| 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 \([p_i] \in π_0^{\mathbf{A}^1}(S)(K(p_i)).\) |
