The Betti numbers of Kunz-Waldi semigroups

Given two coprime numbers p > q p>q , KW semigroups contain p , q p,q and are contained in ⟨ p , q , r ⟩ p,q,r where 2 r = p , q , p + q 2r= p,q, p+q whichever is even. These semigroups were first introduced by Kunz and Waldi [Semigroup Forum 89 (2014), pp. 664–691]. Kunz and Waldi proved that all K W KW semigroups of embedding dimension n ≥ 4 n≥ 4 have Cohen-Macaulay type n − 1 n-1 and first Betti number ( n 2 ) {n 2} . In this paper, we characterize KW semigroups whose defining ideal is generated by the 2 × 2 2× 2 minors of a 2 × n 2× n matrix. In addition, we identify all KW semigroups that lie on the interior of the same face of the Kunz cone C p C_p as a KW semigroup with determinantal defining ideal. Thus, we provide an explicit formula for the Betti numbers of all those KW semigroups.