Welschinger--Witt invariants
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866909770581016576 |
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| author | Brugallé, Erwan Rau, Johannes Wickelgren, Kirsten |
| author_facet | Brugallé, Erwan Rau, Johannes Wickelgren, Kirsten |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_04172 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Welschinger--Witt invariants Brugallé, Erwan Rau, Johannes Wickelgren, Kirsten Algebraic Geometry K-Theory and Homology Symplectic Geometry 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. |
| title | Welschinger--Witt invariants |
| topic | Algebraic Geometry K-Theory and Homology Symplectic Geometry |
| url | https://arxiv.org/abs/2509.04172 |