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abstract: 'Let $X$ be a del Pezzo surface of degree $d$, and assume that $X$ is general if $d=1$. We prove that the spaces ${\mathcal M}_{0,0} \bigl( X, \beta \bigr)$ are either empty or irreducible, if $(d, \beta ) \neq (1, -K_X)$. When $(d, \beta ) = (1, -K_X)$ it is well known that ${\mathcal M}_{0,0} \bigl( X, \beta \bigr)$ consists of twelve reduced points.'
author:
- Damiano Testa
title: |
    The irreducibility of the spaces of\
    rational curves on del Pezzo surfaces
---

Introduction {#introduction .unnumbered}
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Let $X$ be a del Pezzo surface. If the degree of $X$ is one, assume that $X$ is general. Let $\beta \in {\rm H}_2 \bigl( X, {\mathbb Z}\bigr)$ be the class of a curve on $X$. Denote by $R(\beta )$ the subscheme of the linear system $|\beta |$ consisting of the integral nodal curves of geometric genus zero. The Kontsevich mapping space $\overline {\mathcal M}_{0,0} \bigl( X, \beta \bigr)$ is a natural compactification of the space $R(\beta )$. Some care is required, since the mapping spaces in general have more irreducible components than the corresponding spaces $R(\beta )$, arising from degenerate configurations of curves on the surface. In fact it may 