A linear variant of the nearly Gorenstein property

Abstract: We introduce a condition $(\natural)$ for Cohen--Macaulay semi-standard graded rings, motivated by the study of Ehrhart rings. In this context, condition $(\natural)$ is characterized by a Minkowski decomposition of certain lattice polytopes. We show that if $R$ satisfies $(\natural)$, then its Veronese subrings $R^{(k)}$ are nearly Gorenstein for all sufficiently large $k$. This extends previously known results on Ehrhart rings and shows that condition $(\natural)$ provides a valuable framework from both ring-theoretic and combinatorial perspectives.