Geometry of syzygies and veronese surfaces of small degree

Abstract

Syzygies of Veronese embedding has been studied as important examples and problems. In this thesis, we study of the form of elements of the first Koszul cohomology directly for the cases n = 1 and n = 2, d = 2, 3, 4. In [EEL2016], L. Ein, D. Erman and R. Lazarsfeld proved the asymptotic nonvanishing theorem for syzygies of Veronese embedding, and the Betti numbers are known for given cases. We find generators of the Koszul cohomology, and by showing that the number of generators and Betti number are same, we show these elements form a basis. To do this, we explain Hilbert’s syzygy theorem, minimal free resolution, and Betti number following [Eisenbud2005] and [Eisenbud1995], and we treat examples of four points in P2 and seven points in P3 . We also review necessary parts in [EEL2016], and then we find a basis of Koszul cohomology of Veronese embedding for for the cases n = 1 and n = 2, d = 2, 3, 4.