On the quantum cohomology of the plane, old and new

On the quantum cohomology of the plane, old and new Z. Ran Abstract We describe a method for counting maps of curves of given genus (and variable moduli) to P 2 , essentially by splitting the P 2 in two; then specialising to the case of genus 0 we show that the method of quantum cohomology may be viewed as the 'mirror' of the former method where one splits the P 1 rather than the P 2 , and we indicate a proof of the associativity of quantum multiplication based on this idea. Recent work on Mirror Symmetry and Quantum Cohomology has contributed to a revival of interest in problems of a classical nature in Enumerative Geometry (cf. [F] and references therein). These problems involve (holomorphic) maps (1) f : C → X where X is a fixed variety and C is a compact Riemann surface whose moduli are sometimes fixed ('Gromov-Witten') but here will not be, unless otherwise stated. While the case dim X = 1 is not entirely without interest (cf. [D]), the problem begins in earnest with dim X = 2 and naturally the simplest such X is P 2. Here the problem specifically is to count the images f (C) of maps (1) where C has genus g, f (C) has degree d and passes through 3d + g − 1 fixed points in P 2. This problem has already, in essence, been solved in the author's earlier paper [R] by means of a recursive method (we note however that the formula in [R], (3c.1), (3c.3) is trivially misstated and the factor c(˜ K 1 , ˜ K 2) should not be present). Our purpose here is twofold. In Sect.1 we give a partial exposition of the method of [R] and illustrate it on a couple of new examples, namely the curves of degree d and genus g = (d−1)(d−2) 2 − 2 (i.e. with 2 nodes); and the rational quartics. We recover classical formulae due, respectively, to Roberts [Ro] and Zeuthen [Z]. Hopefully, this will help make the method of [R] more accessible. In Sect.2 we show that the method of Kontsevich et al., at least as exposed in [F], may be viewed as none other than the 'dual' of that of [R] for the case of rational curves, 'dual'