Characteristic numbers of rational curves with cusp or prescribed triple contact Joachim Kock

This note pursues the techniques of modified psi classes on the stack of stable maps (cf. [Graber-Kock-Pandharipande]) to give concise solutions to the characteristic number problem of rational curves in P2 or P1×P1 with a cusp or a prescribed triple contact. The classes of such loci are computed in terms of modified psi classes, diagonal classes, and certain codimension-2 boundary classes. Via topological recursions the generating functions for the numbers can then be expressed in terms of the usual characteristic number potentials. Introduction With the advent of stable maps and quantum cohomology (Kontsevich-Manin [11]), there has been a tremendous progress in enumerative geometry. One subject of much research activity has been the characteristic number problem, notably for rational curves. Highlights of these developments include Pandharipande [13], who first determined the simple characteristic numbers of rational curves in projective space; Ernström-Kennedy [5] who computed the numbers for P using stable lifts — a technique that also allowed to determine characteristic numbers including a flag condition, as well as characteristic numbers of cuspidal plane curves; and Vakil [16] who used degeneration techniques to give concise recursions for the characteristic numbers also for elliptic curves. With the notions of modified psi classes and the tangency quantum potential introduced in Graber-Kock-Pandharipande [7], conceptually simpler solutions were given to the characteristic number problem for rational curves in any projective homogeneous space, as well as for elliptic curves in P or P×P. Tangency conditions allow simple expressions in terms of modified psi classes, and then the solutions follow from standard principles in Gromov-Witten theory, e.g. topological recursion. Having settled the question of characteristic numbers of nodal rational curves, a natural next problem to consider is that of cuspidal curves, or to impose higher order contacts, Supported by the National Science Research Council of Denmark. E-mail address: [email protected]