On the compositum of wildly ramified extensions

The ramification filtration on the compositum of two wildly ramified extensions is computed in various cases. Some positive results towards Abhyankar's inertia conjecture have been also proved.     

Harbater-Mumford subvarieties of moduli spaces of covers

We define Harbater-Mumford subvarieties, which are special kinds of closed subvarieties of Hurwitz moduli spaces obtained by fixing some of the branch points. We show that, for many finite groups, finding geometrically irreducible HM-subvarieties defined over is always possible. This provides information on the arithmetic of Hurwitz spaces and applies in particular to the regular inverse Galois problem with (almost all) fixed branch points. Profinite versions of our results can also be stated, providing new tools to study the geometry of modular towers and the regular inverse Galois problem for profinite groups.     

Simplicial ramified covering maps

In this paper we define the concept of a ramified covering map in the category of simplicial sets and we show that it has properties analogous to those of the topological ramified covering maps. We show that the geometric realization of a simplicial ramified covering map is a topological ramified covering map, and we also consider the relation with ramified covering maps in the category of simplicial complexes.     

A characterization of double covers of curves in terms of the ample cone of second symmetric product

We investigate the nef cone spanned by the diagonal and the fibre classes of second symmetric product of a curve of genus g. This 2-dimensional nef cone gives a characterization of double covers of curves of genus . This is a generalization of a result by Debarre [Olivier Debarre, Seshadri constants of abelian varieties, in: The Fano Conference, Univ. Torino, Turin, 2004, pp. 379-394, Proposition 8].     

p-Groups acting on Riemann surfaces

Let p be a prime integer and let $r\ge 3$ be an integer so that $p\ge 5r?7$. We show that a closed Riemann surface S of genus $g\ge 2$ has at most one p-group H of conformal automorphisms so that $S/H$ has genus zero and exactly r cone points. This, in particular, asserts that, for $r=3$ and $p\ge 11$, the minimal field of definition of S coincides with that of $(S,H)$. Another application of this fact, for the case that S is pseudo-real, is that $\mathrm{Aut}(S)/H$ must be either trivial or a cyclic group and that r is necessarily even. This generalizes a result due to Bujalance–Costa for the case of pseudo-real cyclic p-gonal Riemann surfaces.     

Finding rational points on bielliptic genus 2 curves

We discuss a technique for trying to find all rational points on curves of the form Y ²=f ₃ X ⁶+f ₂ X ⁴+f ₁ X ²+f ₀, where the sextic has nonzero discriminant. This is a bielliptic curve of genus 2. When the rank of the Jacobian is 0 or 1, Chabauty's Theorem may be applied. However, we shall concentrate on the situation when the rank is at least 2. In this case, we shall derive an associated family of elliptic curves, defined over a number field ℚα. If each of these elliptic curves has rank less than the degree of ℚα : ℚ, then we shall describe a Chabauty-like technique which may be applied to try to find all the points (x,y) defined over ℚα) on the elliptic curves, for which x∈ℚ. This in turn allows us to find all ℚ-rational points on the original genus 2 curve. We apply this to give a solution to a problem of Diophantus (where the sextic in X is irreducible over ℚ), which simplifies the recent solution of Wetherell. We also present two examples where the sextic in X is reducible over ℚ. Received: 27 November 1998 / Revised version: 4 June 1999     

More about signatures and approximation

Consider a non-singular real algebraic varietyM together with a codimension 1 real algebraic setY M. SupposeY=^–1(0) for a smooth function :M and denote by the signature induced by onM–Y. The following results are proved.For compactM, is induced by a regular functionf R(M) if and only if the setY ^c, where changes sign, is the union of the (d–1)-dimensional parts of some irreducible components ofY if and only if can be approximated by regular functions with the same zero-set. For non-compactM this is true only ifR(M) is a factorial ring. Similar results are proved whenM andY are real analytic instead of algebraic.Dedicated to the memory of our friend Mario RaimondoThe authors are members of GNSAGA of CNR. This work is partially supported by MURST.     

Valuations,intersections et fonctions de Belyi-quelques remarques sur la construction des hauteurs

We present in this article several possibilities to approach the height of an algebraic curve defined over a number field: as an intersection number via the Arakelov theory, as a limit point of the heights of its algebraic points and, finally, using the minimal degree of Belyi functions.     

Classification of real moduli spaces over genus 2 curves

We classify, in terms of simple algebraic equations, the fixed point sets of the moduli space of stable bundles over genus 2 curves with anti-holomorphic involutions.Research supported by SRF of University of Missouri.     

Complex curves of genus three, Kummer surfaces and Quillen metrics

Let C be a smooth, complex, projective curve of genus 3. By choosing an unramified double covering of C, the Abel-Prym map yields an embedding of C into a Kummer surface K when C is non-hyperelliptic. We compute the Quillen metric on the determinant of the cohomologies of with respect to the metric on C induced from the flat Kähler metric on K. For the computation of the Quillen metric, we show the exact self-duality of the Heisenberg-invariant Kummer's quartic surfaces.     

On the torsion of Jacobians of principal modular curves of level 3<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

We demonstrate that the 3-power torsion points of the Jacobians of the principal modular curves X(3ⁿ) are fixed by the kernel of the canonical outer Galois representation of the pro-3 fundamental group of the projective line minus three points. The proof proceeds by demonstrating the curves in question satisfy a two-part criterion given by Anderson and Ihara. Two proofs of the second part of the criterion are provided; the first relies on a theorem of Shimura, while the second uses the moduli interpretation. Received: 30 September 2005     

The Brauer-Manin obstruction for sections of the fundamental group

We introduce the notion of a Brauer-Manin obstruction for sections of the fundamental group extension and establish Grothendieck’s section conjecture for an open subset of the Reichardt-Lind curve.     

Rational classes and divisors on curves of genus 2

We describe a method of looking for rational divisor classes on a curve of genus 2. We have an algorithm to decide if a given class of divisors of degree 3 contains a rational divisor. It is known that the shape of the kernel of Cassel’s morphism (X − T) is related to the existence of rational classes of degree 1. Our key tool is the dual Kummer surface.V. G. L. Neumann supported by CNPq, Brazil     

Local degeneration of the moduli space of vector bundles and factorization of rank two theta functions. I

Supported in part by NSF Mathematics Postdoctoral Fellowship DMS-9007255