Related papers: How many quasiplatonic curves?

We classify compact Riemann surfaces of genus $g$, where $g-1$ is a prime $p$, which have a group of automorphisms of order $\rho(g-1)$ for some integer $\rho\ge 1$, and determine isogeny decompositions of the corresponding Jacobian…

The goal of this work is to give new quantitative results about the distribution of semi-arithmetic hyperbolic surfaces in the moduli space of closed hyperbolic surfaces. We show that two coverings of genus $g$ of a fixed arithmetic surface…

Define $QC(n)$ to be the number of quasiplatonic topological actions of the cyclic group $C_n$ on surfaces of genus at least two. We use formulas of Benim and Wootton to give an explicit formula for $QC(n)$. In addition, we relate the…

We show that up to isomorphism there are exactly twenty pairs $(C,E)$, where $C$ is a genus-$2$ curve over ${\mathbf C}$, where $E$ is an elliptic curve over ${\mathbf C}$, and where for every integer $n>1$ there is a map of degree $n$ from…

For any genus g greater than 1, we construct a family of dimension g+1 of pairs of hyperelliptic curves of genus g whose jacobian are 2^g isogeneous. ----- Pour tout genre g superieur ou egal a 2, nous construisons une famille a g+1…

Let $S$ be a surface isogenous to a product of curves of unmixed type. After presenting several results useful to study the cohomology of $S$ we prove a structure theorem for the cohomology of regular surfaces isogenous to a product of…

We construct infinitely many linearly independent quasi-homomorphisms on the mapping class group of a Riemann surface with genus at least two which vanish on a handlebody subgroup. As a corollary, we disprove a conjecture of Reznikov on…

We give the asymptotic growth of the number of (multi-)arcs of bounded length between boundary components on complete finite-area hyperbolic surfaces with boundary. Specifically, if $S$ has genus $g$, $n$ boundary components and $p$…

In the article we give a self-contained new proof that a normal quasi-ordinary surface germ is analytically isomorphic to a cyclic quotient surface germ.

We show that a F_{q^2}-maximal curve of genus q(q-3)/6 in characteristic three is either a non-reflexive space curve of degree q+1, or it is uniquely determined up to F_{q^2}-isomorphism by a plane model of Artin-Schreier type

It is shown that the number of labelled graphs with n vertices that can be embedded in the orientable surface S_g of genus g grows asymptotically like $c^{(g)}n^{5(g-1)/2-1}\gamma^n n!$ where $c^{(g)}>0$, and $\gamma \approx 27.23$ is the…

We show that the asymptotic growth rate for the minimal cardinality of a set of simple closed curves on a closed surface of genus $g$ which fill and pairwise intersect at most $K\ge 1$ times is $2\sqrt{g}/\sqrt{K}$ as $g \to \infty$ . We…

The present paper gives an explicit classification of the isomorphism classes of non-hyperelliptic genus 4 curves over an algebraically closed field of characteristic 0. A non-hyperelliptic genus 4 curve lies on a quadric in $\mathbb{P^3}$…

We compute odd-degree genus 1 quasimap (and Gromov--Witten) invariants of moduli spaces of Higgs $\mathrm{SL}_2$-bundles on a curve of genus $g\geq2$. We also compute certain invariants for all prime ranks. This proves some parts of…

A Mumford curve of genus g (>1) over a non-archimedean valued field k of positive characteristic has at most max{12(g-1), 2 g^(1/2) (g^(1/2)+1)^2} automorphisms. This bound is sharp in the sense that there exist Mumford curves of arbitrary…

We show that any infinite-type surface without planar ends admits arbitrarily large families of length isospectral hyperbolic structures. If the surface has infinite genus and its space of ends is self-similar, we construct an uncountable…

We study plane curves of type p,q having only nodes as singularities. Every Weierstra\ss semigroup is the Weierstra\ss semigroup of such a curve at its place at infinity for properly chosen p,q. We construct plane curves of type p,q with…

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…

We show that the moduli spaces of non-triple cyclic coverings of genus 2 curves and that of even spin curves of genus 2 are birationally isomorphic via the Prym map. We describe the log canonical models of the moduli space of even spin…

Let $k$ be an algebraically closed field of characteristic $p >0$. Suppose $g \geq 3$ and $0 \leq f \leq g$. We prove there is a smooth projective $k$-curve of genus $g$ and $p$-rank $f$ with no non-trivial automorphisms. In addition, we…