results for various curves and in terms of Clifford indices and coherent systems.
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…
Let $C$ be a Petri general curve of genus $g$ and $E$ a general stable vector bundle of rank $r$ and slope $g-1$ over $C$ with $h^0 (C, E) = r+1$. For $g > (2r+2)(2r+1)$, we show how the bundle $E$ can be recovered from the tangent cone to the theta divisor $Θ_E$ at ${\mathcal O}_C$. We use this to give a constructive proof and a sharpening of Brivio and Verra's theorem that the theta map $SU_C (r) -rightarrow |r Θ|$ is generically injective for large values of $g$.
Projectivity of good moduli spaces of vector bundles on stacky curves
本文利用良模空间存在性判据证明栈曲线上半稳定向量丛模栈具有射影良模空间，并给出线丛无基点性的有效界，从而为抛物丛模空间提供了新的有效构造。
The purpose of this note is to extend Beilinson and Drinfeld's "very good" property to moduli stacks of parabolic vector bundles on curves of genuses $g = 0$ and $g = 1$. Beilinson and Drinfeld show that for $g > 1$ a trivial parabolic structure is sufficient for the moduli stacks to be "very good". We give a necessary and sufficient condition on the parabolic structure for this property to hold in the case of lower genus.
We prove the Kobayashi-Hitchin correspondence between good wild harmonic bundles and polystable good filtered $λ$-flat bundles satisfying a vanishing condition. We also study the correspondence for good wild harmonic bundles with the homogeneity with respect to a group action, which is expected to provide another way to construct Frobenius manifolds.
This paper is devoted to the study of the Higgs bundle associated with the universal abelian variety over the good reduction of a Shimura curve of PEL type. Due to the endomorphism structure, the Higgs bundle decomposes into the direct sum of Higgs subbundles of rank two. They are basically divided into two type: uniformizing type and unitary type. As the first application we obtain the mass formula counting the number of geometric points of the degeneracy locus in the Newton polygon stratification. We show that each Higgs subbundle is Higgs semistable. Furthermore, for each Higgs subbundle of…
In "Quantization of Hitchin's Integrable System and Hecke Eigensheaves", Beilinson and Drinfeld introduced the "very good" property for a smooth complex equidimensional stack. They prove that for a semisimple complex group G, the moduli stack of G-bundles over a smooth complex projective curve X is "very good", as long as X has genus g > 1. In the case of the projective line, when g = 0, this is not the case. However, the result can sometimes be extended to the projective line by introducing additional parabolic structure at a collection of marked points and slightly modifying the definition o…
Moduli of Anti-Invariant Higgs Bundles
本文研究反不变 Higgs 丛的模空间，借助 Alper、Halpern-Leistner 和 Heinloth 的近期结果证明半稳定对象具有分离的好模空间，并顺带给出非 GIT 证明及 Hitchin 系统本征性。
We find an algorithm to compute the cohomology groups of spherical vector bundles on complex projective K3 surfaces, in terms of their Mukai vectors. In many good cases, we give significant simplifications of the algorithm. As an application, when the Picard rank is one, we show a numerical condition that is equivalent to weak Brill-Noether for a spherical vector bundle.
In this paper we continue our study of the moduli space of stable bundles of rank two and degree 1 on a very general quintic surface. The goal in this paper is to understand the irreducible components of the moduli space in the first case in the "good" range, which is $c_2=10$. We show that there is a single irreducible component of bundles which have seminatural cohomology, and conjecture that this is the only component for all stable bundles.
Generalized intersection pairings on moduli spaces of vector bundles over a curve
本文针对带紧合好模空间且稳定部分非空的Artin栈引入广义相交配对概念，通过建立壁交叉公式比较了光滑射影曲线上半稳定丛模栈上四种已知的该类配对构造，并给出低秩情形的显式计算。
We prove formal GAGA for good moduli space morphisms under an assumption of "enough vector bundles" (which holds for instance for quotient stacks). This supports the philosophy that though they are non-separated, good moduli space morphisms largely behave like proper morphisms.
