# Séminaires : Séminaire Variétés Rationnelles

 Equipe(s) : tga, Responsables : Cyril Demarche et Mathieu Florence Email des responsables : Salle : Adresse : Campus Pierre et Marie Curie Description Le séminaire a généralement lieu à Jussieu (Sorbonne Université, Paris), un vendredi par mois, entre 14h30 et 17h. http://math.univ-lyon1.fr/homes-www/gille//sem/sem_variete_archives.html

 Orateur(s) Marc Levine - Essen, Titre Trace maps, quadratic degrees and quadratic curve counting. Date 13/12/2019 Horaire 14:30 à 15:30 Diffusion Résume This is a report on an on-going work with J. Kass, J. Solomon and K. Wickelgren. Let S be a smooth del Pezzo surface over a field k of characteristic ≠ 2, 3 and let D be an effective curve class on S of non-negative self-intersection. Let $M_{0,n}(S, D)$ denote the Kontsevich moduli space of stable maps of genus 0, n-pointed curves to S in the curve class D and take $n=- K_S.D - 1$. Using the geometry of the double point locus for the universal curve over $M_{0,n}(S, D)$, we construct an orientation for a symmetrized version of the evaluation map $M_{0,n}(S, D) \to S_n$. This orientation allows us to define a section $Wel_{S,D}$ of the sheaf of Grothendieck-Witt rings on the unordered configuration space $Sym^n(S)^0$ as the corresponding trace form. The rank of $Wel_{S,D}$ gives the classical curve count and for k a subfield of R, the signature of $Wel_{S,D}$ recovers Welschinger's invariant Wel(p*) for counting real rational curves through a real point configuration p* on S of degree n. Welschinger's theorem, that $Wel(\sum_i p_i)$depends only on the images of the real points $p_i$ in $\pi_0(S(\mathbf{R}))$generalizes as follows: Let K be an extension field of k and let $\sum_i p_i$ be a K-point of $Sym^n(S)^0$. Then the value $Wel(\sum_i p_i)$ depends only on the classes $[p_i] \in π_0^{\mathbf{A}^1}(S)(K(p_i)).$ Salle 15-16-413 Adresse Campus Pierre et Marie Curie