Invariant theory of $\bigwedge^3(9)$ and genus 2 curves

A canonical curve of genus 17

We compute equations for a Hurwitz curve of genus 17 and we conclude that the canonical ideal of any Hurwitz curve of genus 14 or 17 is generated by quadrics.

Families of genus two curves with many elliptic subcovers

We determine all genus 2 curves, defined over $\mathbb C$, which have simultaneously degree 2 and 3 elliptic subcovers. The locus of such curves has three irreducible 1-dimensional genus zero components in $\mathcal M_2$. For each component we find a rational parametrization and construct the equation of the corresponding genus 2 curve and its elliptic subcovers in terms of the parameterization. Such families of genus 2 curves are determined for the first time. Furthermore, we prove that there are only finitely many genus 2 curves (up to $\mathbb C$-isomorphism) defined over $\mathbb Q$, which…

$(p,d)$-elliptic curves of genus two

We study stable curves of arithmetic genus 2 which admit two morphisms of finite degree $p$, resp. $d$, onto smooth elliptic curves, with particular attention to the case $p$ prime.

On the rationality of the moduli of higher spin curves in low genus

The global geometry of the moduli spaces of higher spin curves and their birational classification is largely unknown for g >= 2 and r > 2. Using quite related geometric constructions, we almost complete the picture of the known results in genus g <= 4 showing the rationality of the moduli spaces of even and odd 4-spin curves of genus 3, of odd spin curves of genus 4 and of 3-spin curves of genus 4.

Admissible constants for genus 2 curves

S.-W. Zhang recently introduced a new adelic invariant for curves of genus at least 2 over number fields and function fields. We calculate this invariant when the genus is equal to 2.

Degree even coverings of elliptic curves by genus 2 curves

In this survey we study the genus 2 curves with $(n, n)$-split Jacobian for even $n$.

Uniformizing Tropical Curves I: Genus Zero and One

In tropical geometry, given a curve in a toric variety, one defines a corresponding graph embedded in Euclidean space. We study the problem of reversing this process for curves of genus zero and one. Our methods focus on describing curves by parameterizations, not by their defining equations; we give parameterizations by rational functions in the genus zero case and by non-archimedean elliptic functions in the genus one case. For genus zero curves, those graphs which can be lifted can be characterized in a completely combinatorial manner. For genus one curves, show that certain conditions iden…

Genus 2 curve configurations on Fano surfaces

We study the configurations of genus 2 curves on the Fano surfaces of cubic threefolds. We establish a link between some involutive automorphisms acting on such a surface S and genus 2 curves on S. We give a partial classification of the Fano surfaces according to the automorphism group generated by these involutions and determine the configurations of their genus 2 curves. We study the Fano surface of the Klein cubic threefold for which the 55 genus 2 curves generate a rank 25=h^{1,1} index 2 subgroup of the Néron-Severi group.

Ineffective descent of genus one curves

Raynaud proved in 1968 that etale descent of genus one curves is not effective in general. In this paper we provide an alternative, simplified construction of this phenomenon. Our counterexample is fully explicit.

Extremal effective divisors on the moduli space of n-pointed genus one curves

For every $n\geq 3$, we exhibit infinitely many extremal effective divisors on the moduli space of genus one curves with $n$ marked points.

Finite Hilbert stability of canonical curves, II. The even-genus case

This paper is a sequel to arXiv:1109.4986, where we proved that a general smooth curve of odd genus, canonically or bicanonically embedded, has semistable finite Hilbert points. Here, we prove that a generic canonically embedded curve of even genus has semistable finite Hilbert points. More precisely, we prove that a generic canonically embedded trigonal curve of even genus has semistable finite Hilbert points. Furthermore, we show that the analogous result fails for bielliptic curves. Namely, the Hilbert points of bielliptic curves are asymptotically semistable but become non-semistable below…

Green's canonical syzygy conjecture for generic curves of odd genus

We prove the Green conjecture for generic curves of odd genus. That is we prove the vanishing $K_{k,1}(X,K_X)=0$ for $X$ generic of genus $2k+1$. The curves we consider are smooth curves $X$ on a K3 surface whose Picard group has rank 2. This completes our previous work, where the Green conjecture for generic curves of genus $g$ with fixed gonality $d$ was proved in the range $d\geq g/3$, with the possible exception of the generic curves of odd genus.

