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Register a new userHyperelliptic curves on $(1,4)$ polarised abelian surfaces
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We investigate the number and the geometry of smooth hyperelliptic curves on a general complex abelian surface. We show that the only possibilities of genera of such curves are $2,3,4$ and $5$. We focus on the genus 5 case. We prove that up to translation, there is a unique hyperelliptic curve in the linear system of a general $(1,4)$ polarised abelian surface. Moreover, the curve is invariant with respect to a subgroup of translations isomorphic to the Klein group. We give the decomposition of the Jacobian of such a curve into abelian subvarieties displaying Jacobians of quotient curves and Prym varieties. Motivated by the construction, we prove the statement: every etale Klein covering of a hyperelliptic curve is a hyperelliptic curve, provided that the group of $2$-torsion points defining the covering is non-isotropic with respect to the Weil pairing and every element of this group can be written as a difference of two Weierstrass points.
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Let $K$ be an algebraically closed field of characteristic different from 2, $g$ a positive integer, $f(x)$ a degree $(2g+1)$ polynomial with coefficients in $K$ and without multiple roots, $C: y^2=f(x)$ the corresponding genus $g$ hyperelliptic curve over $K$ and $J$ the jacobian of $C$. We identify $C$ with the image of its canonical embedding into $J$ (the infinite point of $C$ goes to the zero point of $J$). For each point $P=(a,b)in C(K)$ there are $2^{2g}$ points $frac{1}{2}P in J(K)$. We describe explicitly the Mumford represesentations of all $frac{1}{2}P$. The rationality questions for $frac{1}{2}P$ are also discussed.
Let $C$ be a hyperelliptic curve of genus $g>1$ over an algebraically closed field $K$ of characteristic zero and $O$ one of the $(2g+2)$ Weierstrass points in $C(K)$. Let $J$ be the jacobian of $C$, which is a $g$-dimensional abelian variety over $K$. Let us consider the canonical embedding of $C$ into $J$ that sends $O$ to the zero of the group law on $J$. This embedding allows us to identify $C(K)$ with a certain subset of the commutative group $J(K)$. A special case of the famous theorem of Raynaud (Manin--Mumford conjecture) asserts that the set of torsion points in $C(K)$ is finite. It is well known that the points of order 2 in $C(K)$ are exactly the remaining $(2g+1)$ Weierstrass points. One of the authors proved that there are no torsion points of order $n$ in $C(K)$ if $3le nle 2g$. So, it is natural to study torsion points of order $2g+1$ (notice that the number of such points in $C(K)$ is always even). Recently, the authors proved that there are infinitely many (for a given $g$) mutually nonisomorphic pairs $C,O)$ such that $C(K)$ contains at least four points of order $2g+1$. In the present paper we prove that (for a given $g$) there are at most finitely many (up to a isomorphism) pairs $(C,O)$ such that $C(K)$ contains at least six points of order $2g+1$.
We present a computational approach to general hyperelliptic Riemann surfaces in Weierstrass normal form. The surface is either given by a list of the branch points, the coefficients of the defining polynomial or a system of cuts for the curve. A canonical basis of the homology is introduced algorithmically for this curve. The periods of the holomorphic differentials and the Abel map are computed with the Clenshaw-Curtis method in order to achieve spectral accuracy. The code can handle almost degenerate Riemann surfaces. This work generalizes previous work on real hyperelliptic surfaces with prescribed cuts to arbitrary hyperelliptic surfaces. As an example, solutions to the sine-Gordon equation in terms of multi-dimensional theta functions are studied, also in the solitonic limit of these solutions.
Let $K$ be an algebraically closed field of characteristic different from 2, $g$ a positive integer, $f(x)$ a degree $(2g+1)$ polynomial with coefficients in $K$ and without multiple roots, $C:y^2=f(x)$ the corresponding genus $g$ hyperelliptic curve over K, and $J$ the jacobian of $C$. We identify $C$ with the image of its canonical embedding into $J$ (the infinite point of $C$ goes to the identity element of $J$). It is well known that for each $mathfrak{b} in J(K)$ there are exactly $2^{2g}$ elements $mathfrak{a} in J(K)$ such that $2mathfrak{a}=mathfrak{b}$. M. Stoll constructed an algorithm that provides Mumford representations of all such $mathfrak{a}$, in terms of the Mumford representation of $mathfrak{b}$. The aim of this paper is to give explicit formulas for Mumford representations of all such $mathfrak{a}$, when $mathfrak{b}in J(K)$ is given by $P=(a,b) in C(K)subset J(K)$ in terms of coordinates $a,b$. We also prove that if $g>1$ then $C(K)$ does not contain torsion points with order between $3$ and $2g$.
We compute the number of moduli of all irreducible components of the moduli space of smooth curves on Enriques surfaces. In most cases, the moduli maps to the moduli space of Prym curves are generically injective or dominant. Exceptional behaviour is related to existence of Enriques--Fano threefolds and to curves with nodal Prym-canonical model.
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