On the enumeration of rational plane curves with tangency conditions

TL;DR

This paper extends Kontsevich's recursion to count rational plane curves with specified tangency conditions to a smooth cubic, providing new enumerative formulas and generalizations for such geometric problems.

Contribution

It introduces a generalized recursion formula for counting rational curves with tangency conditions, using twisted stable maps, and proves an analogue of Caporaso-Harris's formula.

Findings

Derived a generalized recursion for enumerating rational curves with tangency conditions.

Proved an analogue of Caporaso and Harris's formula for these enumerative problems.

Extended the understanding of curve counting with tangency constraints in algebraic geometry.

Abstract

We use twisted stable maps to answer the following question. Let E\subset P^2 be a smooth cubic. How many rational degree d curves pass through a general points of E, have b specified tangencies with E and c unspecified tangencies, and pass through 3d-1-a-2b-c general points of P^2? The answer is given as a generalization of Kontsevich's recursion. We also investigate more general enumerative problems of this sort, and prove an analogue of a formula of Caporaso and Harris.