Stability and locally exact differentials on a curve

Homology of Hilbert schemes of points on a locally planar curve

Let C be a proper, integral, locally planar curve, and consider its Hilbert schemes of points C^[n]. We define 4 creation/annihilation operators acting on the rational homology groups of these Hilbert schemes and show that the operators satisfy the relations of a Weyl algebra. The action of this algebra is similar to that defined by Grojnowski and Nakajima for a smooth surface. As a corollary, we compute the cohomology of C^[n] in terms of the cohomology of the compactified Jacobian of C together with an auxiliary grading on the latter. This recovers and slightly strenghtens a formula recently…

整合法平面曲线上点的希尔伯特概型的半正交不可分解性

Semiorthogonal indecomposability for Hilbert schemes of points on integral locally planar curves

该研究针对代数闭域上具局部平面奇点的整射影曲线的点希尔伯特概型，证明其相关范畴的半正交不可分解性，建立了相对结论，结果适用于任意特征。

Hilbert schemes of points on a locally planar curve and the Severi strata of its versal deformation

Let C be a locally planar curve. Its versal deformation admits a stratification by the genera of the fibres. The strata are singular; we show that their multiplicities at the central point are determined by the Euler numbers of the Hilbert schemes of points on C. These Euler numbers have made two prior appearances. First, in certain simple cases, they control the contribution of C to the Pandharipande-Thomas curve counting invariants of three-folds. In this context, our result identifies the strata multiplicities as the local contributions to the Gopakumar-Vafa BPS invariants. Second, when C i…

Locally Cohen-Macaulay space curves defined by cubic equations and globally generated vector bundles

We classify globally generated vector bundles with first Chern class $c_1$ at least 4 on the projective 3-space with the property that $E(-c_1+3)$ has a non-zero global section. This (seemingly) technical result allows one to reduce the classification of globally generated vector bundles with $c_1$ at most 7 on the projective 3-space to the classification of stable rank-2 reflexive sheaves with the same properties. The proof is based on a description of the monads of all locally Cohen-Macaulay space curves defined by cubic equations. We extend then this kind of classification to higher dimensi…

Curves in the double plane

We study locally Cohen-Macaulay curves in projective three-space which are contained in a double plane 2H, thus completing the classification of curves lying on surfaces of degree two. We describe the irreducible components of the Hilbert schemes of locally Cohen-Macaulay curves in 2H of given degree and arithmetic genus. We show that these Hilbert schemes are connected. We also discuss the Rao modules of these curves, and liaison and biliaison equivalence classes.

Homology of Hilbert schemes of reducible locally planar curves

Let $C$ be a complex, reduced, locally planar curve. We extend the results of Rennemo arXiv:1308.4104 to reducible curves by constructing an algebra $A$ acting on $V=\bigoplus_{n\geq 0} H_*(C^{[n]}, \mathbb{Q})$, where $C^{[n]}$ is the Hilbert scheme of $n$ points on $C$. If $m$ is the number of irreducible components of $C$, we realize $A$ as a subalgebra of the Weyl algebra of $\mathbb{A}^{2m}$. We also compute the representation $V$ in the simplest reducible example of a node.

局部平面整曲线的紧化雅可比的上同调

Cohomology of compactified Jacobians for locally planar integral curves

本文研究局部平面整曲线紧化雅可比的上同调，探讨Ngô支持定理、perverse滤波及与希尔伯特方案的联系。

Smooth curves specialize to extremal curves

Let $H_{d,g}$ denote the Hilbert scheme of locally Cohen-Macaulay curves of degree $d$ and genus $g$ in projective three space. We show that, given a smooth irreducible curve $C$ of degree $d$ and genus $g$, there is a rational curve $\{[C_t]: t \in \mathbb{A}^1\}$ in $H_{d,g}$ such that $C_t$ for $t \neq 0$ is projectively equivalent to $C$, while the special fibre $C_0$ is an extremal curve. It follows that smooth curves lie in a unique connected component of $H_{d,g}$. We also determine necessary and sufficient conditions for a locally Cohen-Macaulay curve to admit such a specialization to …

基于极大曲线的具有多个恢复集的局部可恢复码

Locally recoverable codes with multiple recovering sets from maximal curves

本文利用具有许多有理点的代数曲线和可分态射，提出了一种构造具有多个恢复集的局部可恢复码（LRCs）的方法，推广并增强了现有框架，纠正了文献中的不准确之处，实现了更好的参数、更广的适用性及大可用性。

The Hilbert schemes of locally Cohen-Macaulay curves in P^3 may after all be connected

Progress on the problem whether the Hilbert schemes of locally Cohen-Macaulay curves in projective 3 space are connected has been hampered by the lack of an answer to a question that was raised by Robin Hartshorne in his paper "On the connectedness of the Hilbert scheme of curves in projective 3 space" Comm. Algebra 28 (2000) and more recently in the open problems list of the 2010 AIM workshop Components of Hilbert Schemes available at http://aimpl.org/hilbertschemes: does there exist a flat irreducible family of curves whose general member is a union of d disjoint lines on a smooth quadric su…

Locally recoverable codes from algebraic curves and surfaces

A locally recoverable code is a code over a finite alphabet such that the value of any single coordinate of a codeword can be recovered from the values of a small subset of other coordinates. Building on work of Barg, Tamo, and Vlăduţ, we present several constructions of locally recoverable codes from algebraic curves and surfaces.

