Homological properties of pinched Veronese rings

Pinched Veronese rings are formed by removing an algebra generator from a subring of polynomial ring. We study the homological properties such rings, including Cohen-Macaulay, Gorenstein, and complete intersection properties. Greco Martino classified Cohen-Macaulayness pinched maximum entry exponent vector monomial; we re-prove their results with semigroup methods correct omission small class examples Cohen-Macaulay rings. When underlying field is prime characteristic, show that exhibit variety F-singularities, F-regular, F-injective, F-nilpotent. also compute upper bounds on Frobenius test exponents computational invariant which controls closure all parameter ideals simultaneously.