We use the G-invariant non-degenerate form on the Steinberg module to Frobenius split the cotangent bundle of a flag variety in good prime characteristics. This was previously only known for the general linear group. Applications are a vanishing theorem for pull back of line bundles to the cotangent bundle (proved for the classical groups and G_2 by Andersen and Jantzen and in characteristic zero by B. Broer (for all groups)), normality and rational singularities for the subregular nilpotent variety and good filtrations of the global sections of pull backs of line bundles to the cotangent bund…
We define the notion of a parahoric group scheme $\mathcal G$ over a smooth projective curve, and formulate four conjectures on the structure of the stack of $\mathcal G$-bundles, which generalize to this case well-known results on $G$-bundles with $G$ a constant reductive group. The conjectures concern the set of connected components, the uniformization by affine flag varieties of twisted loop groups, the Picard groups, and the space of global sections of a dominant line bundle. Since a first version of this paper was circulated, Heinloth [arXiv:0711.4450] has proved a good part of these conj…
On positivity of CM line bundles on the moduli space of klt good minimal models with $κ=1$
研究了具有κ=1的klt良好极小模型模空间上CM线包的正性，证明了其正则化后为 ample，并构造了数值等价类的模空间以克服先前方法的局限性。
We prove a rigidity result for automorphisms of points of certain stacks admitting adequate moduli spaces. It encompasses as special cases variations of the moduli of $G$-bundles on a smooth projective curve for a reductive algebraic group $G$. For example, our result applies to the stack of semistable $G$-bundles, stacks of semistable Hitchin pairs, and stacks of semistable parabolic $G$-bundles. Similar arguments apply to Gieseker semistable $G$-bundles in higher dimensions. We present two applications of the main result. First, we show that in characteristic $0$ every stack of semistable de…
Line bundles on perfectoid covers: case of good reduction
本文研究完美oid空间的Picard群与Picard函子，证明良约化极限覆盖的Picard函子可由特殊纤维表示，并借此解决若干公开问题，同时构造了$\mathbb G_m$的Hodge--Tate谱序列。
On the canonical bundle formula and effective birationality for Fano varieties in char $p>0$
本文研究了正特征下 Fano 型簇的双有理几何与有界性问题，证明了有界性在正规化下不变，给出了更易用的典范丛公式，并证实了具有良好奇点的弱 Fano 簇的有效双有理性。
For a stable irreducible curve $X$ and a torsion free sheaf $L$ on $X$ of rank one and degree $d$, D.S. Nagaraj and C.S. Seshadri ([NS]) defined a closed subset $\Cal U_X(r,L)$ in the moduli space of semistable torsion free sheaves of rank $r$ and degree $d$ on $X$. We prove that $\Cal U_X(r,L)$ is irreducible, when a smooth curve $Y$ specializes to $X$ and a line bundle $\Cal L$ on $Y$ specializes to $L$, the specialization of moduli space of semistable rank $r$ vector bundles on $Y$ with fixed determinant $\Cal L$ has underlying set $\Cal U_X(r,L)$. For rank 2 and 3, we show that there is a …
We describe a satisfactory theory of degeneration of quadratic forms in three variables in the most general setting possible: the quadratic forms are defined on rank 3 vector bundles over an arbitrary scheme and could have values in nontrivial line bundles. Our results extend what is known for good forms; for example we show that the Witt-invariant suffices for classification. We determine explicitly the general, special and usual orthogonal groups and present applications. We indicate examples of rank 4 vector bundles that do not admit any Azumaya structures and of rank 3 vector bundles that …
Let $φ\colon X\rightarrow Y$ be a degree two Galois cover of smooth curves over a local field $F$ of odd characteristic. Assuming that $Y$ has good reduction, we describe a semi-stability criterion for the curve $X$, using the data of the branch locus of the covering $φ$. In the case that $X$ has semi-stable reduction, we describe the dual graph of the minimal regular model of $X$ over $F.$ We do this by adopting the notion of cluster picture defined for hyperelliptic curves for the case where $Y$ is not necessarily a rational curve. Using these results, we describe the variation of the p-adic…
Projectivity of the moduli space of vector bundles on a curve
本文针对亏格 $g\geq 2$ 曲线上半稳定向量丛模空间的射影性，结合好模空间与行列式线丛技术给出现代证明路径。
Let X be an irreducible smooth projective curve of genus at least two over an algebraically closed field k of characteristic p>0. In this paper we study the natural stratification, defined using the absolute Frobenius of X, on the moduli space of vector bundles on X of suitable rank. In characteristic two we provide a complete classification of rank two semi-stable vector bundles whose Frobenius pull-back is not semi-stable. We also obtain fairly good information about the strata of the Frobenius stratification, including the irreducibility and the dimension of each non-empty Frobenius stratum…
In this article we will construct a universal moduli space of stable parabolic vector bundles over the moduli space of marked Deligne-Mumford stable curves $\overline{M}_{_{g, n}}$. The objects that appear over the boundary of $\overline{M}_{_{g, n}}$ i.e., over singular curves will remain vector bundles. The total space and the fibers over $\overline{M}_{_{g, n}}$ will have good singularities.
