A quadratic Abramovich-Bertram formula
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866908416214040576 |
|---|---|
| author | Brugallé, Erwan Wickelgren, Kirsten |
| author_facet | Brugallé, Erwan Wickelgren, Kirsten |
| contents | Quadratic Gromov--Witten invariants allow one to obtain an arithmetically meaningful count of curves satisfying constraints over a field $k$ without assuming that $k$ is the field of complex or real numbers. This paper studies the behavior of quadratic genus $0$ Gromov--Witten invariants during an algebraic analogue of surgery on del Pezzo surfaces. For this, we define and study (twisted) binomial coefficients in the Grothendieck--Witt group, building on work of Serre. We obtain a formula expressing the quadratic genus $0$ Gromov--Witten invariants of surfaces obtained as a smoothing of a given nodal surface in terms of those of the one having the largest Picard group. We give applications to quadratic Gromov--Witten invariants of rational del Pezzo surfaces of degree at least 7, some cubic surfaces, for point constraints defined over quadratic extensions of $k$, as well as an invariance result under a Dehn twist. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_17854 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A quadratic Abramovich-Bertram formula Brugallé, Erwan Wickelgren, Kirsten Algebraic Geometry Algebraic Topology K-Theory and Homology Primary 14N35, 14F42, Secondary 53D45, 19G38 Quadratic Gromov--Witten invariants allow one to obtain an arithmetically meaningful count of curves satisfying constraints over a field $k$ without assuming that $k$ is the field of complex or real numbers. This paper studies the behavior of quadratic genus $0$ Gromov--Witten invariants during an algebraic analogue of surgery on del Pezzo surfaces. For this, we define and study (twisted) binomial coefficients in the Grothendieck--Witt group, building on work of Serre. We obtain a formula expressing the quadratic genus $0$ Gromov--Witten invariants of surfaces obtained as a smoothing of a given nodal surface in terms of those of the one having the largest Picard group. We give applications to quadratic Gromov--Witten invariants of rational del Pezzo surfaces of degree at least 7, some cubic surfaces, for point constraints defined over quadratic extensions of $k$, as well as an invariance result under a Dehn twist. |
| title | A quadratic Abramovich-Bertram formula |
| topic | Algebraic Geometry Algebraic Topology K-Theory and Homology Primary 14N35, 14F42, Secondary 53D45, 19G38 |
| url | https://arxiv.org/abs/2506.17854 |