Abstract. I give a conjectural generating function for the numbers of δ-nodal curves in a linear system of dimension δ on an algebraic surface. It reproduces the results of Vainsencher [V2] for the case δ ≤ 6 and Kleiman-Piene [K-P] for the case δ ≤ 8. The numbers of curves are expressed in terms of five universal power series, three of which I give explicitly as quasimodular forms. This gives in particular the numbers of curves of arbitrary genus on a K3 surface and an abelian surface in terms of quasimodular forms, generalizing the formula of Yau-Zaslow for rational curves on K3 surfaces. The coefficients of the other two power series can be determined by comparing with the recursive formulas of Caporaso-Harris for the Severi degrees in P2. We verify the conjecture for genus 2 curves on an abelian surface. We also discuss a link of this problem with Hilbert schemes of points. 1.

Abstract. We collect various known results (about plane curves and the moduli space of stable maps) to derive new recursive formulas enumerating low genus plane curves of any degree with various behaviors. Recursive formulas are given for the characteristic numbers of rational plane curves, elliptic plane curves, and elliptic plane curves with fixed complex structure. Recursions are also given for the number of elliptic (and rational) plane curves with various “codimension 1 ” behavior (cuspidal, tacnodal, triple pointed, etc., as well as the geometric and arithmetic sectional genus of the Severi variety). We compute the latter numbers for genus 2 and 3 plane curves as well. We rely on results of Caporaso, Diaz, Getzler, Harris, Ran, and especially Pandharipande. 1.

Characteristic numbers of families of maps of nodal curves to P² are de ned as intersection of natural divisor classes. (This definition agrees with the usual definition for families of plane curves.) Simple recursions for characteristic numbers of genus one plane curves of all degrees are computed.

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Abstract. We compute the number of irreducible rational curves of given degree with 1 tacnode or 1 triple point in P 2 or 1 node in P 3 meeting an appropriate generic collection of points and lines. As a byproduct, we also compute the number of rational plane curves of degree d passing through 3d − 2 given points and tangent to a given line. The method is ’classical’, free of Quantum Cohomology. In the past 15 years or so a number of classical problems in the enumerative geometry of curves in P n were solved, first for n = 2, any genus [R1], then for any n, genus 0. The latter development was initiated by Kontsevich-Manin who developed and used the rather substantial machinery of Quantum Cohomology (cf. e.g. [FP]). Subsequently, in a series of papers [R2-R5] the author developed an elementary alternative approach, free of Quantum cohomology, and used it to solve a number of classical enumerative problems for rational, and sometimes elliptic, curves in P n, n ≥ 2. The present paper continues this series. The object here is to enumerate the irreducible rational curves of given degree d in P 2 with one tacnode or one triple point passing through 3d−2 general points (see Theorems 2,5 below), as well as the irreducible rational curves in P 3 with one ordinary node which contain a general points and are incident to 4d − 2a − 1 general lines (see Thm 1 below). Note that the family of 1-tacnodal (resp. 1-triple point, resp. 1-nodal) curves in P 2 (resp. P 2, P 3) is of codimension 1 in the family of all rational curves so that we are effectively computing the ’degree’, in a sense, of certain natural divisors in the family of all rational curves. Indeed by a result of Diaz and Harris [DH] in the case of P 2, the general member of any such divisor, if not nodal, is either 1-cuspidal (which case was enumerated in [R2]) or 1-tacnodal or has 1 triple point. It seems very likely,

In this paper we investigate Murre’s conjecture on the Chow–Künneth decomposition for universal families of smooth curves over spaces which dominate the moduli space Mg, in genus at most 8 and show existence of a Chow–Künneth decomposition. This is done in the setting of equivariant cohomology and equivariant Chow groups to get equivariant Chow–Künneth decompositions.

We study curves with linear series that are exceptional with regard to their secant planes. Working in the framework of an extension of Brill-Noether theory to pairs of linear series, we prove that a general curve of genus g has no exceptional secant planes, in a very precise sense. We also address the problem of computing the number of linear series with exceptional secant planes in a one-parameter family in terms of tautological classes associated with the family. We obtain conjectural generating functions for the that admit d-secant (d−2)-planes. We also describe a strategy for computing the classes of divisors associated to exceptional secant plane behavior in the Picard group of the moduli space of curves in a couple of naturally-arising infinite families of cases, and we give a formula for the number of linear series with exceptional secant planes on a general curve equipped with a one-dimensional family of linear series. tautological coefficients of secant-plane formulas associated to series g 2d−1 m

The problem of describing the birational geometry of the moduli space Mg of complex curves of genus g has a long history. Severi already knew in 1915 that Mg is unirational for g ≤ 10 (cf. [Sev]; see also [AC1] for a modern proof). In the same paper Severi

Real Gromov-Witten invariants on the moduli space of genus 0 stable maps to a smooth rational projective space

Severi varieties and branch curves of abelian surfaces of type (1, 3)