Welschinger--Witt invariants

2025/09/04 by Brugallé, Erwan, Rau, Johannes, Wickelgren, Kirsten
#Algebraic Geometry (math.AG) #FOS: Mathematics #K-Theory and Homology (math.KT) #Symplectic Geometry (math.SG)

paper · doi:10.48550/arxiv.2509.04172

Abstract

Welschinger invariants are signed counts of real rational curves satisfying contraints. Quadratic Gromov--Witten invariants give such counts over general fields of characteristic different from 2 and 3. For rational del Pezzo surfaces over a field, we propose a conjectural relationship between Welschinger and quadratic Gromov--Witten invariants. We construct multivariable unramified Witt invariants, in the sense of Serre, from Welschinger invariants and call them Welschinger--Witt invariants. We show that quadratic Gromov--Witten invariants are also Witt invariants and control their ramification. We then conjecture an equality between these Witt invariants, in particular giving a conjectural computation of all the quadratic Gromov--Witten invariants of k-rational surfaces. We prove this conjecture for k-rational del Pezzo surfaces of degree at least 6.

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Related

• Toward an algebraic theory of Welschinger invariants
• Welschinger invariants of toric Del Pezzo surfaces with non-standard real structures
• A quadratic Abramovich-Bertram formula
• A quadratically enriched count of rational curves
• Appendix to "Welschinger invariant and enumeration of real rational curves"