On irreducibility of the family of ACM curves of degree 8 and genus 4 in $P^4_k$

Let C be an arithmetically Cohen-Macaulay curve of arithmetic genus 4. We prove that the family of such curves of degree 8 in $P^4_k$ is irreducible.

On the genus of a cyclic plane curve over a finite field

Cyclic curves, i.e. curves fixed by a cyclic collineation group, play a central role in the investigation of cyclic arcs in Desarguesian projective planes. In this paper, the genus of a cyclic curve arising from a cyclic k-arc of Singer type is computed.

Computing isogenies between Jacobian of curves of genus 2 and 3

We present a quasi-linear algorithm to compute isogenies between Jacobians of curves of genus 2 and 3 starting from the equation of the curve and a maximal isotropic subgroup of the l-torsion, for l an odd prime number, generalizing the Vélu's formula of genus 1. This work is based from the paper "Computing functions on Jacobians and their quotients" of Jean-Marc Couveignes and Tony Ezome. We improve their genus 2 case algorithm, generalize it for genus 3 hyperelliptic curves and introduce a way to deal with the genus 3 non-hyperelliptic case, using algebraic theta functions.

A Simple Proof of Voisin's Theorem for Canonical Curves of Even Genus

We give a simple proof of Voisin's Theorem for general canonical curves of even genus. This completely determines the terms of the minimal free resolution of the coordinate ring of such curves.

The arithmetic of genus two curves

Genus 2 curves have been an object of much mathematical interest since eighteenth century and continued interest to date. They have become an important tool in many algorithms in cryptographic applications, such as factoring large numbers, hyperelliptic curve cryptography, etc. Choosing genus 2 curves suitable for such applications is an important step of such algorithms. In existing algorithms often such curves are chosen using equations of moduli spaces of curves with decomposable Jacobians or Humbert surfaces. In these lectures we will cover basic properties of genus 2 curves, moduli spaces…

亏格三的Gorenstein曲线奇点

Gorenstein curve singularities of genus three

本文分类了亏格三的孤立Gorenstein曲线奇点的解析芽，并建立了它们与阿贝尔微分分层空间连通分支之间的对应关系。

椭圆曲线的超特殊亏格4二重覆盖

Superspecial genus-$4$ double covers of elliptic curves

本文研究椭圆曲线二重覆盖得到的亏格4曲线，给出显式方程与非奇异性判据，证明相关长多项式在特征不等于2、3时不可约，并在小特征下枚举超特殊亏格4二重覆盖。

Counting the number of trigonal curves of genus 5 over finite fields

The trigonal curves of genus 5 can be represented by projective plane quintics that have 1 singularity of delta invariant 1. Combining this with a partial sieve method for plane curves we count the number of such curves over any finite field. The main application is that this then gives the motivic Euler characteristic of the moduli space of trigonal curves of genus 5.

Khovanskii-finite rational curves of arithmetic genus 2

We study the existence of Khovanskii-finite valuations for rational curves of arithmetic genus two. We provide a semi-explicit description of the locus of degree $n+2$ rational curves in $\mathbb{P}^n$ of arithmetic genus two that admit a Khovanskii-finite valuation. Furthermore, we describe an effective method for determining if a rational curve of arithmetic genus two defined over a number field admits a Khovanskii-finite valuation. This provides a criterion for deciding if such curves admit a toric degeneration. Finally, we show that rational curves with a single unibranched singularity are…

Heegner divisors in the moduli space of genus three curves

S. Kondō used periods of $K3$ surfaces to prove that the moduli space of genus three curves is birational to an arithmetic quotient of a complex 6-ball. In this paper we study Heegner divisors in the ball quotient, given by arithmetically defined hyperplane sections of the ball. We show that the corresponding loci of genus three curves are given by: hyperelliptic curves, singular plane quartics and plane quartics admitting certain rational "splitting curves".

