Tropicalizing abelian covers of algebraic curves
Helminck, Paul Alexander
Universität Bremen: Informatik/Mathematik
Berkovich skeleton, semistable curves, intersection graphs, tropicalization, Laplacian operator, Slope formula, Galois covers, tame covers, abelian covers, Jacobians, superelliptic curves, genus 3 curves, moduli spaces, tropical moduli spaces
In this thesis, we study the Berkovich skeleton of an algebraic curve over a discretely valued field K. We do this using coverings C - P1 of the projective line. To study these coverings, we take the Galois closure of the corresponding injection of function fields K(P1) - K(C), giving a Galois morphism overline C - P1. A theorem by Liu and Lorenzini tells us how to associate to this morphism a Galois morphism of semistable models C - D. That is, we make the branch locus disjoint in the special fiber of D and remove any vertical ramification on the components of Ds. This morphism C - D then gives rise to a morphism of intersection graphs Sigma(C) - Sigma(D). Our goal is to reconstruct Sigma(C) from Sigma(D) and we will do this by giving a set of covering and twisting data. These then give algorithms for finding the Berkovich skeleton of a curve C whenever that curve has a morphism C - P1 with a solvable Galois group. In particular, this gives an algorithm for finding the Berkovich skeleton of any genus three curve. These coverings also give a new proof of a classical result on the semistable reduction type of an elliptic curve, saying that an elliptic curve has potential good reduction if and only if the valuation of the j-invariant is positive.
Tropicalizing abelian covers of algebraic curves
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