Tropical Igusa Invariants
Paul Alexander Helminck

TL;DR
This paper provides a new proof for determining the minimal Berkovich skeleton of genus two curves over non-archimedean fields using Igusa invariants, and interprets these results through tropical geometry.
Contribution
It offers a novel proof applicable to all complete non-archimedean fields and connects Igusa invariants with tropical moduli spaces.
Findings
Criteria for Igusa invariants determining skeletons
Tropicalization map factors through a concrete embedding
Interpretation of Igusa invariants in tropical geometry
Abstract
Let be a smooth geometrically connected projective curve of genus two over a complete non-archimedean field . For discretely valued , the first main theorem in \cite{liu} gives a set of criteria on the Igusa invariants of the curve that determine the minimal Berkovich skeleton of together with its edge lengths and vertex weights. In this paper we use the theory of Berkovich spaces to give a new proof of this theorem that works for arbitrary complete non-archimedean fields. We furthermore interpret the final result in terms of tropical moduli spaces and tropical Igusa invariants. This reformulation shows that the abstract tropicalization map factors through the tropicalization of a concrete embedding of into a weighted projective space.
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Taxonomy
TopicsPolynomial and algebraic computation · Algebraic Geometry and Number Theory · Nonlinear Waves and Solitons
