A Simple Proof that Rational Curves on K3 are Nodal

Hirota簇与有理节点曲线

Hirota Varieties and Rational Nodal Curves

本文研究了由有理节点曲线导出的Hirota簇，定义了其主分支，证明其为不可约分支等价于求解弱Schottky问题，并利用计算工具解决了直到亏格9的该问题。

Two rational nodal quartic threefolds

We prove that the quartic threefolds defined by $$ \sum_{i=0}^{5}x_i=\sum_{i=0}^{5}x_i^4-t\left(\sum_{i=0}^{5}x_i^2\right)^2=0 $$ in $\mathbb{P}^5$ are rational for $t=\frac{1}{6}$ and $t=\frac{7}{10}$.

The integral Chow ring of the stack of at most 1-nodal rational curves

We give a presentation for the stack of rational curves with at most 1 node as the quotient by GL(3) of an open set in a 6-dimensional irreducible representation. We then use equivariant intersection theory to calculate the integral Chow ring of this stack.

Which quartic double solids are rational?

We study the rationality problem for nodal quartic double solids. In particular, we prove that nodal quartic double solids with at most six singular points are irrational, and nodal quartic double solids with at least eleven singular points are rational.

Rigid isotopy classification of generic rational quintics in $\mathbb{R}\mathbb{P}^{2}$

In this article we obtain the rigid isotopy classification of generic rational curves of degre $5$ in $\mathbb{R}\mathbb{P}^{2}$. In order to study the rigid isotopy classes of nodal rational curves of degree $5$ in $\mathbb{R}\mathbb{P}^{2}$, we associate to every real rational nodal quintic curve with a marked real nodal point a nodal trigonal curve in the Hirzebruch surface $Σ_3$ and the corresponding nodal real dessin on $\mathbb{C}\mathbb{P}^{1}/(z\mapsto\bar{z})$. The dessins are real versions, proposed by S. Orevkov, of Grothendieck's dessins d'enfants. The dessins are graphs embedded i…

Positive Rational Nodal Leaves on Surfaces

We consider singular holomorphic foliations on compact complex surfaces with invariant rational nodal curve of positive self-intersection. Then, under some assumptions, we list all possible foliations.

Enumeration of One-Nodal Rational Curves in Projective Spaces

We give a formula computing the number of one-nodal rational curves that pass through an appropriate collection of constraints in a complex projective space. We combine the methods and results from three different papers.

On the geometry of Abel maps for nodal curves

In this paper we give local conditions to the existence of Abel maps for nodal curves that are limits of Abel maps for smooth curves. We use this result to construct Abel maps for any degree for nodal curves with two components.

The degree-2 Abel-Jacobi map for nodal curves - II

We construct a resolution of the degree-2 Abel-Jacobi map for a regular smoothing of a nodal curve.

Nodal curves and components of the Hilbert scheme of curves in $\mathbb {P}^r$ with the expected number of moduli

We study the existence of components with the expected number of moduli of the Hilbert scheme of integral nodal curves $C \subset \mathbb {P}^r$ with prescribed degree, arithmetic genus and number of singular points.

驯顺节点堆叠曲线

Tame nodal stacky curves

本文分析驯顺节点堆叠曲线的性质，特别是扭曲曲线和双扭曲曲线，主要结果包括驯顺堆叠节点的完整分类以及节点堆叠曲线的Picard群和Brauer群的计算。

Degree-2 Abel maps for nodal curves

We present numerical conditions for the existence of natural degree-2 Abel maps for any given nodal curve. Cocoa scripst were written and have so far verified the validity of the conditions for numerous curves.

基变换型Enriques曲面上的有理曲线

Nodal rational curves on Enriques surfaces of base change type

本文提出一种纯几何无格构造方法，证明了基变换型Enriques曲面上双截面通常是节点的，并展示了此类曲面承认形成秩8子群的节点有理曲线集。

The Full Flag Hilbert Scheme of Nodal Curves and the Punctual Hilbert Scheme of Points of the Cusp Curve

In this paper, we show the full flag Hilbert scheme of points of nodal curves has only complete intersection singularities. We study the Hilbert scheme of points of the cusp curve and show the punctual Hilbert scheme is isomorphic to \mathbb{P}^1. Furthermore Hilbert scheme has only one singularty along the punctual one.

