Curves on a Surface
Curves on a Surface
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曲面上的曲线
DOI：
10.1007/978-1-4612-1688-9_2
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发表时间：
1998
影响因子：
0.5
通讯作者：
R. Friedman
中科院分区：
文献类型：
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作者：
R. Friedman
在本书中，除非另有说明，我们所说的曲面总是指复维数为2的连通紧复流形，它是某个N的全纯子流形，因此，“曲面”是光滑（连通）复代数曲面的简称。根据周氏定理，曲面也被描述为N + 1个变量的有限个齐次多项式的零集。对曲面的研究既涉及到曲面的固有几何，也涉及到曲面可能嵌入的几何。就像曲线一样，我们可以按照复杂度递增的顺序来组织这项研究。就一个曲面的外在（合成）几何而言，例如，我们可以尝试研究并最终对一个相对较小程度的曲面进行分类。或者我们可以尝试通过一些内在不变量的复杂度来对曲面进行排序，就像曲线的属一样。这是Kodaira分类的目的，它通过它们的Kodaira维度对表面进行排序。对于这个方案，我们有一个相当完整的曲面的理解除了在Kodaira维2的情况下，一般类型的曲面。我们将涵盖曲面一般理论的大致轮廓。在这一章中，我们将讨论基本不变量、交理论和Riemann-Roch，以及样本除数集的结构。在第三章中，我们将讨论二分几何。
In this book, unless otherwise specified, by surface we shall always mean a connected compact complex manifold of complex dimension 2 which is a holomorphic submanifold of ℙ N for some N. Thus, “surface” is short for smooth (connected) complex algebraic surface. By Chow’s theorem, a surface is also described as the zero set in ℙ N of a finite number of homogeneous polynomials in N + 1 variables. The study of surfaces is concerned both with the intrinsic geometry of the surface and with the geometry of the possible embeddings of the surface in ℙ N . Just as with curves, we could organize this study in order of increasing complexity. In terms of the extrinsic (synthetic) geometry of a surface in ℙ N , we could for instance try to study and eventually classify surfaces in ℙ N of relatively small degree. Or we could attempt to order surfaces by complexity via some intrinsic invariants, by analogy with the genus of a curve. This is the aim of the Kodaira classification, which orders surfaces by their Kodaira dimension. For this scheme, we have a fairly complete understanding of surfaces except in the case of Kodaira dimension 2, general type surfaces. We will cover the broad outlines of the general theory of surfaces. In this chapter, we will discuss the basic invariants, intersection theory and Riemann-Roch, and the structure of the set of ample divisors. In Chapter 3, we will discuss birational geometry.