Gilberto Bini, Gerard van der Geer
For a partition $lambda=\{lambda_1 \geq λ_2 \geq λ_3 \}$ of non-negative integers, we calculate the Euler characteristic of the local system $V_λ$ on the moduli space of genus 3 hyperelliptic curves using a suitable stratification. For some $λ$ of low degree, we make a guess for the motivic Euler characteristic of $V_λ$ using counting of curves over finite fields.
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We study the moduli space of genus 3 hyperelliptic curves via the weighted projective space of binary octavics. This enables us to create a database of all genus 3 hyperelliptic curves defined over $\mathbb Q$, of weighted moduli height $\mathcal h =1$.
Reduction of bielliptic hyperelliptic functions of genus 3
针对亏格3超椭圆函数的约化问题，受方程理论与超椭圆函数可积动力系统应用启发，研究人员将具有2次椭圆曲线态射的亏格3超椭圆曲线的对应超椭圆函数，用Weierstrass椭圆函数和亏格2超椭圆函数表示。
We give a covariant treatment of the quadratic differential identities satisfied by the P-functions on the Jacobian of smooth hyperelliptic curves of genera 1, 2 and 3.
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 $g$.
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.
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.
We compute the cohomological invariants with coefficients in $\mathbb{Z}/p\mathbb{Z}$ of the stack $\mathscr{H}_3$ of hyperelliptic curves of genus $3$ over an algebraically closed field.
In this paper, we study a Howe curve $C$ in positive characteristic $p \geq 3$ which is of genus 3 and is hyperelliptic. We will show that if $C$ is superspecial, then its standard form is maximal or minimal over $\mathbb{F}_{p^2}$ without taking its $\mathbb{F}_{p^2}$-form.
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.
We prove that a certain Brill-Noether locus over a non-hyperelliptic curve $C$ of genus 4, is isomorphic to the \textit{Donagi-Izadi cubic threefold} in the case when the pencils of the two trigonal line bundles of $C$ coincide.
We compute the cohomological invariants of $\mathcal{H}_g$, the moduli stack of smooth hyperelliptic curves, for every odd $g$.
We compute the integral Chow ring of the stack of smooth, non-hyperelliptic curves of genus three. We obtain this result by computing the integral Chow ring of the stack of smooth plane quartics, by means of equivariant intersection theory.
In this paper, we compute the 1-gap sequences of 1-Weierstrass points on non-hyperelliptic smooth projective curves of genus 10. Furthermore, the geometry of such points is classified as flexes, sextactic and tentactic points. Also, an upper bounds for their numbers are estimated.
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.
Let $g$ be an even positive integer, and $p$ be a prime number. We compute the cohomological invariants with coefficients in $\mathbb{Z}/p\mathbb{Z}$ of the stacks of hyperelliptic curves $\mathscr{H}_g$ over an algebraically closed field $k_0$.
We compute the class of the closure of the locus of hyperelliptic curves in the moduli space of stable genus-3 curves in terms of the tautological class $λ$ and the boundary classes $δ_0$ and $δ_1$. The expression of this class is known, but here we compute it directly, by means of Porteous Formula, without resorting to blowups or test curves.
We construct an infinite number of Shimura curves contained in the locus of hyperelliptic Jacobians of genus 3. In the opposite direction, we show that in genus 3 the only possible non-complete (in the moduli space of abelian threefolds) Kuga curves contained in the hyperelliptic locus have the same degeneration data as that of the examples we construct. The locus of genus 3 hyperelliptic Jacobians is a divisor within the moduli space of principally polarized abelian threefolds, and our result demonstrates the techniques we develop more generally for dealing with Shimura curves contained withi…
Klein coverings over hyperelliptic genus 3 curves
本文研究了超椭圆亏格为3的曲线的克莱因覆盖的模空间，证明了普里姆映射的单射性，并展示了其普里姆映射的有限性。
This is the part II of the series under the same title. In part I, using the approach developed by Catanese--Pignatelli arXiv:math/0503294, we gave a structure theorem for hyperelliptic genus 3 fibrations all of whose fibers are 2-connected (arXiv:1209.6278 [math.AG]). In this part II, we shall give a structure theorem for smooth deformation families of these to non-hyperelliptic genus 3 fibrations. As an application, we shall give a set of sufficient conditions for our genus 3 hyperelliptic fibration above to allow deformation to non-hyperelliptic fibrations, and use this to study certain min…
We discuss Weber's formula which gives the quotient of two Thetanullwerte for a plane smooth quartic in terms of the bitangents. In particular, we show how it can easily be derived from the Riemann-Jacobi formula.
