Pencils of quadrics and Gromov-Witten-Welschinger invariants of CP3\mathbb C P^3

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.Comment: 14 pages, 4 figures, minor corrections following referee's suggestion

On the Topology of Real Bundle Pairs over Nodal Symmetric Surfaces

We give an alternative argument for the classification of real bundle pairs over smooth symmetric surfaces and extend this classification to nodal symmetric surfaces. We also classify the homotopy classes of automorphisms of real bundle pairs over symmetric surfaces. The two statements together describe the isomorphisms between real bundle pairs over symmetric surfaces up to deformation.Comment: 19 pages, 3 figure