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  4. Tropicalizing abelian covers of algebraic curves
 
Zitierlink URN
https://nbn-resolving.de/urn:nbn:de:gbv:46-00106622-10

Tropicalizing abelian covers of algebraic curves

Veröffentlichungsdatum
2018-07-11
Autoren
Helminck, Paul Alexander  
Betreuer
Feichtner, Eva-Maria  
Gutachter
Rabinoff, Joseph  
Zusammenfassung
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.
Schlagwörter
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
Institution
Universität Bremen  
Fachbereich
Fachbereich 03: Mathematik/Informatik (FB 03)  
Dokumenttyp
Dissertation
Zweitveröffentlichung
Nein
Sprache
Englisch
Dateien
Lade...
Vorschaubild
Name

00106622-1.pdf

Size

1.4 MB

Format

Adobe PDF

Checksum

(MD5):471b94594288bcf70029aae687745d3c

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