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.
We study degree 2 and 4 elliptic subcovers of hyperelliptic curves of genus 3 defined over $\mathbb C$. The family of genus 3 hyperelliptic curves which have a degree 2 cover to an elliptic curve $E$ and degree 4 covers to elliptic curves $E_1$ and $E_2$ is a 2-dimensional subvariety of the hyperelliptic moduli $\mathcal H_3$. We determine this subvariety explicitly. For any given moduli point $\mathfrak p \in \mathcal H_3$ we determine explicitly if the corresponding genus 3 curve $\mathcal X$ belongs or not to such family. When it does, we can determine elliptic subcovers $E$, $E_1$, and $E_…
We study genus 3 hyperelliptic curves which have an extra involution. The locus $Ł_3$ of these curves is a 3-dimensional subvariety in the genus 3 hyperelliptic moduli $\H_3$. We find a birational parametrization of this locus by affine 3-space. For every moduli point $\p \in \H_3$ such that $|\Aut (\p)|>2$, the field of moduli is a field of definition. We provide a rational model of the curve over its field of moduli for all moduli points $\p \in \H_3$ such that $|\Aut(\p)|>4$. This is the first time that such a rational model of these curves appears in the literature.
In this paper, we list all cyclic automorphisms subgroups $H$ for which there exists a smooth projective non-hyperelliptic sextic curve $C$ with $H\preceq Aut(C)$. Furthermore, we attach to each group a defining equation of a plane sextic curve having exactly this group as cyclic automorphisms subgroup. These equations cover up to isomorphism all plane non-singular sextic curves having some non-trivial automorphism.
In this work we study the Humbert-Edge's curves of type 5, defined as a complete intersection of four diagonal quadrics in $\mathbb{P}^5$. We characterize them using Kummer surfaces and using the geometry of these surfaces we construct some vanishing thetanulls on such curves. In addition, we describe an argument to give an isomorphism between the moduli space of Humbert-Edge's curves of type 5 and the moduli space of hyperelliptic curves of genus 2, and we let see how this argument can be generalized to state an isomorphism between the moduli space of hyperelliptic curves of genus $g=\frac{n-…
A rational function on a real algebraic curve $C$ is called separating if it takes real values only at real points. Such a function defines a covering $\Bbb R C\to\Bbb{RP}^1$. Let $A_1,\dots,A_n$ be connected components of $C$. In a recent paper M. Kummer and K. Shaw defined the separating semigroup of $C$ as the set of all sequences $(d_1(f),\dots,d_n(f))$ where $f$ is a separating function and $d_i(f)$ is the degree of the restriction of $f$ to $A_i$. We describe the separating semigroup for hyperelliptic curves and for genus 3 curves.
The generating function for $S_n$-equivariant Euler characteristics of moduli spaces of pointed hyperelliptic curves for any genus g>1 is calculated. This answer generalizes the known ones for genera 2 and 3 and answers obtained by J. Bergstrom for any genus and n<8 points.
