Related papers: Generalized superelliptic Riemann surfaces

Let $S$ be a regular minimal surface of general type over the field of complex numbers, and $\mathrm{Aut}_\mathbb{Q}(S)$ the subgroup of automorphisms acting trivially on $H^*(S,\mathbb{Q})$. It has been known since twenty years that…

We are solving the classical Riemann-Hilbert problem of rank N>1 on the extended complex plane punctured in 2m+2 points, for NxN quasi-permutation monodromy matrices. Following Korotkin we solve the Riemann-Hilbert problem in terms of the…

We introduce a natural generalisation of holomorphic curves to morphisms of supermanifolds, referred to as holomorphic supercurves. More precisely, supercurves are morphisms from a Riemann surface, endowed with the structure of a…

A generalized Kummer surface $X=Km_{3}(A,G_{A})$ is the minimal resolution of the quotient of a $2$-dimensional complex torus by an order 3 symplectic automorphism group $G_{A}$. A Kummer structure on $X$ is an isomorphism class of pairs…

Generalising an example by Girondo and Wolfart, we use finite group theory to construct Riemann surfaces admitting two or more regular dessins (i.e. orientably regular hypermaps) with automorphism groups of the same order, and in many cases…

We consider a family of genus $g$ hyperelliptic curves as double ramified coverings over the Riemann sphere with the set of branch points of the form $\{0, \infty, x_1, \dots, x_g, u_1, \dots, u_g\}$. The branch point at infinity $P_\infty$…

Let $\mathcal{X}$ be an irreducible, non-singular, algebraic curve defined over a field of odd characteristic $p$. Let $g$ and $\gamma$ be the genus and $p$-rank of $\mathcal{X}$, respectively. The influence of $g$ and $\gamma$ on the…

In positive characteristic, algebraic curves can have many more automorphisms than expected from the classical Hurwitz's bound. There even exist algebraic curves of arbitrary high genus g with more than 16g^4 automorphisms. It has been…

Cais, Ellenberg and Zureick-Brown recently observed that over finite fields of characteristic two, all sufficiently general smooth plane projective curves of a given odd degree admit a non-trivial rational 2-torsion point on their Jacobian.…

In this paper, we will construct an example of a closed Riemann surface $X$ that can be realized as a quotient of a triply periodic polyhedral surface $\Pi \subset \mathbb{R}^3$ where the Weierstrass points of $X$ coincide with the vertices…

For any unoriented loop on a compact connected oriented surface with one boundary component, the generalized Dehn twist along the loop is defined as an automorphism of the completed group ring of the fundamental group of the surface. If the…

An origami (also known as square-tiled surface) is a Riemann surface covering a torus with at most one branch point. Lifting two generators of the fundamental group of the punctured torus decomposes the surface into finitely many unit…

Let K be an algebraically closed field of characteristic 2, and let X be a curve over K of genus g>1 and 2-rank r>0. For 2-subgroups S of the K-automorphism group Aut(X) of X, the Nakajima bound is |S| < 4g-3. For every g=2^h+1>8, we…

In this note we discuss the geometry of Riemannian surfaces having a discrete set of singular points. We assume the conformal structure extends through the singularities and the curvature is integrable. Such points are called \emph{simple…

We study the groups of automorphisms of rational algebraic surfaces that admit a relatively minimal pencil of curves of arithmetic genus one over an algebraically closed field of arbitrary characteristic. In particular, we classify such…

Let X be a rational nonsingular compact connected real algebraic surface. Denote by Aut(X) the group of real algebraic automorphisms of X. We show that the group Aut(X) acts n-transitively on X, for all natural integers n. As an application…

A generalized Kummer surface $X$ obtained as the quotient of an abelian surface by a symplectic automorphism of order 3 contains a $9\mathbf{A}_{2}$-configuration of $(-2)$-curves. Such a configuration plays the role of the…

A compact complex manifold X is said to be rationally cohomologically rigidified if its automorphism group Aut(X) acts faithfully on the cohomology ring H*(X,Q). In this note, we prove that, surfaces of general type with irregularity q>2…

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…

Let $d \geq 1$, $k \geq 2$ and $n\geq d+1$ be integers. A $d$-dimensional smooth complex algebraic variety $M$ is called a generalized Fermat variety of type $(d;k,n)$ if there is a Galois holomorphic branched covering $\pi:M \to {\mathbb…