Introduction

Summary

By a polarized variety we mean a pair (V, L) consisting of a projective variety V and an ample line bundle L on it. We will classify such pairs and describe their structure as precisely as possible.

Needless to say, algebraic varieties are the main object in algebraic geometry. In this book, however, we mainly consider the pair (V, L) rather than the variety V itself. There are several reasons of taking this viewpoint.

First of all, polarization (or the linear system defined by it) is very important for describing the structure of a variety. For example, the projective space ℙⁿ is described by a homogeneous coordinate system, namely a linear parametrization of H⁰(ℙⁿ, O(1)). But it is by no means easy to recognize a projective space without being given a polarization. For beginners it takes some thought to see that a twisted cubic in ℙ³ is isomorphic to ℙ¹, and this is because the polarization O(1) is not given a priori. Another example is the space parametrizing linear ℙ²'s contained in a smooth hyperquadric in ℙ⁵. This is actually isomorphic to ℙ³. Is that obvious to you?

There are polarizations which are not very ample but are useful for this purpose. For example, let f: V → ℙⁿ be a finite double covering. Then L = f^*O(1) is ample, but not very ample.