Tropicalizing abelian covers of algebraic curves
|Other Titles:||Tropikalisierende abelsche Abdeckungen algebraischer Kurven||Authors:||Helminck, Paul Alexander||Supervisor:||Feichtner, Eva-Maria||1. Expert:||Feichtner, Eva-Maria||2. Expert:||Rabinoff, Joseph, Assistant||Abstract:||
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.
|Keywords:||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||Issue Date:||11-Jul-2018||URN:||urn:nbn:de:gbv:46-00106622-10||Institution:||Universität Bremen||Faculty:||FB3 Mathematik/Informatik|
|Appears in Collections:||Dissertationen|
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