Realizability of tropical canonical divisors
Abstract
We use recent results by Bainbridge-Chen-Gendron-Grushevsky-Moeller on compactifications of strata of abelian differentials to give a comprehensive solution to the realizability problem for effective tropical canonical divisors in equicharacteristic zero. Given a pair $(\Gamma, D)$ consisting of a stable tropical curve $\Gamma$ and a divisor $D$ in the canonical linear system on $\Gamma$, we give a purely combinatorial condition to decide whether there is a smooth curve $X$ over a non-Archimedean field whose stable reduction has $\Gamma$ as its dual tropical curve together with a effective canonical divisor $K_X$ that specializes to $D$. Along the way, we develop a moduli-theoretic framework to understand Baker's specialization of divisors from algebraic to tropical curves as a natural toroidal tropicalization map in the sense of Abramovich-Caporaso-Payne.
- Publication:
-
arXiv e-prints
- Pub Date:
- October 2017
- DOI:
- 10.48550/arXiv.1710.06401
- arXiv:
- arXiv:1710.06401
- Bibcode:
- 2017arXiv171006401M
- Keywords:
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- Mathematics - Algebraic Geometry;
- Mathematics - Geometric Topology;
- 14T05;
- 14H10;
- 14H51
- E-Print:
- 36 pages, 10 figures, improved exposition (including a new discussion of algorithmic aspects) as well as a correction in Example 6.5, comments welcome!