Bielliptic surfaces and their geometry
Typ
Examensarbete för masterexamen
Master's Thesis
Master's Thesis
Program
Engineering mathematics and computational science (MPENM), MSc
Publicerad
2024
Författare
Liu, Xudong
Modellbyggare
Tidskriftstitel
ISSN
Volymtitel
Utgivare
Sammanfattning
Geometry is a high concern in modern mathematics. One way to begin the study
is by handling a nice example. The bielliptic surfaces can play such a role. It is
constructed using elliptic curves, some nice curves in some way equivalent to a torus.
The prerequisites of bielliptic surfaces involve algebraic geometry and elliptic curves.
The final result is about the intersection of bielliptic surfaces, so the intersection
theories of surfaces will also be introduced. Works of classification and works of
Néron-Severi lattices are crucial for the study of bielliptic surfaces in the last section.
Algebraic geometry focuses on the method of solving geometry problems in algebraic
ways. The fundamental of the study is abstract algebra. It studies curves,
surfaces, and some other higher-dimension objects like hyperspaces. The key point
is describing geometry structures by zeros of polynomials. Many results are derived
over the complex field, where many nice properties can be found.
The elliptic curve is a kind of algebraic curve of genus one. Weierstrass equations are
the algebraic forms of elliptic curves. The composition law defines an operation on
the elliptic curves. Another important property is that the lattices over the complex
field determine the elliptic curves, which can be derived from the construction of the
Weierstrass ℘-function. Isogenies are introduced as the maps between elliptic curves.
The topic of intersection theory on surfaces concerns the intersection number of two
curves on the given surface, which is the number of intersection points counted with
algebraic multiplicity. The definition of intersection number can be generalized to n
varieties in high dimensions. In the article, the situation of two curves on a surface
is enough. One important result is Bézout’s theorem, a theorem of the intersection
number of plane curves.
The definition of bielliptic surfaces is based on the elliptic curves. With all the
knowledge before, the final result about the intersection number of bielliptic surfaces
can be given.
Beskrivning
Ämne/nyckelord
elliptic curve, bielliptic surfaces, intersection theory, intersection number.