Antonio Campillo

Algebroid curves in positive characteristic

Parametrizations of algebroid curves.- Hamburger-noether expansions of algebroid curves.- Characteristic exponents of plane algebroid curves.- Other systems of invariants for the equisingularity of plane algebroid curves.- Twisted algebroid curves.

THE ALEXANDER POLYNOMIAL OF A PLANE CURVE SINGULARITY AND INTEGRALS WITH RESPECT TO THE EULER CHARACTERISTIC

It was shown that the Alexander polynomial (of several variables) of a (reducible) plane curve singularity coincides with the (generalized) Poincare polynomial of the multi-indexed filtration defined by the curve on the ring of germs of functions of two variables. The initial proof of the result was rather complicated (it used analytical, topological and combinatorial arguments). Here we give a new proof based on the notion of the integral with respect to the Euler characteristic over the projectivization of the space — the notion similar to (and inspired by) the notion of the motivic integration.

On the parameters of algebraic-geometry codes related to Arf semigroups

We compute the order (or Feng-Rao (1994)) bound on the minimum distance of one-point algebraic-geometry codes C/sub /spl Omega//(P, /spl rho//sub t/Q), when the Weierstrass semigroup at the point Q is an Arf 91949) semigroup. The results developed to that purpose also provide the dimension of the improved geometric Goppa codes related to these C/sub /spl Omega// (P, /spl rho//sub t/Q).

Universal Abelian covers of rational surface singularities and multi-index filtrations

In previous papers, the authors computed the Poincaré series of some (multi-index) filtrations on the ring of germs of functions on a rational surface singularity. These Poincaré series were expressed as the integer parts of certain fractional power series, whose interpretation was not given. In this paper, we show that, up to a simple change of variables, these fractional power series are reductions of the equivariant Poincaré series for filtrations on the ring of germs of functions on the universal Abelian cover of the surface. We compute these equivariant Poincaré series.

Cones of Curves and of Line Bundles on Surfaces Associated with Curves Having one Place at Infinity

Let V be a pencil of curves in

Some remarks on pythagorean real curve germs

{\bf P}^2

Symbolic Hamburger-Noether expressions of plane curves and applications to AG codes

with one place at infinity, and

On Generators of the Semigroup of a Plane Curve Singularity

X \longrightarrow {\bf P}^2

Poincaré series of curves on rational surface singularities

the minimal composition of point blow-ups eliminating its base locus. We study the cone of curves and the cones of numerically effective and globally generated line bundles on X. It is proved that all of these cones are regular. In particular, this result provides a new class of rational projective surfaces with a rational polyhedral cone of curves. The surfaces in this class have non-numerically effective anticanonical sheaf if the pencil is neither rational nor elliptic. An application is a global version on X of Zariskis unique factorization theorem for complete ideals. We also define invariants of the semigroup of globally generated line bundles on X depending only on the topology of V at infinity. 2000 Mathematical Subject Classification: primary 14C20; secondary 14E05.

HAMBURGER-NOETHER EXPANSIONS OVER RINGS

Let k be a real closed field. A real AP-curve (over k) is a 1-dimensional, excellent Henselian local real domain with residue field k. A 1-dimensional Noetherian local ring is Arf, if emb dim(B)=mult(B) for every local ring B infinitely near to A [ J. Lipman , Amer. J. Math. 93 (1971), 649–685]. For n≥1, the 2nth Pythagoras number p2n of a commutative ring A is the least p, 1≤p≤+∞, such that any sum of 2nth powers in A is a sum of no more than p2nth powers in A. A main purpose of this paper is to affirm the following conjectures proposed by Ruiz [J. Algebra 94 (1985), no. 1, 126–144]: Let A be a real AP-curve, and let A be Pythagorean (i.e., p2=1). Then (i) A is Arf. (ii) Every local ring infinitely near to A is Pythagorean. Actually, the authors obtain a finer result: For a real AP-curve A, the following assertions are equivalent: (1) A is Arf; (2) A is Pythagorean; (3) p2n=1 for some n; (4) p2n=1 for all n. Here, (2)(1) is exactly Conjecture (i) and (1)(2) reduces Conjecture (ii) to the obvious fact that, if A is Arf, every local ring infinitely near to A is Arf too. Of course, the result contains some additional insight into the study of Pythagorass numbers, even of higher order, of real curve germs.