The Koszul property of pinched Veronese varieties

Let $K$ be an arbitrary field. Let $n,d\ge 2$ be positive integers. Let $V(n,d)$ be the set of all lattice points $\mathbf{b}=(b_1,\mathrm{.}\mathrm{.}\mathrm{.},b_n)$ in ${\mathbb{N}}^n$ such that ${\sum }_{i=1}^n{b}_i=d$. Let $\Gamma = V(n,d) \setminus { \mathbf a }$forsomeelement$\mathbf a \in V(n,d)$. In this paper we prove that the semigroup ring $K[\mathrm{\Gamma }]$ is Koszul unless $d\ge 3$ and $\mathbf{a}=(0,\mathrm{.}\mathrm{.}\mathrm{.},0,2,d-2)$ or one of its permutations. This generalizes results of Caviglia, Conca, and Tancer.