Bend, Break and Count Preliminary Version: 27.2.97

The purpose of this paper is establish and apply an enumerative formula or 'method' dealing with a family C = { ¯ C y : y ∈ Y } of rational curves on a variety S, e.g. a rational surface. Significantly, the family C is not assumed to be the family of 'all' rational curves of given homology class: rather, we require only that it be sufficiently large (n = dim Y ≥ 3) and well-behaved as regards deformation theory and codimension-1 degenerations. The formula computes the 'degree' d(C), i.e. the number of curves ¯ C y through n general points of S, in terms of analogous degree-type numbers attached to the (codimension-1) boundary component's Z of Y , which parametrize the reducible curves (there are several such numbers depending on how the components of the reducible curve are 'weighted'). Some comments are in order on connections with quantum cohomology. While the author denies any first-hand knowledge or understanding of the latter, its algebro-geometric aspect has been represented as essentially equivalent to certain recursive formulae for counting rational curves, which are contained in the associa-tivity formula for quantum multiplication. It has seemed to the author that these recursions, at least, are largely a matter of taking advantage of the 'slack' in the problem, i.e. the large number of deformation parameters for rational curves (on 'convex' varieties). This viewpoint suggests a connection with Mori's bend-and-break technique, a small part of which is the observation that once a rational curve 'bends' sufficiently (on a surface, this means moving in a 3-parameter family) it will 'break', i.e. admit a reducible limit. Our general formula (Theorem 1, Sect. 1) is merely a quantitative version of this idea. As already indicated, it applies to any given(good enough) family and accordingly does not rely on existence of (compact) moduli spaces for (reparametrization classes of) stable maps as in [FP]. The proof is a completely elementary argument involving (multi-) sections and fibre components on a birationally ruled surface. Now for better or worse, the effect of Theorem 1 is to shift the difficulty elsewhere, namely to 'enumerating' the boundary components Z, which in principle is a lower-degree problem, but not necessarily well-behaved. The simplest Z are of 'product type' and parametrize a pair of independently varying curves plus a point of their intersection: these are unproblematic. However, there are others, such as those parametrizing a pair of …