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Title: Algebraic covers
Author: Dias, Eduardo Manuel
ISNI:       0000 0004 5921 8987
Awarding Body: University of Warwick
Current Institution: University of Warwick
Date of Award: 2016
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The main goal of this thesis is the description of the section ring of a surface R(S,L) = O∞n=0 H0(S,nL) where L is an ample base point free divisor defining a covering map φL: S -> P2 such that φ*OS = OP2 O Ω1P2 O Ω1P2 O Op2(-3). This is an abelian surface with a polarization of type (1,3) which was studied before in [BL94, Cas99, Cas12]. Given a covering map φ: X -> Y, following the methods introduced by Miranda for general d covers, in chapter 3 we will define a cover homomorphism that will induce a commutative and associative multiplication in φ*OX. Chapter 4 focuses in the OP2-modules Hom (S2Ω1P2,Ω1P2) that will be used to define a commutative multiplication for our surface. Chapter 5 is about the associative condition. It is a computational method based on the paper [Rei90]. In the last chapter we use the ring R(S,L) to prove that the moduli space of abelian surfaces with a polarization of type (1,3) and canonical level structure is rational. We will also show how to use the same method to find models for covering maps such that φ*OS = OP2 O Ω1P2(-m1) O Ω1P2(-m2) O OP2(-m1-m2-3). The last section contains new problems whose goal is to construct and study algebraic varieties given by the vanishing of a high codimensional Gorenstein ideal.
Supervisor: Not available Sponsor: Fundação para a Ciência e a Tecnologia (FCT)
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: QA Mathematics