Pencils of quadrics and Gromov-Witten-Welschinger invariants of mathbb C P³

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.