Abstract:
Given two coprime numbers $p < q$, KW semigroups contain $p,\ q$ and are contained in $\langle p,\ q,\ r \rangle$ where $2r= p,\ q,\ p+q$ whichever is even. Kunz and Waldi proved that all $KW$ semigroups of embedding dimension $n\geq 4$ have Cohen-Macaulay type $n-1$ and first Betti number ${n \choose 2}$. In this talk, we give a characterization of the KW semigroups whose defining ideal is generated by the $2\times 2$ minors of a matrix. In addition, we identify all KW semigroups that lie on the interior of the same face of the Kunz cone $\mathcal{C}_p$ as a KW semigroup with determinantal defining ideal. Thus, we provide an explicit formula for the Betti numbers of all such semigroup rings: $\beta_0 = 1$, and $\beta_i = i {n \choose i+1}$ for $i=1,\ldots,n-1$. This talk is based on a joint work with S. Singh and H. Srinivasan.