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