Pencils of quadrics and Gromov-Witten-Welschinger invariants of $\mathbb C P^3$

TL;DR

This paper derives a formula for Gromov-Witten-Welschinger invariants of complex projective 3-space with mixed real and conjugate point constraints, using pencils of quadrics to relate invariants across different geometries.

Contribution

It introduces a novel method based on pencils of quadrics to compute mixed real and complex enumerative invariants of C P^3, connecting them to invariants of simpler geometries.

Findings

Derived a formula for Gromov-Witten-Welschinger invariants of C P^3.

Connected invariants of C P^3 to those of C P^1 C P^1 and elliptic curves.

Provided a new computational approach for mixed real and complex enumerative invariants.

Abstract

We establish a formula for the Gromov-Witten-Welschinger invariants of $\mathbb CP^3$ with mixed real and conjugate point constraints. The method is based on a suggestion by J. Koll\'ar that, considering pencils of quadrics, some real and complex enumerative invariants of $\mathbb CP^3$ could be computed in terms of enumerative invariants of $\mathbb CP^1\times\mathbb CP^1$ and of elliptic curves.