Related papers: A computational note about Fricke-Macbeath's curve

Let H(N) denote the Hurwitz class number. It is known that if $p$ is a prime, then {equation*} \sum_{|r|<2\sqrt p}H(4p-r^2) = 2p. {equation*} In this paper, we investigate the behavior of this sum with the additional condition $r\equiv…

In this short note, we propose a definition of complete Hurwitz schemes (and stacks) in mixed characteristic. We follow an idea of R. Pandharipande, and define the complete Hurwitz stack as a substack of stable maps of degree d of nodal…

The defect of a curve over a finite field is the difference between the number of rational points on the curve and the Weil-Serre upper bound for the number of points on the curve. We present algorithms for constructing curves of genus 5,…

For any nonconstant f,g in C(x) such that the numerator H(x,y) of f(x)-g(y) is irreducible, we compute the genus of the normalization of the curve H(x,y)=0. We also prove an analogous formula in arbitrary characteristic when f and g have no…

We show that the number of simple closed geodesics of length bounded by L on a hyperbolic surface of genus g with c cusps and b boundary components grows roughly like L^{6g+2b+2c-6}. This has been conjectured for some time.

In 1891, Hurwitz introduced the enumeration of genus $g$, degree $d$, branched covers of the Riemann sphere with simple ramification over prescribed points and no branching elsewhere. He showed that for fixed degree $d$, the enumeration…

We derive a closed-form expression for all genus 1 Hurwitz numbers, and give a simple new graph-theoretic interpretation of Hurwitz numbers in genus 0 and 1. (Hurwitz numbers essentially count irreducible genus g covers of the sphere, with…

In this note we discuss techniques for determining the automorphism group of a genus $g$ hyperelliptic curve $\X_g$ defined over an algebraically closed field $k$ of characteristic zero. The first technique uses the classical $GL_2…

We characterise smooth curves in a smooth cubic threefold whose blow-ups produce a weak-Fano threefold. These are curves $C$ of genus $g$ and degree $d$, such that (i) $2(d-5) \le g$ and $d\le 6$; (ii) $C$ does not admit a 3-secant line in…

In this work we study the Humbert-Edge's curves of type 5, defined as a complete intersection of four diagonal quadrics in $\mathbb{P}^5$. We characterize them using Kummer surfaces and using the geometry of these surfaces we construct some…

Let k be a perfect field of characteristic p > 0, and let G be a finite group. We consider the pointed G-curves over k associated by Harbater, Katz, and Gabber to faithful actions of G on k[[t]] over k. We use such "HKG G-curves" to…

In 1922, Mordell conjectured that the set of rational points on a smooth curve $C$ over $\mathbb{Q}$ with genus $g \ge 2$ is finite. This has been proved by Faltings in 1983. However, Coleman determined in 1985 an upper bound of…

We compute the class of arithmetic genus two Teichmueller curves in the Picard group of pseudo-Hilbert modular surfaces, distinguished according to their torsion order and spin invariant. As an application, we compute the number of genus…

Let $\mathcal{C}$ be a smooth, projective, genus $g\geq 2$ curve, defined over $\mathbb{C}$. Then $\mathcal{C}$ has \emph{many automorphisms} if its corresponding moduli point $p \in \mathcal{M}_g$ has a neighborhood $U$ in the complex…

We are motivated by cone spherical metrics on compact Riemann surfaces of positive genus to solve a special case of the Hurwitz problem. Precisely speaking, letting $d,\,g$ and $\ell$ be three positive integers and $\Lambda$ be the…

In this note, an analogous statement to the Nagell-Lutz theorem does not hold for the Jacobian of a certain curve of genus 2 over $\mathbb{C}(t)$. As a by-product, we give a (2, 3, 6) quasi-torus decomposition for the dual curve of a smooth…

In this paper, we examine superspecial genus-2 curves $C: y^2 = x(x-1)(x-\lambda)(x-\mu)(x-\nu)$ in odd characteristic $p$. As a main result, we show that the difference between any two elements in $\{0,1,\lambda,\mu,\nu\}$ is a square in…

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…

The Hasse-Weil-Serre bound is improved for curves of low genera over finite fields with discriminant in {-3,-4,-7,-8,-11,-19} by studying optimal curves.

In this paper we prove that there are no hyperelliptic supersingular curves over F_2bar of genus 2^n-1 for any integer n>1. Let g be a natural number, and h=floor(log_2(g+1)+1). Let X be a hyperelliptic curve over F_2bar of genus g>2 and…