A positive proportion of locally soluble hyperelliptic curves over $\mathbb Q$ have no point over any odd degree extension

A hyperelliptic curve over $\mathbb Q$ is called "locally soluble" if it has a point over every completion of $\mathbb Q$. In this paper, we prove that a positive proportion of hyperelliptic curves over $\mathbb Q$ of genus $g\geq 1$ are locally soluble but have no points over any odd degree extension of $\mathbb Q$. We also obtain a number of related results. For example, we prove that for any fixed odd integer $k > 0$, the proportion of locally soluble hyperelliptic curves over $\mathbb Q$ of genus $g$ having no points over any odd degree extension of $\mathbb Q$ of degree at most $k$ tends …

On the equidistribution of totally geodesic submanifolds in compact locally symmetric spaces and application to boundedness results for negative curves and exceptional divisors

We prove an equidistribution result for totally geodesic submanifolds in a compact locally symmetric space. In the case of Hermitian locally symmetric spaces, this gives a convergence theorem for currents of integration along totally geodesic subvarieties. As a corollary, we obtain that on a complex surface which is a compact quotient of the bidisc or of the 2-ball, there is at most a finite number of totally geodesic curves with negative self intersection. More generally, we prove that there are only finitely many exceptional totally geodesic divisors in a compact Hermitian locally symmetric …

On the motive of the Quot scheme of finite quotients of a locally free sheaf

Let $X$ be a smooth variety, $E$ a locally free sheaf on $X$. We express the generating function of the motives $[\textrm{Quot}_X(E,n)]$ in terms of the power structure on the Grothendieck ring of varieties. This extends a recent result of Bagnarol, Fantechi and Perroni for curves, and a result of Gusein-Zade, Luengo and Melle-Hernández for Hilbert schemes. We compute this generating function for curves and we express the relative motive $[\textrm{Quot}_{\mathbb A^d}(\mathscr{O}^{\oplus r}) \to \textrm{Sym}\, \mathbb A^d]$ as a plethystic exponential.

On locally analytic vectors of the completed cohomology of modular curves

We study the locally analytic vectors in the completed cohomology of modular curves and determine the eigenvectors of a rational Borel subalgebra of $\mathfrak{gl}_2(\mathbb{Q}_p)$. As applications, we prove a classicality result for overconvergent eigenforms of weight one and give a new proof of the Fontaine-Mazur conjecture in the irregular case under some mild hypotheses. For an overconvergent eigenform of weight $k$, we show its corresponding Galois representation has Hodge-Tate-Sen weights $0,k-1$ and prove a converse result.

加权形式的初始理想与局部Cohen-Macaulay曲线的亏格

Initial ideals of weighted forms and the genus of locally Cohen-Macaulay curves

研究局部Cohen-Macaulay曲线的最大亏格问题，证明当曲线度数d等于s或d≥2s-1时，预测的亏格最大值成立，基于对非标准格雷度多项式环中特定齐次形式的初始理想猜想的证明。

Questions of Connectedness of the Hilbert Scheme of Curves in ${\mathbb P}^3$

We review the present state of the problem, for each degree $d$ and genus $g$, is the Hilbert scheme of locally Cohen--Macaulay curves in ${\mathbb P}^3$ connected?

Curves on a Double Surface

Let F be a smooth surface in a smooth projective threefold T, and let X=2F be the first infinitesimal neighborhood of X in T. A locally Cohen-Macaulay curve C in X gives rise to two effective divisors on F, namely the curve part P of the intersection of C and F, and the curve R residual in C to this intersection. We show that a general deformation of R on F lifts to a deformation of C on X when a certain cohomology group vanishes. In our paper "Hilbert Schemes of Degree Four Curves" we use this result to prove the connectedness of the Hilbert schemes H(4,g) of locally Cohen-Macaulay space curv…

Faisceaux cohérents sur les courbes multiples

Multiple primitive curves have been defined and studied by A. Banica and O. Forster. We introduce new invariants for coherent sheaves on these curves and define quasi locally free sheaves, which play the same role as locally free sheaves on smooth varieties. A more detailed study of torsion free sheaves is given for primitive curves of multiplicity 2.

