Classification of degree two curves in the symmetric square with positive self-intersection
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Meritxell Sáez
Abstract
We give a precise classification of the pairs (C, B͠) with C a smooth curve of genus g and B͠ ⊂ C(2) a curve of degree two and positive self-intersection. We prove that there are no such pairs if g < pa(B͠) < 2g−1. We study the singularities and self-intersection of any degree two curve in C(2). Moreover, we give examples of curves with arithmetic genus in the Brill–Noether range and positive self-intersection on C × C.
Acknowledgements
The author thanks Rita Pardini for all the time and useful suggestions given during the author’s stay at the University of Pisa and afterwards. Many thanks also to Gian Pietro Pirola for suggesting to look into C × C for curves with arithmetic genus in the Brill–Noether range. The most sincere gratitude to Miguel Angel Barja and Joan Carles Naranjo for the multiple discussions and the huge amount of time devoted to the development of this article, to the anonymous referee for many suggestions that greatly improved the exposition, and finally to the Universitat de Barcelona for the research grant and their hospitality afterwards.
Funding: The author has been partially supported by the Proyecto de Investigación MTM2012-38122-C03-02.
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Articles in the same Issue
- Frontmatter
- Negative refraction and tiling billiards
- Classification of degree two curves in the symmetric square with positive self-intersection
- On lattice coverings by simplices
- On CMC hypersurfaces in 𝕊n+1 with constant Gauß–Kronecker curvature
- Fully truncated simplices and their monodromy groups
- Lie algebras of conservation laws of variational partial differential equations
- Feet in orthogonal-Buekenhout–Metz unitals
- Pseudo-metric 2-step nilpotent Lie algebras
Articles in the same Issue
- Frontmatter
- Negative refraction and tiling billiards
- Classification of degree two curves in the symmetric square with positive self-intersection
- On lattice coverings by simplices
- On CMC hypersurfaces in 𝕊n+1 with constant Gauß–Kronecker curvature
- Fully truncated simplices and their monodromy groups
- Lie algebras of conservation laws of variational partial differential equations
- Feet in orthogonal-Buekenhout–Metz unitals
- Pseudo-metric 2-step nilpotent Lie algebras
