We study the Frobenius complexity of Hibi rings over fields of characteristic p > 0. In particular, for a certain class of Hibi rings (which we call ω (−1) -level), we compute the limit of the Frobenius complexity as p → ∞.

The trace of the canonical module of a Cohen-Macaulay ring describes its non-Gorenstein locus. We study the trace of the canonical module of a Segre product of algebras, and we apply our results to compute the non-Gorenstein locus of toric rings. We provide several sufficient and necessary conditions for Hibi rings and normal semigroup rings to be Gorenstein on the punctured spectrum.

We prove that if f f is a reduced homogeneous polynomial of degree d d , then its F F -pure threshold at the unique homogeneous maximal ideal is at least 1 d − 1 \frac {1}{d-1} . We show, furthermore, that its F F -pure threshold equals 1 d − 1 \frac {1}{d-1} if and only if f ∈ m [ q ] f\in \mathfrak m^{[q]} and d = q + 1 d=q+1 , where q q is a power of p p . Up to linear changes of coordinates (over a fixed algebraically closed field), we classify such “extremal singularities”, and show that there is at most one with isolated singularity. Finally, we indicate several ways in which the projective hypersurfaces defined by such forms are “extremal”, for example, in terms of the configurations of lines they can contain.

Let P be a finite poset, K a field, and O(P ) (resp. C (P )) the order (resp. chain) polytope of P . We study the non-Gorenstein locus of E K [O(P )] (resp. E K [C (P )]), the Ehrhart ring of O(P ) (resp. C (P )) over K, which are each normal toric rings associated P . In particular, we show that the dimension of non-Gorenstein loci of E K [O(P )] and E K [C (P )] are the same. Further, we show that E K [C (P )] is nearly Gorenstein if and only if P is the disjoint union of pure posets P 1 , . . . , P s with |rankP i − rankP j | ≤ 1 for any i and j.