Bézout's theorem in P^1 x P^1

Sibma, Lisanne (2022) Bézout's theorem in P^1 x P^1. Bachelor's Thesis, Mathematics.

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Abstract

B´ezout’s theorem states that, given two projective curves, the number of intersection points is at most the product of their degrees. Moreover, we have equality if we work over an algebraically closed field. However, when looking at the intersection of two affine curves, we often find that the number of intersection points we expect by B´ezout’s theorem is higher than we can perceive by plotting them, even when working over an algebraically closed field. The latter shows that the hypothesis that we work over a projective space is essential to have a uniform result, i.e., one that depends only on the degrees of the curves. We can go from the affine space to the projective space by compactifying the affine space. One possible compactification of the affine plane is the projective plane P 2 . It is natural to wonder what kind of results one gets when considering other compactifications, such as P 1 ×P 1 . In contrast to P 2 , where we have one line at infinity, in P 1 ×P 1 we have two lines at infinity. This will change the intersection behaviour of curves. Using divisors on surfaces we will discover a version of B´ezout’s theorem in P 1 × P 1 . Before this can be done, we will learn about plane curves and their intersection behaviour to prove B´ezout’s theorem in the classical sense. At the end of the paper, the two versions of B´ezout’s theorem are compared and we will shed a light on how the techniques used to find B´ezout’s theorem in P 1 × P 1 can be used to find a version of B´ezout’s theorem on even more surfaces

Item Type:	Thesis (Bachelor's Thesis)
Supervisor name:	Salgado Guimaraes da Silva, C. and Top, J.
Degree programme:	Mathematics
Thesis type:	Bachelor's Thesis
Language:	English
Date Deposited:	01 Mar 2022 09:54
Last Modified:	20 Feb 2024 13:46
URI:	https://fse.studenttheses.ub.rug.nl/id/eprint/26490

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