积分节点曲线上的核sheaf

Kernel Sheaf on Integral Nodal Curves

本文研究了积分节点曲线上秩为一的无 torsion sheaf 生成子空间所获得的核sheaf 的稳定性问题。

结点代数曲线上主$\Bbb C^*$-丛的模栈的上同调

Cohomology of Moduli Stacks of Principal $\Bbb C^*$-Bundles Over Nodal Algebraic Curves

研究结点曲线上主$\Bbb C^*$-丛模栈，用陈类确定其有理上同调代数。

Divisibilities among nodal curves

We prove that there are no effective or anti-effective classes of square -1 or -2 arising from nodal curves on smooth algebraic surfaces by way of divisibility. This general fact has interesting applications to Enriques and K3 surfaces. The proof relies on specific properties of root lattices and their duals.

Rationality of Moduli space over reducible curve

Let $M(2,\textbf{\underline{w}},χ)$ be the moduli space of rank $2$ torsion-free sheaves over a reducible nodal curve with each component having utmost two nodal singularities. We show that in each component of $M(2,\textbf{\underline{w}},χ)$, the closure of rank $2$ vector bundles with fixed determinant is rational.

Deformation of Locally Free Sheaves and Hitchin Pairs over Nodal Curve

In this article, we study the deformation theory of locally free sheaves and Hitchin pairs over a nodal curve. As a special case, the infinitesimal deformation of these objects gives the tangent space of the corresponding moduli spaces, which can be used to calculate the dimension of the corresponding moduli space. We show that the deformation of locally free sheaves and Hitchin pairs over a nodal curve is equivalent to the deformation of generalized parabolic bundles and generalized parabolic Hitchin pairs over the normalization of the nodal curve respectively.

A Torelli Type theorem for Nodal curves

The moduli space of Gieseker vector bundles is a compactification of moduli of vector bundles on a nodal curve. This moduli space has only normal crossing singularity and it provides a flat degeneration. We prove a Torelli type theorem for a nodal curve using the moduli space of stable Gieseker vector bundles of fixed rank (strictly greater than $1$) and fixed degree such that rank and degree are co-prime.

节点曲线上的 Chevalley-Weil 公式

The Chevalley-Weil formula on nodal curves

本文研究了代数闭域上节点曲线全纯微分空间在有限自同构群作用下的特征子空间，在商曲线附加条件下计算了 Chevalley-Weil 公式并给出实例。

算术亏格为一的不可约节点曲线上希格斯对模空间的几何

A geometry of the moduli space of Higgs pairs on an irreducible nodal curve of arithmetic genus one

本文利用希钦映射和椭圆曲线希格斯丛模空间的平坦退化，刻画了算术亏格为一的不可约节点曲线上希格斯对模空间的几何结构。

The space of nodal curves of type p,q with given Weierstraßsemigroup

We continue the investigation of curves of type p,q started in [KKW]. We study the space of such curves and the space of nodal curves with prescribed Weierstraßsemigroup. A necessary and sufficient criterion for a numerical semigroup to be a Weierstraßsemigroup is given. We find a class of Weierstraßsemigroups which apparently has not yet been described in the literature.

The degree-2 Abel--Jacobi map for nodal curves - I

Let $f\col\C\ra B$ be a regular local smoothing of a nodal curve. In this paper, we find a modular description of the Abel--Néron map having values in Esteves's fine compactified Jacobian and extending the degree-2 Abel--Jacobi map of the generic fiber of $f$

Genus three curves and 56 nodal sextic surfaces

Catanese and Tonoli showed that the maximal cardinality for an even set of nodes on a sextic surface is 56 and they constructed such nodal surfaces. In this paper we give an alternative, rather simple, construction for these surfaces starting from a non-hyperelliptic genus three curve. We illustrate our method by giving explicitly the equation of such a sextic surface starting from the Klein curve.

About the surjectivity of the Wahl map of nodal curves on K3 surfaces

I correct an error in my article arXiv:0911.5474, concerning the non-surjectivity of the Wahl map of nodal curves on K3 surfaces. I prove that a modification of the Wahl map is indeed non-surjective for nodal curves with a low number of nodes, on arbitrary K3 surfaces. Along the way, I also obtain a lower bound of independent interest for the multiple point Seshadri constants of K3 surfaces with cyclic Picard group.

