arXiv 0705.09062007-05-07math.AG

Remarks on families of singular curves with hyperelliptic normalizations

Andreas Leopold Knutsen

备注：: 18 pages

英文摘要：

We give restrictions on the existence of families of curves on smooth projective surfaces $S$ of nonnegative Kodaira dimension all having constant geometric genus $g \geq 2$ and hyperelliptic normalizations. In particular, we prove a Reider-like result whose proof is ``vector bundle-free'' and relies on deformation theory and bending-and-breaking of rational curves in $\Sym^2(S)$. We also give examples of families of such curves.

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。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

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

相关论文

Remarks on Collino cycles and hyperelliptic Johnson homomorphisms

A Collino cycle is a higher cycle on the Jacobian of a hyperelliptic curve. The universal family of Collino cycles naturally gives rise to a normal function, whose induced monodromy relates to the hyperelliptic Johnson homomorphism. Colombo computed this monodromy explicitly and made this relation precise. We recast this in the perspective of relative completion. In particular, we use Colombo's result to construct Collino classes, which are cohomology classes of hyperelliptic mapping class groups with coefficients in a certain symplectic representation. We also determine the dimension of their…

2023-01-13math.AG

关于泛超椭圆曲线截面的注记

Remarks on the sections of universal hyperelliptic curves

该文研究亏格不小于3的泛超椭圆曲线截面障碍，借助超椭圆映射类群相对完备化与构形空间基本群幂单完备化李代数的障碍，证明对应Birman正合序列不分裂。

2023-06-09math.AG

Singular curves on a K3 surface and linear series on their normalizations

In this paper, we study the Brill-Noether theory of the normalizations of singular, irreducible curves on a $K3$ surface. We introduce a {\em singular} Brill-Noether number $ρ_{sing}$ and show that if the Picard group of the K3 surface is ${\mathbb Z} [L]$, there are no $g^r_d$'s on the normalizations of irreducible curves in $|L|$, provided that $ρ_{sing} <0$. We give examples showing the sharpness of this result. We then focus on the case of {\em hyperelliptic normalizations}, and classify linear systems $|L|$ containing irreducible nodal curves with hyperelliptic normalizations, for $ρ_{sin…

2006-09-19math.AG

Some remarks on the hyperelliptic moduli of genus 3

In 1967, Shioda \cite{Shi1} determined the ring of invariants of binary octavics and their syzygies using the symbolic method. We discover that the syzygies determined in \cite{Shi1} are incorrect. In this paper, we compute the correct equations among the invariants of the binary octavics and give necessary and sufficient conditions for two genus 3 hyperelliptic curves to be isomorphic over an algebraically closed field $k$, $\ch k \neq 2, 3, 5, 7$. For the first time, an explicit equation of the hyperelliptic moduli for genus 3 is computed in terms of absolute invariants.

2012-09-06math.AG

Compactifying moduli of hyperelliptic curves

We construct a new compactification of the moduli space H_g of smooth hyperelliptic curves of genus g. We compare our compactification with other well-known remarkable compactifications of H_g .

2007-06-01math.AG

Jacobian Nullwerte, Periods and Symmetric Equations for Hyperelliptic Curves

We propose a solution to the hyperelliptic Schottky problem, based on the use of Jacobian Nullwerte and symmetric models for hyperelliptic curves. Both ingredients are interesting on its own, since the first provide period matrices which can be geometrically described, and the second have remarkable arithmetic properties.

2006-10-10math.AGmath.NT

更多相关论文收起更多论文

Remarks on Xiao's approach of Slope inequalities

We prove the slope inequality for a relative minimal surface fibration in positive characteristic via Xiao's approach. We also prove a better low bound for the slope of non-hyperelliptic fibrations.

2016-08-11math.AG

On families of rational curves in the Hilbert square of a surface (with an Appendix by Edoardo Sernesi)

Under natural hypotheses we give an upper bound on the dimension of families of singular curves with hyperelliptic normalizations on a surface S with p_g(S) >0 via the study of the associated families of rational curves in Hilb^2(S). We use this result to prove the existence of nodal curves of geometric genus 3 with hyperelliptic normalizations, on a general K3 surface, thus obtaining specific 2-dimensional families of rational curves in its Hilbert square. We describe two infinite series of examples of general, primitively polarized K3's such that their Hilbert squares contain a IP^2 or a thr…

2007-04-11math.AG

$(φ,Γ)$-modules associés aux courbes hyperelliptiques lisses

In 2003, Kedlaya gave an algorithm to compute the zeta function associated to a hyperelliptic curve over a finite field, by computing the rigid cohomology of the curve. Edixhoven remarked that it is actually possible to compute the crystalline cohomology of the curve, which is a lattice in the rigid cohomology. Following a method of Wach, we first explain how to use this lattice to compute the $(φ,Γ)$-module associated to an hyperelliptic curve. We also explain an alternative way to get the $(φ,Γ)$-module mod $p$ that relies on the Deligne-Illusie morphism.

2014-01-02math.AG

Jacobians, Anti-affine groups and torsion points

We give criteria for the Jacobian of a singular curve $X$ with at most ordinary $n$-point singularities to be anti-affine. In particular, for the case of curves with single ordinary double point we exhibit a relation with torsion divisors. If the geometric genus of the singular curve is atleast 3 and the normalization is non-hyperelliptic and non-bielliptic, then except for finitely many cases the Jacobian of $X$ is anti-affine. Furthermore, if the normalization is a general curve of genus atleast 3 then the Jacobian of $X$ is always anti-affine.

