Combinatorial patterns in syzygies
Affine semigroup rings are the coordinate rings of not necessarily normal toric varieties. They include the coordinate rings of the Segre-Veronese embeddings of projective spaces, and special projections of those. The study of affine semigroup rings lies in the intersection of commutative algebra, algebraic geometry and combinatorics. In this thesis, we study the syzygies for certain classes of affine semigroup rings.
The Betti numbers of affine semigroup rings can be computed as the dimensions of homology groups of certain simplicial complexes. Therefore, the study of the Betti numbers of affine semigroup rings can be translated into some combinatorial problems. The idea of using combinatorial topology to study syzygies originated from the work of Hochster, Reisner and Stanley in the seventies and eighties and since then have been an active area of research and proved to be useful in lots of cases.
In the first chapter, we introduce the problems concerned in our dissertation, and their relations to topology of simplicial complexes and representation theory of symmetric groups. We also include some background material from combinatorial commutative algebra, algebraic geometry, and representation theory.
In the second chapter, we use combinatorial and representation theoretic methods arising from work of Karaguerian, Reiner and Wachs to reduce the study of the syzygies of Veronese varieties to the study of homology groups of matching complexes. In turn we use combinatorial methods to show the vanishings of certain homology groups of these matching complexes, giving a lower bound on the length of the linear part of the resolution of the Veronese varieties. In the case of third Veronese embeddings, we carry out the computation to prove the Ottaviani-Paoletti conjecture.
In the final chapter, we study a conjecture of Herzog and Srinivasan and a higher analog. The conjecture says that the Betti numbers of affine monomial curves under translations are eventually periodic. We prove the conjecture, and use it to study the analogous question for higher dimensional affine semigroup rings under translations and some other consequences.