Toward an algebraic theory of Welschinger invariants

Abstract

Let $S$ be a smooth del Pezzo surface over a field $k$ of characteristic $\ne 2,3$. We define an invariant in the Grothendieck-Witt ring $GW(k)$ for "counting" rational curves in a curve class $D$ of fixed positive degree (with respect to the anti-canonical bundle $-K_S$) and containing a collection of distinct closed points $\mathfrak{p}={\sum }_i{p}_i$ of total degree $r:=-D\cdot K_S-1$ on $S$. This recovers Welschinger's invariant 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 $Sy{m}^r(S{)}^0(k)$, where $Sy{m}^r(S{)}^0$ is the open subscheme of $Sy{m}^r(S)$ parametrizing geometrically reduced 0-cycles.