Extensions and rank-2 vector bundles on irreducible nodal curves

We generalize Bertram's work on rank two vector bundles to an irreducible projective nodal curve C. We use extensions of a line bundle L by O_C and the associated `forgetful' map to study a compactification of the moduli space of semi-stable vector bundles of rank 2 and determinant L on C.

Nodal curves with general moduli on K3 surfaces

We investigate the modular properties of nodal curves on a low genus K3 surface. We prove that a general genus g curve C is the normalization of a d-nodal curve X sitting on a primitively polarized K3 surface S of degree 2p-2, for p any integer between 3 and 11 and g = p - d between 2 and p. The proof is based on a local deformation-theoretic analysis of the map from the stack of pairs (S,X) to the moduli space of curves of genus g that associates to X the isomorphism class [C] of its normalization.

Rigid isotopy classification of real degree-4 planar rational curves with only real nodes (An elementary approach)

We prove an elementary method to classify, up to rigid isotopy, all nodal degree 4 real rational curves in $\mathbb{RP}^2$ that have only real double points. We show how to associate a chord diagram to a nodal real degree 4 planar rational curve and prove that such curves are defined up to rigid isotopy by their chord diagrams.

Singular principal bundles on reducible nodal curves

Studying degenerations of moduli spaces of semistable principal bundles on smooth curves leads to the problem of constructing and studying moduli spaces on singular curves. In this note, we will see that the moduli spaces of $δ$-semistable pseudo bundles on a nodal curve constructed by the first author become, for large values of $δ$, the moduli spaces for semistable singular principal bundles. The latter are reasonable candidates for degenerations and a potential basis of further developments as on irreducible nodal curves. In particular, we find a notion of semistability for principal bundle…

Weierstraßsemigroups and nodal curves of type p,q

We study plane curves of type p,q having only nodes as singularities. Every Weierstraßsemigroup is the Weierstraßsemigroup of such a curve at its place at infinity for properly chosen p,q. We construct plane curves of type p,q with explicitly given nodes and determine their Weierstraßsemigroups. Many such semigroups are found.

节点恩里克斯曲面是雷耶线汇

Nodal Enriques surfaces are Reye congruences

本文证明了任何包含光滑有理曲线的经典恩里克斯曲面均可实现为雷耶线汇，揭示了这两类代数几何对象之间的对应关系。

Hodge conjecture for the moduli space of semi-stable sheaves over a nodal curve

In this article, we prove the Hodge conjecture for a desingularization of the moduli space of rank 2, semi-stable, torsion-free sheaves with fixed odd degree determinant over a very general irreducible nodal curve of genus at least 2. We also compute the algebraic Poincare polynomial of the associated cohomology ring.

Nodal Curves on K3 Surfaces

In this paper, we study the Severi variety $V_{L,g}$ of genus $g$ curves in $|L|$ on a general polarized K3 surface $(X,L)$. We show that the closure of every component of $V_{L,g}$ contains a component of $V_{L,g-1}$. As a consequence, we see that the general members of every component of $V_{L,g}$ are nodal.

The Wahl map of one-nodal curves on K3 surfaces

We consider a general primitively polarized K3 surface $(S,H)$ of genus $g+1$ and a 1-nodal curve $\widetilde C\in |H|$. We prove that the normalization $C$ of $\widetilde C$ has surjective Wahl map provided $g=40,42$ or $\ge 44$.

Nodal Uniformization of G-bundles

We give a survey of uniformization results for principal bundles on curves. We provide a proof of uniformization for nodal curves; this result is a special case of work of Belkale and Fakhruddin for uniformization on singular curves. We use the uniformization result to give a construction of a compactification of the moduli of G bundles on a family of nodal curves. Along the way we also explain how to use equivariant compactifactions of a group to compactify the moduli stack of bundles over a fixed nodal curve.

