Brill-Noether theory of curves on $\mathbb{P}^1 \times \mathbb{P}^1$: tropical and classical approach
Abstract
The gonality sequence $(d_r)_{r\geq1}$ of a smooth algebraic curve comprises the minimal degrees $d_r$ of linear systems of rank $r$. We explain two approaches to compute the gonality sequence of smooth curves in $\mathbb{P}^1 \times \mathbb{P}^1$: a tropical and a classical approach. The tropical approach uses the recently developed Brill--Noether theory on tropical curves and Baker's specialization of linear systems from curves to metric graphs. The classical one extends the work of Hartshorne on plane curves to curves on $\mathbb{P}^1 \times \mathbb{P}^1$.
- Publication:
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arXiv e-prints
- Pub Date:
- September 2017
- DOI:
- 10.48550/arXiv.1709.07254
- arXiv:
- arXiv:1709.07254
- Bibcode:
- 2017arXiv170907254C
- Keywords:
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- Mathematics - Algebraic Geometry