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.

arXiv comment36 pages, 10 figures, improved exposition (including a new discussion of algorithmic aspects) as well as a correction in Example 6.5, comments welcome!

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Cited by in corpus (7)

• Compactification of strata of abelian differentials
• A smooth compactification of the space of genus two curves in projective space via logarithmic geometry and Gorenstein curves
• Tropicalized quartics and canonical embeddings for tropical curves of genus 3
• Gromov-Hausdorff limits of flat Riemannian surfaces
• Reduction and lifting problem for differential forms on Berkovich curves
• Tropical double ramification loci
• Voronoi tilings, toric arrangements and degenerations of line bundles III