We shall study the structure of hyperelliptic fibrations of genus 3, from the view point given by Catanese and Pignatelli in arXiv:math/0503294. In this part I, we shall give a structure theorem for such fibrations for the case of f : S \to B with all fibers 2-connected. We shall also give, for the case of B projective line, sufficient conditions for the existence of such fibrations from the view point of our structure theorem, prove the uniqueness of the deformation type and the simply connectedness of S for some cases, and give some examples including those with minimal regular S with geomet…
We compute the intersection Betti numbers of the GIT model of the moduli space of Brill-Noether-Petri general curves of genus 4. This space was shown to be the final non-trivial log canonical model for the moduli space of stable genus four curves, under the Hassett-Keel program. The strategy of the cohomological computation relies on a general method developed by Kirwan to calculate the cohomology of GIT quotients of projective varieties, based on the equivariantly perfect stratification of the unstable points studied by Hesselink and others and a partial resolution of singularities.
We give an algorithm to compute $(\ell,\ell,\ell)$-isogenies from the Jacobians of genus three hyperelliptic curves to the Jacobians of non-hyperelliptic curves. An important application is to reduce the discrete logarithm problem in the Jacobian of a hyperelliptic curve to the corresponding problem in the Jacobian of a non-hyperelliptic curve.
The Horikawa index and the local signature are introduced for relatively minimal fibered surfaces whose general fiber is a non-hyperelliptic curve of genus $4$ with unique trigonal structure.
Upper bounds on the genus of hyperelliptic Albanese fibrations
本文研究一般型极小非正则曲面上由阿尔巴内塞映射诱导的超椭圆纤维化，证明了其亏格满足二次上界，并构造例子说明该上界不能改进为线性，同时在特殊情形下得到亏格不超过14。
When the genus $g$ is even, we extend the computation of mod 2 cohomological invariants of $\mathcal{H}_g$ to non algebraically closed fields, we give an explicit functorial description of the invariants and we completely describe their multiplicative structure. In the Appendix, we show that the cohomological invariants of the compactification $\overline{\mathcal{H}}_g$ are trivial, and use our methods to give a very short proof of a result by Cornalba on the Picard group of the compactification $\overline{\mathcal{H}}_g$ and extend it to positive characteristic
Hyperelliptic genus 3 curves with involutions and a Prym map
本文刻画了含两个额外对合的亏格3超椭圆曲线，给出显式方程，并证明在固定椭圆商曲线时Prym映射为2:1，同时构造了不与对偶同构的极化阿贝尔曲面族。
In this paper we study bielliptic curves of genus 3 defined over an algebraically closed field $k$ and the intersection of the moduli space $\M_3^b$ of such curves with the hyperelliptic moduli $\H_3$. Such intersection $§$ is an irreducible, 3-dimensional, rational algebraic variety. We determine the equation of this space in terms of the $Gl(2, k)$-invariants of binary octavics as defined in \cite{hyp_mod_3} and find a birational parametrization of $§$. We also compute all possible subloci of curves for all possible automorphism group $G$. Moreover, for every rational moduli point $\p \in §$…
Hyperelliptic classes are rigid and extremal in genus two
本文证明了亏格二模空间上带标记 Weierstrass 点、共轭点对和自由点的超椭圆曲线轨迹类在有效锥中是刚性和极端的，推广了 Chen 和 Tarasca 的结果，并构造了任意高余维数的无限族刚性和极端类。
We consider the moduli space $\Hh_{g,n}$ of $n$-pointed smooth hyperelliptic curves of genus $g$. In order to get cohomological information we wish to make $\s_n$-equivariant counts of the numbers of points defined over finite fields of this moduli space. We find recurrence relations in the genus that these numbers fulfill. Thus, if we can make $\s_n$-equivariant counts of $\Hh_{g,n}$ for low genus, then we can do this for every genus. Information about curves of genus 0 and 1 is then found to be sufficient to compute the answers for $\Hh_{g,n}$ for all $g$ and for $n \leq 7$. These results ar…
Enumeration Algorithm for Genus-4 Superspecial Hyperelliptic Curves with Automorphism Group $Q_8$
本文提出枚举自同构群为四元数群 $Q_8$ 的亏格 4 超特殊超椭圆曲线的算法，并通过 Magma 实现得到特征 $p<10000$ 下的同构类数量，进而提出关于同构类个数与 $p$ 模 8 关系的猜想。
Non-existence of Shimura curves of Mumford type generically in the non-hyperelliptic locus
研究证明在非双曲曲线的Teichmüller族中，不存在具有严格最大Higgs场的Shimura曲线，特别是Mumford型Shimura曲线不在该族中。
The affine ring A of the affine Jacobian variety of a hyperelliptic curve of genus 3 is studied as a D-module. The conjecture on the minimal D-free resolution previously proposed is proved in this case. As a by-product a linear basis of A is explicitly constructed in terms of derivatives of Klein's hyperelliptic pe functions.
