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On moduli of hyperelliptic curves of genus two

Valk, M. van der (2011) On moduli of hyperelliptic curves of genus two. Master's Thesis / Essay, Mathematics.

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Abstract

A curve C, over the complex number field, of genus g is a hyperelliptic curve of genus g if there exists a holomorphic map from C onto the Riemann sphere of degree two. The curve C is non-hyperelliptic if there does not exists such a map. First, we study hyperelliptic curves of given genus. This results in three equivalent definitions of them. Next, we restrict ourselves to the case of hyperelliptic curves of genus two. Among others, we prove that all curves of genus two are hyperelliptic and that a hyperelliptic curve of genus two can be identified with an unordered six tuple which consists of distinct points of the Riemann sphere. Here curves are taken over the comlex number field. The latter observation provides a natural notion of isomorphic hyperelliptic curves of genus two. This suggests to study hyperelliptic curves of genus two all at once instead of study them separately. We are therefore looking for a space, say M, such that a point of M corresponds to an isomorphism class of hyperelliptic curves of genus two. Such space is called a moduli space (of hyperelliptic curves of genus two). We will first obtain M as set parameterized by the so-called invariants of the space of binary sextics. Once obtained M as set we will give it the structure of a variety (scheme). We will only sketch the idea, and sketch some features, of moduli spaces since a mathematically rigorous treatment needs a good understanding of Category Theory and beyond.

Item Type: Thesis (Master's Thesis / Essay)
Degree programme: Mathematics
Thesis type: Master's Thesis / Essay
Language: English
Date Deposited: 15 Feb 2018 07:46
Last Modified: 15 Feb 2018 07:46
URI: http://fse.studenttheses.ub.rug.nl/id/eprint/9712

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