ERC starting
Spin Curves Enumeration
Overview. Enumerative geometry of curves is the branch of mathematics that aims at counting complex curves, or Riemann surfaces, satisfying specific algebraic constraints.
This field has ancient roots, but gained significant interest in the 90s when physicists showed that solutions to enumerative problems of curves could be used to describe several theoretical models in string theory and quantum gravity.
In mathematics, the modern approach to enumerative problems involves the study of moduli spaces of curves, and the main difficulty is to describe the structure of their cohomology rings. In the present project, we intend to use spin curves to adress this problem. A spin curve is an enhancement of a curve obtained by choosing a square root of the co-tangent space. Curves have only one topological invariant (the genus), but spin curves carry an extra sign + or -, that allows for the definition of more refined enumerative problems. These refinements provide new classes and techniques to investigate the cohomology of moduli spaces of curves.
When and where? The project started in January 2025 at Laboratoire Analyse, Géometrie et Modélisation (AGM) in Cergy-Pontoise. I moved in September 2025 to Institut Mathématique de Toulouse, where the project is now developping until December 2029.
Members attached to the project.
PhD students:
- G. Politopoulos (thesis: 2021-2025), co-supervised by D. Holmes, Leiden Universiteit.
- L. Constant (thesis: 2023-present)
- S. Velstra (thesis: 2025-present), co-supervised by P. Georgieva, IMJ-PRG.
- Q. Shi (thesis: 2026-present), co-supervised by T. Blomme, IMT.
- X. Yan (2025-2027).
- A. Bud (2025-2027).
Events.
Upcoming:
- Gromov-Witten Theory and Cohomological Field Theories, 10-14 May 2027, Aussois (France)
Conference organized with T. Blomme, A. Bud, F. Carocci, and X. Yan. - Enumerative and arithmetic aspects of logarithmic geometry, 21-25 June 2027, Toulouse (France)
Conference organized with N. Borne, J. Gilibert, D. Holmes, and W. Niziol.
- Spin structures in enumerative geometry, 16-18 June 2025 (Leiden) link
Workshop organized with D. Holmes, M. Kool, and S. Shadrin.
(Pre-)Publications.
-
DR cycles and strata of differentials with spin parity (D. Holmes, G. Politopoulos, and A. Sauvaget).
arXiv:2509.03650, 2025. arXiv -
Spin refinement of moduli spaces of residueless meromorphic differentials and the BKP hierarchy (David Klompenhouwer, and Stijn Velstra).
2506.02540, 2025. arXiv
(Front page painting : Automne , Sonia Delaunay, 1965)
adrien.sauvaget@math.cnrs.fr
Institut de mathématiques de Toulouse, bureau 1R2-111
118 route de Narbonne,
31400 Toulouse (France)