In this paper we generalize the formula of Frobenius-Stickelberger and the formula of Kiepert to genus-three case. The latter is well-known determinant expression for any division polynomial of any elliptic curve.
The multiplicity-one theorem for the superspeciality of hyperelliptic curves
本文通过Lauricella系统类型D推广了超特殊性同时方程的多重性一定理，适用于任意亏格的超椭圆曲线。
We consider a rational surface with a relatively minimal fibration. Picard number of a such fibred surface is bounded in terms of the genus of a general fibre. When Picard number is the maximum for any given genus, we characterize a such fibred surface whose Mordell-Weil group is trivial by singular fibres. Furthermore, we describe the defining equation explicitly.
Let $p$ be an odd prime number and be an integer coprime to $p$. We survey an algorithm for computing explicit rational representations of $(\ell,...,\ell)$-isogenies between Jacobians of hyperelliptic curves of arbitrary genus over an extension $K$ of the field of $p$-adic numbers $\mathbb{Q}\_p$. The algorithm has a quasi-linear complexity in $\ell$ as well as in the genus of the curves.
We study the Abramovich--Vistoli moduli space of genus zero orbifold stable maps to [Sym^2 P^2], the stack symmetric square of P^2. This compactifies the moduli space of stable maps from hyperelliptic curves to P^2, and we show that all genus zero Gromov--Witten invariants are determined from trivial enumerative geometry of hyperelliptic curves. We also show how the genus zero Gromov--Witten invariants can be used to determine the number of hyperelliptic curves of degree d and genus g interpolating 3d + 1 generic points in P^2. Comparing our method to that of Graber for calculating the same nu…
We describe an algorithm to compute the zeta function of any non-hyperelliptic genus 3 plane curve $C$ over a finite field with automorphism group $G = \mathbb{Z} / 2 \mathbb{Z}$. This algorithm computes in the Monsky-Washnitzer cohomology of~the curve. Using the relation between the Monsky-Washnitzer cohomology of $C$ and its quotient $E := C/G$, the computation splits into 2 parts: one in a subspace of the Monsky-Washnitzer cohomology and a second which reduces to the point counting on an elliptic curve $E$. The former corresponds to the dimension $2$ abelian surface $\mathrm{ker}(\mathrm{Ja…
We give an explicit solution to the problem of differentiation of hyperelliptic functions in genus 4 case. We describe explicitly the polynomial Lie algebras and polynomial dynamical systems connected to this problem.
This paper provides a probabilistic algorithm to determine generators of the m-torsion subgroup of the Jacobian of a hyperelliptic curve of genus two.
We describe explicitly all hyperelliptic limits of quadrics through smooth canonical curves of genus $g$ in ${\mathbb P}^{g-1}$. Also, we construct an open embedding of the blow up of a ${\rm PGL}_g$-bundle over the moduli space of curves of genus $g$ along the hyperelliptic locus into the blow up of the canonical Hilbert scheme of ${\mathbb P}^{g-1}$ along the closure of the locus of canonical ribbons, which are certain double thickenings of rational normal curves introduced and studied by Bayer and Eisenbud.
