Eventual Nonstandard Koszulness Fails for Veronese Subrings of Weighted Polynomial Rings

Abstract

By a result of Backelin, Veronese subrings of a standard $\mathbb{Z}$-graded algebra are eventually Koszul. Davis, Erman, and Martinova recently conjectured that the analogous statement holds for Veronese subrings of polynomial rings with positive nonstandard $\mathbb{Z}$-gradings. We disprove this conjecture. For the weighted polynomial ring $\mathbb{K}[x_1,x_2,x_3,x_4]$ with weights $(1,4,7,9)$, we prove that the $(9k+12)$-th Veronese subring is not nonstandard Koszul for every $k\ge 1$. Our counterexamples arise from certain arrangements of lattice points, which we call cubic obstruction configurations. Each such configuration produces a minimal cubic generator in the defining ideal of the corresponding associated graded ring. This construction yields counterexamples in $n$ variables for all $n\ge 4$. Moreover, we prove that the set of primitive four-variable weight vectors for which eventual nonstandard Koszulness fails has positive density. For a fixed three-variable grading, our cubic obstruction can occur at only finitely many Veronese indices, leaving that case open.

7 pages. comments welcome!