Related papers: Rational points in the moduli space of genus two

We compute cohomology of the moduli space of genus three curves with level two structure and some related spaces. In particular, we determine the cohomology groups of the moduli space of plane quartics with level two structure as…

We compute the Mori cone of curves of the moduli space \M_{g,n} of stable n-pointed curves of genus g in the case when g and n are relatively small. For instance, we show that for g<14 every curve in \M_g is numerically equivalent to an…

Let $S$ be a closed Riemann surface of genus $g\geq 2$ and let ${\rm Aut}(S)$ be its group of conformal automorphisms. It is well known that if either: (i) ${\rm Aut}(S)$ is trivial or (ii) $S/{\rm Aut}(S)$ is an orbifold of genus zero with…

We describe an algorithm to compute the number of points over finite fields on a broad class of modular curves: we consider quotients $X_H/W$ for $H$ a subgroup of $\GL_2(\mathbb Z/n\mathbb Z)$ such that for each prime $p$ dividing $n$, the…

We compile a complete list of isomorphism class representatives of curves of genus 6 over $\mathbb{F}_2$. We use explicit descriptions of canonical curves in each stratum of the Brill--Noether stratification of the moduli space…

The genus of a maximal curve over a finite field with r^2 elements is either g_0=r(r-1)/2 or less than or equal to g_1=(r-1)^2/4. Maximal curves with genus g_0 or g_1 have been characterized up to isomorphism. A natural genus to be studied…

Given an integer $\gamma\geq 2$ and an odd prime power $q$ we show that for every large genus $g$ there exists a non-singular curve $C$ defined over $\mathbb{F}_q$ of genus $g$ and gonality $\gamma$ and with exactly $\gamma(q+1)$…

Let $X$ be an ordinary smooth curve defined over an algebraically closed field of characteristic 2. The absolute Frobenius induces a rational map $F$ on the moduli space $M_X$ of rank 2 vector bundles with fixed trivial determinant. If the…

We provide new upper bounds on N_q(g), the maximum number of rational points on a smooth absolutely irreducible genus-g curve over F_q, for many values of q and g. Among other results, we find that N_4(7) = 21 and N_8(5) = 29, and we show…

For a smooth projective curve C with genus g >= 2 and a degree 1 line bundle L on C, let M := SU_{C}(r;L) be the moduli space of stable vector bundles of rank r and with the fixed determinant L. In this paper, we study the small rational…

Let $M(2,\textbf{\underline{w}},\chi)$ be the moduli space of rank $2$ torsion-free sheaves over a reducible nodal curve with each component having utmost two nodal singularities. We show that in each component of…

We study moduli spaces $M_X(r,c_1,c_2)$ parametrizing slope semistable vector bundles of rank $r$ and fixed Chern classes $c_1, c_2$ on a ruled surface whose base is a rational nodal curve. We show that under certain conditions, these…

We investigate low degree rational cohomology groups of smooth compactifications of moduli spaces of curves with level structures. In particular, we determine $H^k(\sgbar, \Q)$ for $g \ge 2$ and $k \le 3$, where $\sgbar$ denotes the moduli…

Using an Euclidean approach, we prove a new upper bound for the number of closed points of degree 2 on a smooth absolutely irreducible projective algebraic curve defined over the finite field $\mathbb F\_q$.This bound enables us to provide…

We study the question of whether algebraic curves of a given genus g defined over a field K must have points rational over the maximal abelian extension K^{ab} of K. We give: (i) an explicit family of diagonal plane cubic curves with…

The dimension of the moduli space of smooth pointed curves with prescribed Weierstrass semigroup at the market point is computed for three families of symmetric semigroups of multiplicity six. We also collect the dimensions of such moduli…

Given a smooth cubic hypersurface $X$ over a finite field of characteristic greater than 3 and two generic points on $X$, we use a function field analogue of the Hardy-Littlewood circle method to obtain an asymptotic formula for the number…

A rational map $\phi: \mathbb{P}^1 \to \mathbb{P}^1$ along with an ordered list of fixed and critical points is called a totally marked rational map. The space of totally marked degree two rational maps, $Rat^{tm}_2$ can be parametrized by…

Motivated by several recent results on the geometry of the moduli spaces $\bar{\Cal M}_{g,n}$ of stable curves of genus $g$ with $n$ marked points, here we determine their birational structure for small values of $g$ and $n$ by exploiting…

We give equations for 13 genus-2 curves over $\overline{\mathbb{Q}}$, with models over $\mathbb{Q}$, whose unpolarized Jacobians are isomorphic to the square of an elliptic curve with complex multiplication by a maximal order. If the…