Realizability of tropical canonical divisors

Martin Moeller; Martin Ulirsch; Annette Werner

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arXiv · 1710.06401

Realizability of tropical canonical divisors

Martin Moeller·Martin Ulirsch·Annette Werner

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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.

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Martin Moeller, Martin Ulirsch, Annette Werner. 2017-10-17. Realizability of tropical canonical divisors. https://arxiv.org/abs/1710.06401

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