Algebraic Stories — Lefschetz properties of graded algebras

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $S=\mathbb C[x_1,\ldots,x_n]$ with its usual total-degree grading. For a homogeneous quotient $R$, define \[\operatorname{Hilb}_R(t)=\sum_{e\geq0}(\dim_{\mathbb C}R_e)t^e. \] A generic tuple of forms has its coefficient vector in a suitable nonempty Zariski open subset of the product of the prescribed homogeneous pieces.For a partition $\mu=(\mu_1,\ldots,\mu_q)$ of $e\geq1$, a graded algebra $A$ has the $\mu$-Lefschetz property if, for generic linear forms $\ell_1,\ldots,\ell_q$, the maps \[A_i\xrightarrow{\ \times\ell_1^{\mu_1}\cdots\ell_q^{\mu_q}\ } A_{i+e} \] have rank $\min\{\dim A_i,\dim A_{i+e}\}$ for every $i\geq0$. Let $R=S/(f_1,\ldots,f_r)$, with the $f_j$ generic homogeneous forms of arbitrary prescribed positive degrees.

Statement recorded in the supplied source.

Does every such $R$ have the $\mu$-Lefschetz property for every partition $\mu$? [S2]

Short English statement

Does multiplication by any generic product of powers of linear forms have maximal rank in every degree of a generic quotient?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 1500014 (AMR-014-0014). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 R. Fröberg, S. Lundqvist, A. Oneto and B. Shapiro, Algebraic Stories from One and from the Other Pockets, arXiv:1801.01692 (2018), §2.3, Problem I. https://arxiv.org/abs/1801.01692.