Bend, break and count (Q1806258)

This paper develops and applies recursive methods for enumerating rational curves on a surface. The title refers to the methodology: A rational curve which can be ``bent'' sufficiently (i.e. admits a sufficiently large family of deformations) will ``break'' (i.e. admit a reducible limit). Although similiar calculations are apparently available by quantum cohomology, the methods in this paper are more elementary and proceed from pushing the problem onto the codimension 1 boundary of the total space of a family of rational curves. The author then applies this to, among other things, counting curves on del Pezzo surfaces and counting plane curves of geometric genus \(g=2\) and fixed moduli [the case \(g=1\) having been already dealt with by \textit{R. Pandharipande}, Proc. Am. Math. Soc. 125, No.~12, 3471-3479 (1997; Zbl 0883.14031)]. The author acknowledges that some of his results overlap with results of \textit{S. Katz, Z. Qin} and \textit{Y. Ruan} [J. Algebr. Geom. 7, No.~3, 569-587 (1998; Zbl 0946.14031)] and \textit{L. Caporaso} and \textit{J. Harris} [Compos. Math. 113, No.~2, 155-208; 209-236 (1998; Zbl 0930.14036)]. This article is the first of two: See also part II by \textit{Z. Ran} [Math. Proc. Camb. Philos. Soc. 127, 7-12 (1999)].