The moduli and the global period mapping of surfaces with : a counterexample to the global Torelli problem
The moduli space of curves of genus two covering elliptic curves.
The moduli space of Riemann surfaces is Kähler hyperbolic.
The Moduli Spaces of Vector Bundles over an Algebraic Curve.
The monodromy conjecture for zeta functions associated to ideals in dimension two
The monodromy conjecture states that every pole of the topological (or related) zeta function induces an eigenvalue of monodromy. This conjecture has already been studied a lot. However in full generality it is proven only for zeta functions associated to polynomials in two variables.In this article we work with zeta functions associated to an ideal. First we work in arbitrary dimension and obtain a formula (like the one of A’Campo) to compute the “Verdier monodromy” eigenvalues associated to an...
The Monodromy of a Curve with Ordinary Double Points.
The Monodromy of a Degenerating Family of Curves.
The Monodromy of Reducible Plane Curves.
The monodromy of Weierstrass points.
The Mordell conjecture revisited
The Mumford conjecture
The Mumford Conjecture asserts that the rational cohomology of the stable moduli space of Riemann surfaces is a polynomial algebra on the Mumford-Morita-Miller characteristic classes; this can be reformulated in terms of the classifying space derived from the mapping class groups. The conjecture admits a topological generalization, inspired by Tillmann’s theorem that admits an infinite loop space structure after applying Quillen’s plus construction. The text presents the proof by Madsen and...
The Normal Bundle of a Curve on a Quadric.
The number of curves of genus two with elliptic differentials.
The number of elliptic curves over Q with conductor N.
The Number of Twisted Cubic Curves on the General Quintic Threefold.
The optimality of the Bounded Height Conjecture
In this article we show that the Bounded Height Conjecture is optimal in the sense that, if is an irreducible subvariety with empty deprived set in a power of an elliptic curve, then every open subset of does not have bounded height. The Bounded Height Conjecture is known to hold. We also present some examples and remarks.
The -adic gamma function and the congruences of Atkin and Swinnerton-Dyer
The -part of Tate-Shafarevich groups of elliptic curves can be arbitrarily large
In this paper we show that for every prime the dimension of the -torsion in the Tate-Shafarevich group of can be arbitrarily large, where is an elliptic curve defined over a number field , with bounded by a constant depending only on . From this we deduce that the dimension of the -torsion in the Tate-Shafarevich group of can be arbitrarily large, where is an abelian variety, with bounded by a constant depending only on .
The -rank stratification of Artin-Schreier curves
We study a moduli space for Artin-Schreier curves of genus over an algebraically closed field of characteristic . We study the stratification of by -rank into strata of Artin-Schreier curves of genus with -rank exactly . We enumerate the irreducible components of and find their dimensions. As an application, when , we prove that every irreducible component of the moduli space of hyperelliptic -curves with genus and -rank has dimension . We also determine all pairs for...
The Picard group of a coarse moduli space of vector bundles in positive characteristic
Let C be a smooth projective curve over an algebraically closed field of arbitrary characteristic. Let M r,Lss denote the projective coarse moduli scheme of semistable rank r vector bundles over C with fixed determinant L. We prove Pic(M r,Lss) = ℤ, identify the ample generator, and deduce that M r,Lss is locally factorial. In characteristic zero, this has already been proved by Drézet and Narasimhan. The main point of the present note is to circumvent the usual problems with Geometric Invariant...