Related papers: Genus 2 Curves in Small Characteristic
We study the problem of efficiently constructing a curve C of genus 2 over a finite field F for which either the curve C itself or its Jacobian has a prescribed number N of F-rational points. In the case of the Jacobian, we show that any…
Let X be an irreducible smooth projective curve of genus at least two over an algebraically closed field k of characteristic p>0. In this paper we study the natural stratification, defined using the absolute Frobenius of X, on the moduli…
Let k=F_q be a finite field of characteristic 2. A genus 3 curve C/k has many involutions if the group of k-automorphisms admits a C_2\times C_2 subgroup H (not containing the hyperelliptic involution if C is hyperelliptic). Then C is an…
Using Galois-Stiefel-Whitney classes of theta characteristics we show that over a totally real base field the moduli stack of smooth genus $g$ curves and the moduli stack of principally polarized abelian varieties of dimension $g$ have…
We calculate the automorphism group of the Kummer surface associated with a curve of genus 2 or the product of two elliptic curves in characteristic two under the assumption that the Kummer surface is a $K3$ surface. Moreover we discuss the…
We present three families of pairs of geometrically non-isomorphic curves whose Jacobians are isomorphic to one another as unpolarized abelian varieties. Each family is parametrized by an open subset of P^1. The first family consists of…
Dans cette note on decrit le champ des courbes hyperelliptiques lisses defines sur un corps algebriquement clos de caracteristique deux comme un champ quotient. On en deduit le groupe de Picard. ----- In this note we describe the stack…
We show the geometric syzygy conjecture in positive characteristic. Specifically, if C is a general smooth curve of genus g defined over an algebraically closed field of characteristic p, then all linear syzygy spaces are spanned by…
In this paper we study genus-$4$ curves obtained as double covers of elliptic curves. Firstly we shall give explicit defining equations of such curves with explicit criterion for whether it is nonsingular, and show the irreducibility of the…
Let $X$ be an ordinary smooth curve defined over an algebraically closed field of characteristic 2. The absolute Frobenius induces a rational map $F$ on the moduli space $M_X$ of rank 2 vector bundles with fixed trivial determinant. If the…
In this paper I prove that for any prime $p$ there is a constant $C_p>0$ such that for any $n>0$ and for any $p$-power $q$ there is a smooth, projective, absolutely irreducible curve over $\mathbb{F}_q$ of genus $g\leq C_p q^n$ without…
We investigate the geometry of \'etale $4:1$ coverings of smooth complex genus 2 curves with the monodromy group isomorphic to the Klein four-group. There are two cases, isotropic and non-isotropic depending on the values of the Weil…
We study the algebraic curve over $\mathbb{F}_{q^2}$ defined by $y^{q+1} = x^n(x^n+1)$, where $n$ is a positive integer coprime to the characteristic. We first prove (when $q$ is odd) that the nonsingular model of this curve is…
In this paper, we are concerned with the computation of the $p$-rank and $a$-number of singular curves and their smooth model. We consider a pair $X, X'$ of proper curves over an algebraically closed field $k$ of characteristic $p$, where…
We give equations for 13 genus-2 curves over $\overline{\mathbb{Q}}$, with models over $\mathbb{Q}$, whose unpolarized Jacobians are isomorphic to the square of an elliptic curve with complex multiplication by a maximal order. If the…
This paper examines the relationship between the automorphism group of a hyperelliptic curve defined over an algebraically closed field of characteristic two and the 2-rank of the curve. In particular, we exploit the wild ramification to…
Explicit models of families of genus 2 curves with multiplication by $\sqrt D$ are known for $D= 2, 3, 5$. We obtain generic models for genus 2 curves over $\mathbb Q$ with real multiplication in 12 new cases, including all fundamental…
In the present note we show that any surface of general type with $p_g=2$,$q=1$ and non-birational bicanonical map has a pencil of curves of genus 2. Combining this result with previous ones, one obtains thatan irregular surface $S$ of…
Our concern is a nonsingular plane curve defined over a finite field of q elements which includes all the rational points of the projective plane over the field. The possible degree of such a curve is at least q+2. We prove that nonsingular…
In this paper we study genus 2 function fields K with degree 3 elliptic subfields. We show that the number of Aut(K)-classes of such subfields of K is 0,1,2, or 4. Also we compute an equation for the locus of such K in the moduli space of…