Strasbourg, 12 mai 2011

Lieu : Salle de conférences IRMA

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10h30 Accueil, café.

11:00 Broccoli curves and the tropical invariance of Welschinger numbers.

Hannah Markwig (Universität des Saarlandes)

Welschinger invariants count real rational curves on a toric Del Pezzo surface belonging to an ample linear system and passing through a generic conjugation invariant set of points P, weighted with ±1, depending on the nodes of the curve. They can be determined via tropical geometry, i.e. one can define a count of certain tropical curves (which we refer to as Welschinger curves) and prove a Correspondence Theorem stating that this tropical count equals the Welschinger invariant. It follows from the Correspondence Theorem together with the fact that the Welschinger invariants are independent of P that the corresponding tropical count of Welschinger curves is also independent of the chosen points. However, if P consists of not only real points but also pairs of complex conjugate points, no proof of this tropical invariance within tropical geometry has been known so far.

We introduce broccoli curves, certain tropical curves of genus zero which are similar to Welschinger curves. We prove that the numbers of broccoli curves through given (real or complex conjugate) points are independent of the chosen points. In the toric Del Pezzo situation we show that broccoli invariants equal the numbers of Welschinger curves, thus providing a proof of the invariance of Welschinger numbers within tropical geometry.

Joint work with Andreas Gathmann and Franziska Schroeter.

14:00 Asymptotical enumeration of real curves with fixed cogenus.

Benoît Bertrand (Institut de mathématiques de Toulouse / IUT de Tarbes)

The number of real curves of fixed cogenus c and passing throught d(d+3)/2 -c real points in generic position depends on the configuration of points and is bounded by the number of complex such curves. One says that the problem is maximal if there is a configuration such that all solutions are real. I will introduce a signed version of floor diagramms and use them to prove that for a fixed cogenus c the above problem is asymptotically maximal when d tends towards infinity and that it is maximal in cogenus 1.

15h Pause café.

15h30 Puiseux power series solutions for systems of equations.

Lucía López de Medrano Álvarez (Instituto de Matemáticas de la UNAM, Cuernavaca)

We give an algorithm to compute term by term multivariate Puiseux series expansions of series arising as local parametrizations of zeroes of systems of algebraic equations at singular points. The algorithm is an extension of Newton?s method for plane algebraic curves replacing the Newton polygon by the tropical variety of the ideal generated by the system. As a corollary we deduce a property of tropical varieties of quasi-ordinary singularities.

Joint work with F. Aroca and G. Illardi.