2022-05-19math.AGmath.GR

属五 Howe 曲线的平面六次模型的显式构造，II

Explicit construction of a plane sextic model for genus-five Howe curves, II

本文构造了属五非超椭圆 Howe 曲线的显式平面六次模型，确定其奇点一般为四个二重点，并结合已有结果证明存在具有不同二重点数的曲线族。

2024-01-01math.AGmath.NT

Remarks on projective normality for certain Calabi-Yau and hyperkähler varieties

We prove some results on effective very ampleness and projective normality for some varieties with trivial canonical bundle. In the first part we prove an effective projective normality result for an ample line bundle on regular smooth four-folds with trivial canonical bundle. More precisely we show that for a regular smooth fourfold with trivial canonical bundle, $A^{\otimes 15}$ is projectively normal for $A$ ample. In the second part we emphasize on the projective normality of multiples of ample and globally generated line bundles on certain classes of known examples (upto deformation) of p…

2019-09-30math.AG

On a Conjecture on Linear Systems

In a remark to Green's conjecture, Paranjape and Ramanan analyzed the vector bundle $E$ which is the pullback by the canonical map of the universal quotient bundle $T_{\Pp^{g-1}}(-1)$ on $\Pp^{g-1}$ and stated a more general conjecture and proved it for the curves with Clifford Index $1$ (trigonal and plane quintic). In this paper, we state the conjecture for general linear systems and obtain the results for the case of hyperelliptic curves.

2016-04-12math.AG

A remark on 1-forms and plane curve singularities

We study the pull-back of regular 1-forms on a complex irreducible plane curve singularity under the normalization morphism.

2017-09-06math.ACmath.AG

On Prym varieties for the coverings of some singular plane curves

Let $k$ be a field of characteristic zero containing a primitive $n$-th root of unity. Let $C^0_n$ be a singular plane curve of degree $n$ over $k$ admitting an order $n$ automorphism, $n$ nodes as the singularities, and $C_n$ be its normalization. In this paper we study the factors of Prym variety $\mbox{Prym}(\widetilde{C}_n/C_n)$ associated to the double cover $\widetilde{C}_n$ of $C_n$ exactly ramified at the points obtained by the blow-up of the singularities. We provide explicit models of some algebraic curves related to the construction of $\mbox{Prym}(\widetilde{C}_n/C_n)$ as a Prym va…

2016-09-13math.AG

Some remarks on flatness of morphisms over smooth base spaces

In this paper we study flatness of the restriction on some special subgerms (e.g. the reduction and the unmixed part) of the total space of a flat morphism over a smooth base space. We give a relationship between reducedness of the total space and that of the generic fibers of a flat morphism over a reduced Cohen-Macaulay base space. Moreover, we study flatness of the composition of a flat morphism over a smooth base space and the normalization of the total space of that morphism.

2014-08-13math.AG

Some remarks about equations defining coincident root loci

Consider the projective variety $X_λ$ of binary forms of degree $d$ whose linear factors are distributed according to the partition $λ$ of $d$. We determine minimal sets of local generators of the fiber product of $X_λ$ with its normalization, and we show that the local Jacobian matrices of this product contain the product of the identity matrix of maximal rank with a unit. We use this to fill a gap in a crucial proof in Chipalkatti's "On equations defining Coincident Root Loci". Also, we give a new description of the singular locus of $X_λ$ and a criterion for the smoothness of $X_λ$.

2011-08-23math.AG

A remark on the intersection of plane curves

Let $D$ be a very general curve of degree $d=2\ell-ε$ in $\mathbb{P}^2$, with $ε\in \{0,1\}$. Let $Γ\subset \mathbb{P}^2$ be an integral curve of geometric genus $g$ and degree $m$, $Γ\neq D$, and let $ν: C\to Γ$ be the normalization. Let $δ$ be the degree of the \emph{reduction modulo 2} of the divisor $ν^*(D)$ of $C$. In this paper we prove the inequality $4g+δ\geqslant m(d-8+2ε)+5$. We compare this with similar inequalities due to Geng Xu and Xi Chen. Besides, we provide a brief account on genera of subvarieties in projective hypersurfaces.

2019-01-07math.AG

Hill曲线的变形与KdV和sin-Gordon方程中的等周期性

Deformations of the Hill curves and isoperiodicity in the KdV and the sine-Gordon equations

研究通过等周期变形方法，解决KdV和sin-Gordon方程中周期解的变形问题。

2025-12-05math-phmath.AGmath.AP

通过下降方法研究非正则光滑曲线

Non-smooth regular curves via a descent approach

本文提出一种基于下降方法的新途径，通过零p-曲率可积联络扩展Cartier定理，并引入微分次数等局部不变量，完成了特征3域上亏格为3的非正则光滑曲线的分类。

2025-01-29math.AG

The geometry of eight points in projective space: Representation theory, Lie theory, dualities

This paper deals with the geometry of the space (GIT quotient) M_8 of 8 points in P^1, and the Gale-quotient N'_8 of the GIT quotient of 8 points in P^3. The space M_8 comes with a natural embedding in P^{13}, or more precisely, the projectivization of the S_8-representation V_{4,4}. There is a single S_8-skew cubic C in P^{13}. The fact that M_8 lies on the skew cubic C is a consequence of Thomae's formula for hyperelliptic curves, but more is true: M_8 is the singular locus of C. These constructions yield the free resolution of M_8, and are used in the determination of the "single" equation …

2011-03-27math.AG