Focal Varieties of Curves of Genus 6 and 8

In this paper we give a simple Torelli type theorem for curves of genus 6 and 8 by showing that these curves can be reconstructed from their Brill-Noether varieties. Among other results, it is shown that the focal variety of a general, canonical and nonhyperelliptic curve of genus 6 is a hypersurface.

Cohomology of moduli spaces of curves of genus three via point counts

In this article we consider the moduli space of smooth $n$-pointed non-hyperelliptic curves of genus 3. In the pursuit of cohomological information about this space, we make $\mathbb{S}_n$-equivariant counts of its numbers of points defined over finite fields for $n \leq 7$. Combining this with results on the moduli spaces of smooth pointed curves of genus 0, 1 and 2, and the moduli space of smooth hyperelliptic curves of genus 3, we can determine the $\mathbb{S}_n$-equivariant Galois and Hodge structure of the ($\ell$-adic respectively Betti) cohomology of the moduli space of stable curves of…

Genus six curves, K3 surfaces, and stable pairs

A general smooth curve of genus six lies on a quintic del Pezzo surface. In \cite{AK11}, Artebani and Kondō construct a birational period map for genus six curves by taking ramified double covers of del Pezzo surfaces. The map is not defined for special genus six curves. In this paper, we construct a smooth Deligne-Mumford stack $\mathfrak{P}_0$ parametrizing certain stable surface-curve pairs which essentially resolves this map. Moreover, we give an explicit description of pairs in $\mathfrak{P}_0$ containing special curves.

Hyperelliptic genus 4 curves on abelian surfaces

We study smooth curves on abelian surfaces, especially for genus 4, when the complementary subvariety in the Jacobian is also a surface. We show that up to translation there is exactly one genus 4 hyperelliptic curve on a general (1, 3)-polarised abelian surface. We investigate these curves and show that their Jacobians contain a surface and its dual as complementary abelian subvarieties.

Irreducibility and stable rationality of the loci of Weierstrass points on curves of genus at most six

We show that for any numerical semigroup H of genus g at most 6, the locus of Weierstrass points on curves of genus g with Weierstrass semigroup H is irreducible and that for all but possibly two semigroups it is stably rational.

Fourier-Mukai partners of singular genus one curves

The objective of the paper is to prove that, as it happens for smooth elliptic curves, any Fourier-Mukai partner of a projective reduced Gorenstein curve of genus one and trivial dualising sheaf, is isomorphic to itself.

The genus of projective curves on complete intersection surfaces

We bound the genus of a projective curve lying on a complete intersection surface in terms of its degree and the degrees of the defining equations of the surface on which it lies.

Green-Lazarsfeld's Gonality Conjecture for a Generic Curve of Odd Genus

The gonality conjecture predicts that the gonality of a curve can be read off Koszul cohomology of line bundles of sufficiently large degree. We verify this conjecture for generic curves of odd genus. The even-genus case was previously solved in the joint work arXiv:math.AG/0301261.

Explicit Solution By Radicals, Gonal Maps and Plane Models of Algebraic Curves of Genus 5 or 6

We give explicit computational algorithms to construct minimal degree (always $\le 4$) ramified covers of $\Prj^1$ for algebraic curves of genus 5 and 6. This completes the work of Schicho and Sevilla (who dealt with the $g \le 4$ case) on constructing radical parametrisations of arbitrary genus $g$ curves. Zariski showed that this is impossible for the general curve of genus $\ge 7$. We also construct minimal degree birational plane models and show how the existence of degree 6 plane models for genus 6 curves is related to the gonality and geometric type of a certain auxiliary surface.

Log canonical models and variation of GIT for genus four canonical curves

We discuss GIT for canonically embedded genus four curves and the connection to the Hassett-Keel program. A canonical genus four curve is a complete intersection of a quadric and a cubic, and, in contrast to the genus three case, there is a family of GIT quotients that depend on a choice of linearization. We discuss the corresponding VGIT problem and show that the resulting spaces give the final steps in the Hassett-Keel program for genus four curves.

