虧格數和非正則數均為二的一般形曲面的研究

Abstract

這篇文章主要是在探討代數曲面的分類理論。特別地我們考慮虧格數與非正則數均為二的一般形曲面，到目前為止，對於這一種曲面的分類還是不完全的。我所做的事是在於真正構造出實際的例子。在這篇文章中我們考慮兩種構造曲面的方法。第一種做法是考慮與兩條曲線乘積同源的曲面。這一類的曲面有一些好的性質，所以在討論上是相對容易的。佩納吉尼做出了一張列表，把這類的曲面完全分類完畢了。第二種做法是去考慮在阿貝爾的曲面上的有限覆蓋。陳榮凱與黑肯利用阿貝爾曲面上的三次覆蓋造出了一個例子。我依循他們的做法，利用阿貝爾曲面上的四次覆蓋造出了另外一個這樣的曲面。

This article is about the classification of algebraic surfaces. We focus on the surfaces of general type with pg = q = 2. This kind of surfaces are not completely classified yet.We construct several examples in this article.In this paper I discuss two methods for constructing a surface of general type with pg = q = 2. The first one is by considering a surface which is isogenus to a product of curves. This kind of surfaces is rather easy to work out. Penegini gives a complete list of such surfaces. The other way to construct a surface with given invariant is to consider a finite covering of an abelain surface. Chen and Hacon construct a surface with K2 = 5 by considering a triple cover of an abelian surface. Using the similar method, I construct a quadruple covering over an abelian surface to get a surface with K2 = 6.