# A remark on Gromov-Witten-Welschinger invariants of $\mathbb{C} P^3\#\overline{\mathbb{C} P}^3$

## TL;DR

This paper extends the formula for Gromov-Witten-Welschinger invariants from complex projective space to a blow-up of it, linking real and complex enumerative invariants through geometric constructions.

## Contribution

It generalizes existing invariants formulas to a new class of complex surfaces, connecting invariants of blow-ups with those of simpler surfaces using pencils of quadrics.

## Key findings

- Formulas for invariants of  P^3 P^3A0# P^3 P^3A0 with explicit computations.
- Expresses invariants of the blow-up as combinations of invariants of  P^2 P^2A0 blown up at two real points.
- Provides tools for calculating real enumerative invariants in more complex geometries.

## Abstract

We generalize the formula of Gromov-Witten-Welschinger invariants of $\mathbb{C} P^3$ established by E. Brugall\'e and P. Georgieva in [BG16b] to $\mathbb{C} P^3\#\overline{\mathbb{C} P}^3$. Using pencils of quadrics, some real and complex enumerative invariants of $\mathbb{C} P^3\#\overline{\mathbb{C} P}^3$ can be written as the combination of the enumerative invariants of the blow up of $\mathbb{C} P^2$ at two real points.