Enumerative geometry of plane curves of low genus

We collect various known results (about plane curves and the moduli space of stable maps) to derive new recursive formulas enumerating low genus plane curves of any degree with various behaviors. Recursive formulas are given for the characteristic numbers of rational plane curves, elliptic plane curves, and elliptic plane curves with fixed complex structure. Recursions are also given for the number of elliptic (and rational) plane curves with various "codimension 1" behavior (cuspidal, tacnodal, triple pointed, etc., as well as the geometric and arithmetic sectional genus of the Severi variety…

关于亏格16曲线模空间的Kodaira维数

On the Kodaira dimension of the moduli space of curves of genus 16

本文证明了亏格16曲线模空间不是一般型，解决了该维数下的一个长期未决问题。

Boundary complexes of moduli spaces of curves in higher genus

Given a collection of boundary divisors in the moduli space of stable genus-zero n-pointed curves, Giansiracusa proved that their intersection is nonempty if and only if all pairwise intersections are nonempty. We give a complete classification of the pairs (g,n) for which the analogous statement holds in the moduli space of n-pointed curves of genus g.

Birational contraction of genus two tails in the moduli space of genus four curves I

We show that for $α\in (2/3, 7/10)$, the log canonical model $\bar M_4(α)$ of the pair $(\bar M_4, αδ)$ is isomorphic to the moduli space $\bar M_4^{hs}$ of h-semistable curves, and that there is a birational morphism $Ξ: \bar M_4^{hs} \to \bar M_4(2/3)$ that contracts the locus of curves $C_1\cup_p C_2$ consisting of genus two curves meeting in a node $p$ such that $p$ is a Weierstrass point of $C_1$ or $C_2$. To obtain this morphism, we construct a compact moduli space $\bar M_{2,1}^{hs}$ of pointed genus two curves that have nodes, ordinary cusps and tacnodes as singularity, and prove that …

The 2-Weierstrass points of genus 3 hyperelliptic curves with extra automorphisms

For each group $G$, $(|G| > 2)$ \, which acts as a full automorphism group on a genus 3 hyperelliptic curve, we determine the family of curves which have 2-Weierstrass points. Such families of curves are explicitly determined in terms of the absolute invariants of binary octavics. The 1-dimensional families that we discover have the property that they contain only genus 0 components.

Rational cohomology of the moduli space of trigonal curves of genus 5

We compute the rational cohomology of the moduli space of trigonal curves of genus 5. We do this by considering their natural embedding in the first Hirzebruch surface and by using Gorinov-Vassiliev's method.

Automorphism groups of superspecial curves of genus $4$ over $\mathbb{F}_{11}$

In this paper, we explicitly determine the automorphism group of every nonhyperelliptic superspecial curve of genus $4$ over $\mathbb{F}_{11}$. Our algorithm determining automorphism groups works for any nonhyperelliptic curves of genus $4$ over finite fields. With this computation, we show the compatibility between the enumeration of superspecial curves of genus $4$ over $\mathbb{F}_{11}$ obtained computationally by the first and second authors in 2017 and an enumeration by Galois cohomology theory.

16亏格曲线的模空间是单有理的

The moduli space of curves of genus 16 is uniruled

该研究运用非阿贝尔布里尔-诺特理论，证明16亏格曲线的模空间是单有理的，这是目前已知该类结果的最高亏格。

Weil-Petersson volumes of moduli spaces of curves and the genus expansion in two dimensional gravity

A formula for the generating function of the Weil-Petersson volumes of moduli spaces of pointed curves that is identical to the genus expansion of the free energy in two dimensional gravity is obtained. The contribution of arbitrary genus is expressed in terms of the Bessel function $J_0$.

Quantum cohomology of moduli spaces of genus zero stable curves

We investigate the (small) quantum cohomology ring of the moduli spaces of stable n-pointed curves of genus 0. In particular, we determine an explicit presentation in the case n=5 and we outline a computational approach to the case n=6.

