Computing The Invariants of Intersection Algebras of Principal Monomial Ideals

We continue the study of intersection algebras $\mathcal B = \mathcal B_R(I, J)$ of two ideals $I, J$ in a commutative Noetherian ring $R$. In particular, we exploit the semigroup ring and toric structures in order to calculate various invariants of the intersection algebra when $R$ is a polynomial ring over a field and $I,J$ are principal monomial ideals. Specifically, we calculate the $F$-signature, divisor class group, and Hilbert-Samuel and Hilbert-Kunz multiplicities, sometimes restricting to certain cases in order to obtain explicit formul{\ae}. This provides a new class of rings where formul{\ae} for the $F$-signature and Hilbert-Kunz multiplicity, dependent on families of parameters, are provided.