(The) algebraic and geometric methods for syzygies of veronese embeddings and their asymptotic behaviors

Abstract

A syzygy of a given projective variety explains how the homogeneous ideal of the projective variety is generated. And Koszul cohomology, or graded Betti numbers, give basic information for the syzygy. Then, the vanishing and non-vanishing of Koszul cohomology implies some geometric property of the projective variety. In this thesis, we will study syzygies of Veronese embeddings. First, we introduce the definition of Koszul cohomology. And then, we prove M. Green’s Vanishing Theorem and Duality Theorem, which are basic tools for computation of Koszul cohomology. Furthermore, Vector Bundle Technique will be suggested so that one can compute the Kouszl cohomology by using the sheaf cohomology. Second, we will define property Np for a vanishing property of the Koszul cohomology. And we see a result of G. Ottaviani and R. Paoletti. It says that graded Betti numbers of Veronese embeddings are not so simple in the view of property $N_p$. Also, we suggest a geometric interpretation of the result. Finally, we will introduce a notion of asymptotic syzygies, and a recent result of L. Ein and R. Lazarsfeld.