Lefschetz properties and square-free Gröbner degenerations
演讲者
Hongmiao Yu
时间
2026年05月07日 15:00 至 16:00
地点
A6-101
线上
Zoom 518 868 7656
(BIMSA)
摘要
The algebraic Lefschetz properties are inspired by the Hard Lefschetz Theorem from the cohomology of projective varieties. In this talk, we study the weak and strong Lefschetz properties for the Stanley–Reisner ring $R/in(I_t)$, where $I_t$ is the ideal of a polynomial ring R generated by the $t$-minors of an $m \times n$ matrix of indeterminates, and $in(I_t)$ denotes the initial ideal of $I_t$ with respect to a diagonal monomial order.
We show that when $I_t$ is generated by maximal minors, that is, when $t=\min\{m,n\}$, the algebra $R/in(I_t)$ has the strong Lefschetz property for all $m,n$. In contrast, for $t < \min\{m,n\}$, we establish a bound such that $R/in(I_t)$ fails to satisfy the weak Lefschetz property whenever the product $mn$ exceeds this bound.
As an application, these results yield counterexamples that provide a negative answer to a question posed by Murai regarding the preservation of Lefschetz properties under square-free Gröbner degenerations.
We show that when $I_t$ is generated by maximal minors, that is, when $t=\min\{m,n\}$, the algebra $R/in(I_t)$ has the strong Lefschetz property for all $m,n$. In contrast, for $t < \min\{m,n\}$, we establish a bound such that $R/in(I_t)$ fails to satisfy the weak Lefschetz property whenever the product $mn$ exceeds this bound.
As an application, these results yield counterexamples that provide a negative answer to a question posed by Murai regarding the preservation of Lefschetz properties under square-free Gröbner degenerations.