Moduli of $\mathcal{G}$-bundles under nonconnected group schemes and nondensity of essentially finite bundles
该研究证明了光滑射影曲线上非连通约化群概形的主$\u003cmathcal{G}\u003e$-丛存在射影良好模空间，且模栈可分解为有限子栈，进而证明除环面群或亏格小于2的曲线外，0次半稳定$G$-丛模空间中本质有限$G$-丛不稠密。
Non-pluripolar products on vector bundles and Chern--Weil formulae
本文在向量丛奇异度量上建立非多重次调和积理论，引入 $\mathcal{I}$-良好奇点并推导 Chern–Weil 型公式，进而用 Riemann–Zariski 空间上的相交理论重新表述该公式。
We develop a theory of arithmetic characteristic classes of (fully decomposed) automorphic vector bundles equipped with an invariant hermitian metric. These characteristic classes have values in an arithmetic Chow ring constructed by means of differential forms with certain log-log type singularities. We first study the cohomological properties of log-log differential forms, prove a Poincaré lemma for them and construct the corresponding arithmetic Chow groups. Then we introduce the notion of log-singular hermitian vector bundles, which is a variant of the good hermitian vector bundles introdu…
Moduli spaces of twisted equivariant G-bundles over a curve
本文构造了紧黎曼曲面上多稳定(θ,c)-扭曲Γ-等变G-丛的粗模空间，推广了Ramanathan的G-丛模空间构造，并给出非连通群Ĝ-丛模空间的GIT构造。
In char $k = p >0$, A. Langer proved a strong restriction theorem (in the style of H. Flenner) for semistable sheaves to a very general hypersurface of degree $d$, on certain varieties, with the condition that `char $k > d$'. He remarked that to remove this condition, it is enough to answer either of the following questions affirmatively: {\it For the syzygy bundle $\sV_d$ of ${\mathcal O}(d)$, is $\sV_d$ semistable for arbitrary $n, d$ and $p = {char} k$?, or is there a good estimate on $μ_{max}(\sV_d^*)$?} Here we prove that (1) the bundle $\sV_d$ is semistable, for a certain infinite set of…
We prove that, for a good meromorphic flat bundle with poles along a divisor with normal crossings, the restriction of the irregularity complex to each natural stratum of this divisor only depends on the formal flat bundle along this stratum. This answers a question raised by J.-B. Teyssier. This version takes into account the published erratum.
Cohomological $χ$-independence for moduli of one-dimensional sheaves and moduli of Higgs bundles
本文证明了一维半稳定层和半稳定Higgs丛模空间的交上同调独立于欧拉特征，证实了相关猜想，并给出了结合Ngô支撑定理等工具的证明。
This paper considers curves in Grassmannians which are themselves immersed in projective space by the Plucker map. It is shown that for a generic vector bundle of high enough degree, the image curve lies in a proper linear subvariety of this projective space and satisfies good conditions on syzygies as a curve in this subspace. For very small degree and generic vector bundle, the curve is non- degenerate.
Vanishing results for the coherent cohomology of automorphic vector bundles over the Siegel variety in positive characteristic
本文证明了正特征Siegel簇上自守向量丛的凝聚上同调在特定权条件下的消失结果，并提供了Sage算法以显式计算。
In this paper we deal with polarizations on a nodal curve $C$ with smooth components. Our aim is to study and characterize a class of polarizations, which we call "good", for which depth one sheaves on $C$ reflect some properties that hold for vector bundles on smooth curves. We will concentrate, in particular, on the relation between the $\underline{w}$-stability of $\mathcal{O}_C$ and the goodness of $\underline{w}$. We prove that these two concepts agree when $C$ is of compact type and we conjecture that the same should hold for all nodal curves.
The cone conjecture for vector-valued Siegel automorphic forms
本文证明了特征$p$下秩为$n$的Siegel模簇上自守向量丛的有效锥由$G$-zip叠编码，证明了零阶上同调群在zip锥外消失，并在$n=3$时给出显式消失条件，同时定义了平凡型自守形式。
Hitchin systems, higher Gaudin operators and $r$-matrices
我们适应希钦的可积系统的情况下，穿孔曲线。在$\CC P^{1}$和$SL_{n}$ -丛的情况下，它们等价于Garnier研究的系统。相应的量子系统由B.费金，E. Frenkel和N.高丁系统的列舍提欣。利用R. Goodman和N. Wallach关于仿射代数临界水平包络代数的中心。最后，我们构造了一个打孔椭圆曲线上Hitchin系统的动力学$r$ -矩阵和$GL_{n}$ -丛，以及（对于$n=2$）相应的量子系统。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。