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Parametrizations over Q of cubic surfaces

Pannekoek, R. (2009) Parametrizations over Q of cubic surfaces. Master's Thesis / Essay, Mathematics.

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Abstract

Given any smooth cubic surface S defined over a number field K, it is a well-known fact that there exists a birational map from S to the projective plane. If we pose the additional requirement that f be defined over K, however, the assertion may no longer be true. In the 1970s, Manin and Swinnerton-Dyer formulated a necessary and sufficient criterion for S to allow a birational map to the projective plane over K. First, I will discuss their criterion and show that it is a pretty strong restriction on S. Also, I will give several examples of cubic surfaces in order to give some idea which cases actually occur. After this, I will elaborate on the fact that there are several cases in which a special sort of birational map can be found; I will show how these cases overlap and that they do not exhaust the class of all birationally trivial cubic surfaces. Finally, I will give examples of explicit birational and rational maps that I have been able to construct.

Item Type: Thesis (Master's Thesis / Essay)
Degree programme: Mathematics
Thesis type: Master's Thesis / Essay
Language: English
Date Deposited: 15 Feb 2018 07:28
Last Modified: 15 Feb 2018 07:28
URI: http://fse.studenttheses.ub.rug.nl/id/eprint/8513

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