Related papers: Fields of moduli of classical Humbert curves

To a family of smooth projective cubic surfaces one can canonically associate a family of abelian fivefolds. In characteristic zero, we calculate the Hodge groups of the abelian varieties which arise in this way. In arbitrary characteristic…

For any almost-simple group $G$ over an algebraically closed field $k$ of characteristic zero, we describe the automorphism group of the moduli space of semistable $G$-bundles over a connected smooth projective curve $C$ of genus at least…

For every $n\ge 6$, we give an example of a finite subset of $\mathbb{P}^{2}$ of degree $n$ which does not descend to any Brauer-Severi surface over the field of moduli. Conversely, for every $n\le 5$ we prove that a finite subset of degree…

Given a smooth plane curve $\overline{C}$ of genus $g\geq 3$ over an algebraically closed field $\overline{k}$, a field $L\subseteq\overline{k}$ is said to be a \emph{plane model-field of definition for $\overline{C}$} if $L$ is a field of…

We give upper bounds of the numbers of holomorphic sections of Veech holomorphic families of Riemann surfaces. The numbers depend only on the topological types of base Riemann surfaces and fibers. We also show a relation between types of…

It is conjectured that there exist finitely many isomorphism classes of simple endomorphism algebras of abelian varieties of GL_2-type over \Q of bounded dimension. We explore this conjecture when particularized to quaternion endomorphism…

We show that the coarse moduli space $\cR_5$ of \'etale double covers of curves of genus~5 over the complex numbers is unirational. We give two slightly different arguments, one purely geometric and the other more computational.

Motivated by a question by D. Mumford : can a computer classify all surfaces with $p_g = 0$ ? we try to show the complexity of the problem. We restrict it to the classification of the minimal surfaces of general type with $p_g = 0, K^2 = 8$…

We give a classification of maximal elements of the set of finite groups that can be realized as the full automorphism groups of polarized abelian surfaces over finite fields.

In this paper we determine the topology of the moduli space $\mathcal{MS}_{1,1}(\vartheta)$ of surfaces of genus one with a Riemannian metric of constant curvature $1$ and one conical point of angle $2\pi\vartheta$. In particular, for…

We show that the higher genus 4-point superstring amplitude is strongly constrained by the geometry of moduli space of Riemann surfaces. A detailed analysis leads to a natural proposal which satisfies several conditions. The result is based…

This paper describes connected components of the strata of holomorphic abelian differentials on marked Riemann surfaces with prescribed degrees of zeros. Unlike the case for unmarked Riemann surfaces, we find there can be many connected…

There is a canonical identification, due to the author, of a convex real projective structure on an orientable surface of genus g and a pair consisting of a conformal structure together with a holomorphic cubic differential on the surface.…

We study the enumerative geometry of algebraic curves on abelian surfaces and threefolds. In the abelian surface case, the theory is parallel to the well-developed study of the reduced Gromov-Witten theory of K3 surfaces. We prove complete…

We extend Hitchin's results on "The self-duality equations on a Riemann surface" (Proc. LMS (3), vol. 55, 1987) to orbifold Riemann surfaces. We prove existence results for orbifold solutions of the Yang-Mills-Higgs equations and construct…

We study closures of GL_2(R)-orbits on the total space of the Hodge bundle over the moduli space of curves under the assumption that they are algebraic manifolds. We show that, in the generic stratum, such manifolds are the whole stratum,…

Using the L^2 norm of the Higgs field as a Morse function, we study the moduli spaces of U(p,q)-Higgs bundles over a Riemann surface. We require that the genus of the surface be at least two, but place no constraints on (p,q). A key step is…

We construct configuration spaces for cyclic covers of the projective line that admit extra automorphisms and we describe the locus of curves with given automorphism group. As an application we provide examples of arbitrary high genus that…

Translation surfaces can be defined in an elementary way via polygons, and arise naturally in in the study of various basic dynamical systems. They can also be defined as Abelian differentials on Riemann surfaces, and have moduli spaces…

This article surveys some recent work of the author on Hilbert modular fourfolds X. After some preliminaries on the cohomology and special, codimension 2 cycles Z on X of Hirzebruch-Zagier type, a proof of the Tate conjecture for X over…