Gromov-Witten invariants in family and quantum cohomology

TL;DR

This paper constructs a family version of the moduli space of stable maps, defines Gromov-Witten invariants in this context, and applies it to recover classical enumerative geometry results about rational curves in P3.

Contribution

It introduces a new family moduli space of stable maps and defines Gromov-Witten invariants for it, extending classical theories and enabling new enumerative computations.

Findings

Constructed a family moduli space of stable maps.

Defined Gromov-Witten invariants in the family setting.

Reproduced classical enumerative formulas for rational curves in P3.

Abstract

A moduli space of stable maps to the fibers of a fiber bundle is constructed. The new moduli space is a family version of the classical moduli space of stable maps to a non-singular complex projective variety. The virtual cycle for this moduli space is also constructed, and an analogue of Gromov-Witten invariants is defined. As an application, we recover the formula for the number of rational degree d curves in P3, whose image lies in a plane in P3 (known as planar curves in P3), intersecting r general lines while passing through given s general points, where r + 2s = 3d + 2, firstly proved by R. Mukherjee, R. Kumar Singh and the fourth named author.