Abstract

This is report on part of a program to give refinements of numerical invariants arising in enumerative geometry to invariants living in the Grothendieck-Witt ring over the base-field. Here we define an invariant in the Grothendieck-Witt ring for ``counting'' rational curves. More precisely, for a del Pezzo surface S over a field k and a positive degree curve class $D$ (with respect to the anti-canonical class $-K_S$), we define a class in the Grothendiek-Witt ring of k, whose rank gives the number of rational curves in the class D containing a given collection of distinct closed points $\mathfrak{p}=\sum_ip_i$ of total degree $-D\cdot K_S-1$. This recovers Welschinger's invariants in case $k=\mathbb{R}$ by applying the signature map. The main result is that this quadratic invariant depends only on the $\mathbb{A}^{1$-connected component containing $\mathfrak{p}$ in $Sym}{3d-1}(S)^{0(k)$, where $Sym}{3d-1}(S)^{0$ is the open subscheme of $Sym}{3d-1}(S)$ parametrizing geometrically reduced 0-cycles.