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 graded algebra $A=\bigoplus_{i\geq0}A_i$, multiplication by a generic linear form $\ell\in A_1$ has maximal rank if its rank is the minimum of the dimensions of its domain and codomain. The weak Lefschetz property (WLP) requires maximal rank for $\ell:A_i\to A_{i+1}$ for all $i\geq0$. The strong Lefschetz property (SLP) requires maximal rank for $\ell^e:A_i\to A_{i+e}$ for every $i\geq0$ and $e\geq1$.Fix positive degrees $d_1,\ldots,d_r$ and generic $f_j\in S_{d_j}$, with $r>n$, and put $R=S/(f_1,\ldots,f_r)$.

Statement recorded in the supplied source.

Does $R$ have both the WLP and the SLP for every choice of $n,r$ and the generator degrees? [S2]

Short English statement

Do all quotients by more generic forms than variables have both weak and strong Lefschetz properties?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 1500012 (AMR-014-0012). 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 G. https://arxiv.org/abs/1801.01692.