The maximum genus problem for locally Cohen-Macaulay space curves

Let $P_{\text{MAX}}(d,s)$ denote the maximum arithmetic genus of a locally Cohen-Macaulay curve of degree $d$ in $\mathbb{P}^3$ that is not contained in a surface of degree $<s$. A bound $P(d, s)$ for $P_{\text{MAX}}(d,s)$ has been proven by the first author in characteristic zero and then generalized in any characteristic by the third author. In this paper, we construct a large family $\mathcal{C}$ of primitive multiple lines and we conjecture that the generic element of $\mathcal{C}$ has good cohomological properties. With the aid of \emph{Macaulay2} we checked the validity of the conjecture…

椭圆曲线上的局部非原始点

Locally imprimitive points on elliptic curves

研究在椭圆曲线上的局部非原始点现象，通过构造非平凡例子展示全局原始点可能在局部失效的情况。

Locally recoverable codes on algebraic curves

A code over a finite alphabet is called locally recoverable (LRC code) if every symbol in the encoding is a function of a small number (at most r) other symbols. A family of linear LRC codes that generalize the classic construction of Reed-Solomon codes was constructed in a recent paper by I. Tamo and A. Barg. In this paper we extend this construction to codes on algebraic curves. We give a general construction of LRC codes on curves and compute some examples, including asymptotically good families of codes derived from the Garcia- Stichtenoth towers. The local recovery procedure is performed …

Normal bundles of rational curves in projective spaces

We determine the splitting (isomorphism) type of the normal bundle of a generic genus-0 curve with 1 or 2 components in any projective space, as well as the (sometimes nontrivial) way the bundle deforms locally with a general deformation of the curve. We deduce an enumerative formula for divisorial loci of smooth rational curves whose normal bundle is of non-generic splitting type.

Smoothable locally Cohen--Macaulay and non Cohen--Macaulay multiple structures on curves

In this article we show that a wide range of multiple structures on curves arise whenever a family of embeddings degenerates to a morphism $φ$ of degree $n$. One could expect to see, when an embedding degenerates to such a morphism, the appearance of a locally Cohen-Macaulay multiple structure of certain kind (a so-called rope of multiplicity $n$). We show that this expectation is naive and that both locally Cohen-Macaulay and non Cohen-Macaulay multiple structures might occur in this situation. In seeing this we find out that many multiple structures can be smoothed. When we specialize to the…

Stability of sheaves of locally closed and exact forms

For any smooth projective variety $X$ of dimension $n$ over an algebraically closed field $k$ of characteristic $p>0$ with $μ(Ω^1_X)>0$. If ${\rm T}^{\ell}(Ω^1_X)$ ($0<\ell<n(p-1)$) are semi-stable, then the sheaf $B^1_X$ of exact 1-forms is stable. When $X$ is a surface with $μ(Ω^1_X)>0$ and $Ω^1_X$ is semi-stable, the sheaf $B^2_X$ of exact 2-forms is also stable. Moreover, under the same condition, the sheaf $Z^1_X$ of closed 1-forms is stable when $p>3$, and $Z^1_X$ is semi-stable when $p=3$.

Codes with locality from cyclic extensions of Deligne-Lusztig curves

Recently, Skabelund defined new maximal curves which are cyclic extensions of the Suzuki and Ree curves. Previously, the now well-known GK curves were found as cyclic extensions of the Hermitian curve. In this paper, we consider locally recoverable codes constructed from these new curves, complementing that done for the GK curve. Locally recoverable codes allow for the recovery of a single symbol by accessing only a few others which form what is known as a recovery set. If every symbol has at least two disjoint recovery sets, the code is said to have availability. Three constructions are descr…

Incidence systems on Cartesian powers of algebraic curves

We show that a reduct of the Zariski structure of an algebraic curve which is not locally modular interprets a field, answering a question of Zilber's.

不可约射影曲线上的稳定性条件

Stability conditions on irreducible projective curves

本文研究了不可约射影曲线上的稳定性条件，对光滑曲线进行了分类，并提出正则弱稳定性条件的概念以完善稳定性流形的边界。

Osculating spaces and diophantine equations (with an appendix by Pietro Corvaja and Umberto Zannier)

This paper deals with some classical problems about the projective geometry of complex algebraic curves. We call \textit{locally toric} a projective curve that in a neighbourhood of every point has a local analytical parametrization of type $(t^{a_1},...,t^{a_n})$, with $a_1,..., a_n$ relatively prime positive integers. In this paper we prove that the general tangent line to a locally toric curve in $\bP^3$ meets the curve only at the point of tangency. This result extends and simplifies those of the paper \cite{kaji} by H.Kaji where the same result is proven for any curve in $\bP^3$ such that…

Set theoretic complete intersection for curves in a smooth affine algebra

It is proved in this paper that a locally complete intersection curve in a smooth affine C-algebra with trival conormal bundle is a set theoretic complete intersection if its corresponding class in the Grothendieck Group is torsion.

覆盖构造的局部刚性与曲线模空间的Weil–Petersson子簇

Local rigidity of covering constructions and Weil--Petersson subvarieties of the moduli space of curves

本文证明了曲线模空间$\u039c_{g,n}$的Weil–Petersson全测地子簇及覆盖构造是局部刚性的，该结论由一类轨形映射的一般刚性结果推导得到。