Enumeration of rational plane curves tangent to a smooth cubic

Abstract

We use twisted stable maps to compute the number of rational degree d plane curves having prescribed contacts to a smooth plane cubic.

Relative and orbifold Gromov�Witten invariants

“…Cadman and Chen computed the orbifold invariants for rational curves with tangency to a smooth plane cubic [CC08,Theorem 3.5] and observed their agreement with Gathmann's earlier calculation of the relative invariants [Gat03,Section 2,table,p. 409], even when neither invariant is enumerative.…”

“…409], even when neither invariant is enumerative. Thus we arrive at the coincidence observed by Cadman and Chen [CC08], already described in Section 1.1. Our general comparison result, Theorem 1.1, explains this coincidence and generalizes it to rational curves in arbitrary targets.…”

Structures in genus‐zero relative Gromov–Witten theory

“…In this paper, those large-age markings are studied and translated into markings with negative contact orders. We remark that a very special case of orbifold invariants with large-age markings was studied in [3,Section 5] in details. This paper in fact provides two equivalent definitions of relative Gromov-Witten theory with negative contact.…”

Lectures on Gromov–Witten Invariants of Orbifolds

“…Computations in Gromov-Witten theory of stacks beyond orbifold cohomology are not as numerous. C. Cadman [11,12] computed Gromov-Witten invariants of r (P 2 , C) with C a smooth cubic, and derived the number of rational plane curves of degree d with tangency conditions to the cubic C.…”