Bielliptic surfaces and their geometry - Liu, Xudong (My University, 2024)
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.
Keywords: elliptic curve, bielliptic surfaces, intersection theory, intersection number.