Geometry on nodal curves II: cycle map and intersection calculus

We study the relative Hilbert scheme of a family of nodal (or smooth) curves via its (birational) cycle map, going to the relative symmetric product. We show the cycle map is the blowing up of the discriminant locus, which consists of cycles with multiple points. We derive an intersection calculus for Chern classes of tautological bundles on the relative Hilbert scheme, which has applications to enumerative geometry.

Modular properties of nodal curves on K3 surfaces

In this paper we partially address two issues: - The first is a rigidity property for pairs (S,C) consisting of a general projective K3 surface S, and a curve C obtained as the normalization of a nodal, hyperplane section of S. We prove that a non-trivial deformation of such a pair (S,C) induces a non-trivial deformation of C; - The second question concerns the Wahl map of curves C as above. We prove that the Wahl map of the normalization of a nodal curve contained in a general projective K3 surface is non-surjective. In both cases, we impose upper bounds on the number of nodes of the hyperpla…

Nodal curves with a contact-conic and Zariski pairs

In this present paper, we study the splitting of nodal plane curves with respect to contact conics. We define the notion of splitting type of such curves and show that it can be used as an invariant to distinguish the embedded topology of plane curves. We also give a criterion to determine the splitting type in terms of the configuration of the nodes and tangent points. As an application, we construct sextics and contact conics with prescribed splitting types, which give rise to new Zariski-triples.

Cycle map on Hilbert schemes of nodal curves

We study the structure of the relative Hilbert scheme for a family of nodal (or smooth) curves via its natural cycle map to the relative symmetric product. We show that the cycle map is the blowing up of the discriminant locus, which consists of cycles with multiple points. We discuss some applications and connections, notably with birational geometry and intersection theory on Hilbert schemes of smooth surfaces. Revised version corrects some minor errors.

结点曲线的 Hodge 到 de Rham 退化

Hodge to de Rham degeneration of nodal curves

本文研究结点曲线的非交换 Hodge–de Rham 退化，计算其 Hochschild 与负循环同调，并分类可提升的 Hochschild 类，为范畴枚举不变量计算提供基础。

A note on Hilbert schemes of nodal curves

We study completely the Hilbert scheme and punctual Hilbert scheme of a nodal curve, and the relative Hilbert scheme of a family of curves acquiring a node. The results are then extended, less completely, to flag Hilbert schemes, parametrizing chains of subschemes. We find, notably, that if the total space $X$ of a family $X/B$ is smooth (over an algebraically closed field $\k$), then the relative Hilbert scheme $Hilb_m(X/B)$ is smooth over $\k$ and the flag Hilbert schemes are normal and locally complete intersection, but generally singular.

Comment on: On the irreducibility of the Severi variety of nodal curves in a smooth surface, by E. Ballico

In this short note, I point out that results of Ballico and Kool--Shende--Thomas together imply that on $K3$, Enriques, and Abelian surfaces, if $L$ is a very ample and $(2p_a(L)-2g-1)$-spanned line bundle, then the equigeneric Severi variety $V_{g}(L)$ of all curves in $|L|$ having genus $g$ is non-empty, irreducible, of the expected dimension, and its general member is a $(p_a(L)-g)$-nodal curve.

Coherent systems and BGN extensions on nodal reducible curves

Let $(C,\underline{w})$ be a polarized nodal reducible curve. In this paper we consider coherent systems of type $(r,d,k)$ on $C$ with $k < r$. We prove that the moduli spaces of $(\underline{w},α)$-stable coherent systems stabilize for large $α$ and we generalize several results known for the irreducible case when we chose a good polarization. Then, we study in details the components of moduli spaces containing coherent systems arising from locally free sheaves.

Families of nodal curves in P^r with the expected number of moduli

Let V^{r}_{d,g, δ} be the Hilbert scheme of nodal curves in P^r of degree d and arithmetic genus g with δnodes. Under suitable numerical assumptions on d and g, for every 0 \le δ\le g we construct an irreducible component of V^{r}_{d,g, δ} having the expected number of moduli.

Twisted vector bundles on pointed nodal curves

We establish a striking connection between Abramovich's and Vistoli's twisted bundles and Gieseker vector bundles. This note grew out of an attempt to understand a recent draft of Seshadri.