The integral Chow ring of the stack of hyperelliptic curves of even genus

Let $g$ be an even positive integer. In this paper we compute the integral Chow ring of the stack of smooth hyperelliptic curves of genus $g$.

Weierstrass semigroups on double covers of genus four curve

Let C be a complete non-singular irreducible curve of genus 4 over an algebraically closed field of characteristic 0. We determine all possible Weierstrass semigroups of ramification points on double covers of C which have genus greater than 11. Moreover, we construct double covers with ramification points whose Weierstrass semigroups are the possible ones.

Arithmetic genus of integral space curves

We give an estimation for the arithmetic genus of an integral space curve, which are not contained in a surface of degree $k-1$. Our main technique is the Bogomolov-Gieseker type inequality for $\mathbb{P}^3$ proved by Macri.

The Chow ring of the stack of hyperelliptic curves of odd genus

We find a new presentation of the stack of hyperelliptic curves of odd genus as a quotient stack and we use it to compute its integral Chow ring by means of equivariant intersection theory.

Rational curves on genus one fibrations

In this paper we look for necessary and sufficient conditions for a genus one fibration to have rational curves. We show that a projective variety with log terminal singularities that admits a relatively minimal genus one fibration $X\rightarrow B$ does contain vertical rational curves if and only if it not isomorphic to a finite étale quotient of a product $\tilde{B}\times E$ over $B$. Many sufficient conditions for the existence of rational curves in a variety that admits a genus one fibration are proved in this paper.

The normal bundle of canonical genus 8 curves

We study the stability of the normal bundle of canonical genus $8$ curves and prove that on a general curve the bundle is stable. The proof rests on Mukai's description of these curves as linear sections of a Grassmannian $\mathrm{G}(2,6)$. This is the next case of a conjecture by M. Aprodu, G. Farkas, and A. Ortega: the general canonical curve of every genus $g \geq 7$ should have stable normal bundle. We also give some more evidence for this conjecture in higher genus.

On the equivariant algebraic Jacobian for curves of genus two

We present a treatment of the algebraic description of the Jacobian of a generic genus two plane curve which exploits an SL2(k) equivariance and clarifes the structure of E.V.Flynn's 72 defining quadratic relations. The treatment is also applied to the Kummer variety.

Surfaces of general type with p_g=q=0 having a pencil of hyperelliptic curves of genus 3

We prove that the bicanonical map of a surfaces of general S type with p_g=q=0 is non birational if there exists a pencil on S whose general member is an hyperelliptic curve of genus 3.

On vanishing of Abelian integrals and contraction of genus two curves

The paper is withdrawn.

Genus 2 curves that admit a degree 5 map to an elliptic curve

We continue our study of genus 2 curves $C$ that admit a cover $ C \to E$ to a genus 1 curve $E$ of prime degree $n$. These curves $C$ form an irreducible 2-dimensional subvariety $Ł_n$ of the moduli space $\M_2$ of genus 2 curves. Here we study the case $n=5$. This extends earlier work for degree 2 and 3, aimed at illuminating the theory for general $n$. We compute a normal form for the curves in the locus $Ł_5$ and its three distinguished subloci. Further, we compute the equation of the elliptic subcover in all cases, give a birational parametrization of the subloci of $Ł_5$ as subvarieties …

The class of the d-elliptic locus in genus 2

We compute the rational Chow class of the locus of genus 2 curves admitting a d-to-1 map to a genus 1 curve, recovering a result of Faber-Pagani when d=2. The answer exhibits quasi-modularity properties similar to those in the Gromov-Witten theory of a fixed genus 1 curve. Along the way, we give a classification of Harris-Mumford admissible covers of a genus 1 curve by a genus 2 curve, which may be of independent interest. As an application of the main calculation, we compute the number of d-elliptic curves in a very general family of genus 2 curves obtained by fixing five branch points of the…

The moduli of curves of genus 6 and K3 surfaces

We prove that the coarse moduli space of curves of genus 6 is birational to an arithmetic quotient of a bounded symmetric domain of type IV by giving a period map to the moduli space of some lattice-polarized K3 surfaces.