Defining Equations for Certain Types of Polarized Varieties

Abstract

Introduction In this paper we improve a result of Mumford [5]. To be explicit, we fix our notation and terminology 1 . Every variety is assumed to be defined over an algebraically closed field K . For line bundles L, M on a variety V we denote by R(L, M) the kernel of the natural multiplication homomorphism Γ(L)⊗Γ(M)→Γ(L+M) . A line bundle L on V is said to be simply generated if Γ(tL)⊗Γ(L)→((t+1)L) is surjective for every t ≧1. L is said to be quadratically presented if' it is simply generated and if the natural homomorphism R(sL, tL) ⊗Γ(L)→R(sL, (t+1)L) is surjective for all s,t ≧1. Now we state the following Theorem (Mumford). Let L be a line bundle on a smooth curve C of genus g. Then L is simply generated if deg L >≧2 g+1, and L is quadratically presented if deg L≧3g+1. We improve the above result in the following three ways. First, we can weaken the assumption that C is smooth. Second, we show that L is quadratically presented if deg L ≧2 g +2. Third, in the complex case, we give a higher dimensional version of these results, an embedding theorem and a structure theorem for certain types of polarized varieties, which will play an important role in our study of polarized varieties (see [la]). As an example of applications, we give in § 5 a criterion characterizing smooth hypercubics.