AI 中文总结
该研究否定了Davis等人关于带正非标准ℤ-分次多项式环的Veronese子环具有最终非标准Koszul性的猜想，通过构造三次障碍构型在n≥4元变量中得到反例，并证明相关权向量集具正密度，三变量情形仍开放。
AI 中文摘要
根据Backelin的结果，标准ℤ-分次代数的Veronese子环是最终Koszul的。Davis、Erman和Martinova最近推测，对于具有正非标准ℤ-分次的多项式环的Veronese子环，类似结论也成立。我们否定了这一猜想。对于权为(1,4,7,9)的加权多项式环𝕂[x₁,x₂,x₃,x₄]，我们证明对所有k≥1，第(9k+12)个Veronese子环都不是非标准Koszul的。我们的反例来自某些格点排列，称之为三次障碍构型。每个此类构型都会在对应相伴分次环的定义理想中产生一个极小三次生成元。该构造对所有n≥4的n元变量都产生反例。此外，我们证明，最终非标准Koszul性不成立的本原四变量权向量集具有正密度。对于固定的三变量分次，我们的三次障碍仅能出现在有限个Veronese指标处，该情形仍待解决。
英文摘要
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 $k\geq1$. 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 $n\geq4$. 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.
Comments7 pages. comments welcome!
