Related papers: On cyclic covers of the projective line

We study genus 3 hyperelliptic curves which have an extra involution. The locus $\L_3$ 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…

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…

We determine conditions that guarantee that a hyperelliptic or plane curve over a field of characteristic not equal to 2 can be defined over its field of moduli. We also give new examples of curves not definable over their fields of moduli.

A superelliptic curve $\X$ of genus $g\geq 2$ is not necessarily defined over its field of moduli but it can be defined over a quadratic extension of it. While a lot of work has been done by many authors to determine which hyperelliptic…

In this long survey article we show that the theory of elliptic and hyperelliptic curves can be extended naturally to all superelliptic curves. We focus on automorphism groups, stratification of the moduli space $\mathcal{M}_g$, binary…

In this note, we introduce the notion of an unramified strongly cyclic covering for a cyclic curve, a class that has similar properties to, and contains, unramified double covers of hyperelliptic curves. We determine several of their basic…

In this paper we determine automorphism groups of cyclic algebraic curves defined over finite fields of any characteristic.

We consider families of cyclic covers of the projective line, where we fix the covering group and the local monodromies and we vary the branch points. We prove that there are precisely twenty such families that give rise to a special…

The purpose of this paper is to study hyperelliptic curves with extra involutions. The locus $\L_g$ of such genus $g$ hyperelliptic curves is a $g$-dimensional subvariety of the moduli space of hyperelliptic curves $\H_g$. We discover a…

We study automorphism groups of fibered surfaces for finite cyclic covering fibrations of an elliptic surface. We estimate the order of a finite subgroup of automorphism groups in terms of the genus of the fiber, the genus of the base…

Let F be an algebraically closed field with char(F) not equal to 2, let F/K be a Galois extension, and let X be a hyperelliptic curve defined over F. Let \iota be the hyperelliptic involution of X. We show that X can be defined over its…

To every covering of curves, we associate several varieties having the same field of moduli and same fields of definition. We deduce examples of curves having Q (the field of rationals) as field of moduli, that admit models over any…

We study the Picard groups of moduli spaces of smooth complex projective curves that have a group of automorphisms with a prescribed topological action. One of our main tools is the theory of symmetric mapping class groups. In the first…

Let G be a finite group, and $g \geq 2$. We study the locus of genus g curves that admit a G-action of given type, and inclusions between such loci. We use this to study the locus of genus g curves with prescribed automorphism group G. We…

Let $\L_g^G$ denote the locus of hyperelliptic curves of genus $g$ whose automorphism group contains a subgroup isomorphic to $G$. We study spaces $\L_g^G$ for $G \iso \Z_n, \Z_2{\o}\Z_n, \Z_2{\o}A_4$, or $SL_2(3)$. We show that for $G \iso…

A fine moduli space is constructed, for cyclic-by-$\mathsf{p}$ covers of an affine curve over an algebraically closed field $k$ of characteristic $\mathsf{p}>0$. An intersection of finitely many fine moduli spaces for cyclic-by-$\mathsf{p}$…

Let $\mathcal X_g$ be a genus $g\geq 2$ superelliptic curve, $F$ its field of moduli, and $K$ the minimal field of definition. In this short note we construct an equation of the curve $\mathcal X_g$ over its minimal field of definition $K$…

The Prym map assigns to each covering of curves a polarized abelian variety. In the case of unramified cyclic covers of curves of genus two, we show that the Prym map is ramified precisely on the locus of bielliptic covers. The key…

Let $X$ be a smooth projective algebraic curve of genus $g\geq 2$ defined over a field $K$. We show that $X$ can be defined over its field of moduli if it has odd signature, i.e. if the signature of the covering $X\to X/\Aut(X)$ is of type…

We study genus $g$ hyperelliptic curves with reduced automorphism group $A_5$ and give equations $y^2=f(x)$ for such curves in both cases where $f(x)$ is a decomposable polynomial in $x^2$ or $x^5$. For any fixed genus the locus of such…