For the evaluation and inversion of abelian integrals we show that the image of the Abel-Jacobi map of genus less than 5 hyperelliptic curve in its Jacobian is the intersection of shifted theta divisors with specified shifts. Therefore the image is a solution of a (slightly overdetermined) set of equations in the Jacobian.
Let $K$ be an algebraically closed field of characteristic different from $2$, $g$ a positive integer, $f(x)\in K[x]$ a degree $2g+1$ monic polynomial without repeated roots, $C_f: y^2=f(x)$ the corresponding genus g hyperelliptic curve over $K$, and $J$ the jacobian of $C_f$. We identify $C_f$ with the image of its canonical embedding into $J$ (the infinite point of $C_f$ goes to the zero of group law on $J$). It is known (arXiv:1809.03061 [math.AG]) that if $g>1$ then $C_f(K)$ does not contain torsion points, whose order lies between $3$ and $2g$. In this paper we study torsion points of ord…
One of the big questions in the area of curves over finite fields concerns the distribution of the numbers of points: Which numbers occur as the number of points on a curve of genus $g$? The same question can be asked of various subclasses of curves. In this article we classify the possibilities for the number of points on genus 4 hyperelliptic supersingular curves over finite fields of order $2^n$, $n$ odd.
Let $X$be a complex hyperelliptic curve of genus two equipped with the canonical metric $ds^2$. We study mean field equations on complex hyperelliptic curves and show that the Gaussian curvature function of $(X,ds^2)$ determines an explicit solution to a mean field equation.
We discuss a family of multi-term addition formulae for Weierstrass functions on specialized curves of genus one and two with many automorphisms. In the genus one case we find new addition formulae for the equianharmonic and lemniscate cases, and in genus two we find some new addition formulae for a number of curves, including the Burnside curve.
We exhibit a non-hyperelliptic curve C of genus 3 such that the class of the Ceresa cycle [C]-[-C] in the intermediate Jacobian of JC is torsion.
Unirational quasi-hyperelliptic surfaces in characteristic five
本文研究特征5下的单有理拟超椭圆曲面，对其奇异纤维分类，给出算术亏格和典范除子自交数公式，并作为推论通过确定奇异纤维组合、定义方程和截面分类有理拟超椭圆曲面。
We describe a computation of rational points on genus 3 hyperelliptic curves $C$ defined over $\mathbb{Q}$ whose Jacobians have Mordell-Weil rank 1. Using the method of Chabauty and Coleman, we present and implement an algorithm in Sage to compute the zero locus of two Coleman integrals and analyze the finite set of points cut out by the vanishing of these integrals. We run the algorithm on approximately 17,000 curves from a forthcoming database of genus 3 hyperelliptic curves and discuss some interesting examples where the zero set includes global points not found in $C(\mathbb{Q})$.
It is well known that, fixed an even, unimodular, positive definite quadratic form, one can construct a modular form in each genus; this form is called the theta series associated to the quadratic form. Varying the quadratic form, one obtains the ring of stable modular forms. We show that the differences of theta series associated to specific pairs of quadratic forms vanish on the locus of hyperelliptic Jacobians in each genus. In our examples, the quadratic forms have rank 24, 32 and 48. The proof relies on a geometric result about the boundary of the Satake compactification of the hyperellip…
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 .
In this work we give an explicit solution to the problem of differentiation of hyperelliptic functions in genus $3$ case. It is a genus $3$ analogue of the result of F. G. Frobenius and L. Stickelberger. Our method is based on the series of works by V. M. Buchstaber, D. V. Leikin and V. Z. Enolskii. We describe a polynomial map $p\colon \mathbb{C}^{3g} \to \mathbb{C}^{2g}$. For $g = 1,2,3$ we describe $3g$ polynomial vector fields in $\mathbb{C}^{3g}$ projectable for $p$ and their polynomial Lie algebras. We obtain the corresponding derivations of the field of hyperelliptic functions.
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