Brauer group and birational type of moduli spaces of torsionfree sheaves on a nodal curve

Let U^{'s}_{L}(n,d) be the moduli space of stable vector bundles of rank $n$ and fixed determinant L of degree d on a nodal curve Y. The moduli space of semistable vector bundles of rank n and degree $d$ will be denoted by U'_Y(n,d). We calculate the Brauer groups of U^{'s}_{L}(n,d)$. We study the question of rationality of $U^{'s}_{L}(n,d)$ and $U'_Y(n,d)$.

Moduli of nodal curves on smooth surfaces of general type

In this paper we focus on the problem of computing the number of moduli of the so called Severi varieties (denoted by V(|D|, δ)), which parametrize universal families of irreducible, δ-nodal curves in a complete linear system |D|, on a smooth projective surface S of general type. We determine geometrical and numerical conditions on D and numerical conditions on δensuring that such number coincides with dim(V(|D|, δ). As related facts, we also determines some sharp results concerning the geometry of some Severi varieties.

Generalized parabolic structures over smooth curves with many components and principal bundles over reducible nodal curves

Let $Y_{1},\dots,Y_{l}$ be smooth irreducible projective curves and let $Y$ be its disjoint union. Given a semisimple reductive algebraic group $G$ and a faithful representation $ρ:G\hookrightarrow \textrm{SL}(V)$ we construct a projective moduli space of $(\underlineκ,δ)$-(semi)stable singular principal $G$-bundles with generalized parabolic structure of type $\underline{e}$. In case $Y$ is the normalization of a connected and reducible projective nodal curve $X$, there is a closed subscheme coarsely representing the subfunctor corresponding to descending bundles. We prove that the descent op…

On the degree-1 Abel map for nodal curves

Let $C$ be a nodal curve and $L$ be an invertible sheaf on $C$. Let $α_{L}:C\dashrightarrow J_{C}$ be the degree-$1$ rational Abel map, which takes a smooth point $Q\in C$ to $\left[ m_{Q}\otimes L\right] $ in the Jacobian of $C$. In this work we extend $α_{L}$ to a morphism $\overlineα_{L}:C\rightarrow\overline{J}_{E}^{P}$ taking values on Esteves' compactified Jacobian for any given polarization $E$. The maps $\overlineα_{L}$ are limits of Abel maps of smooth curves of the type $α_{L}$.

Brill-Noether locus of rank 1 and degree g-1 on a nodal curve

In this paper we consider the Brill-Noether locus $W_{\underline d}(C)$ of line bundles of multidegree $\underline d$ of total degree $g-1$ having a nonzero section on a nodal reducible curve $C$ of genus $g\geq2$. We give an explicit description of the irreducible components of $W_{\underline d}(C)$ for a semistable multidegre $\underline d$. As a consequence we show that, if two semistable multidegrees of total degre $g-1$ on a curve with no rational components differ by a twister, then the respective Brill-Noether loci have isomorphic components.

Abel Maps for nodal curves via tropical geometry

We consider Abel maps for regular smoothing of nodal curves with values in the Esteves compactified Jacobian. In general, these maps are just rational, and an interesting question is to find an explicit resolution. We translate this problem into an explicit combinatorial problem by means of tropical and toric geometry. We show that the solution of the combinatorial problem gives rise to an explicit resolution of the Abel map. We are able to use this technique to construct and study all the Abel maps of degree one.

On the irreducibility of the Severi variety of nodal curves in a smooth surface

Let $X$ be a smooth projective surface and $L\in \mathrm{Pic}(X)$. We prove that if $L$ is $(2k-1)$-spanned, then the set $\tilde{V}_k(L)$ of all nodal and irreducible $D\in |L|$ with exactly $k$ nodes is irreducible. The set $\tilde{V}_k(L)$ is an open subset of a Severi variety of $|L|$, the full Severi variety parametrizing all integral $D\in |L|$ with geometric genus $g(L)-k$.

A Note on Parabolic Bundles on Nodal Curves

Mehta and Seshadri have proved that the set of equivalence classes of irreducible unitary representations of the fundamental group of a punctured compact Riemann surface, can be identified with equivalence classes of stable parabolic bundles of parabolic degree zero on the compact Riemann surface. In this note, we discuss the Mehta-Seshadri correspondence over an irreducible projective curve with at most nodes as singularities.