Algebraic Stories — Hilbert functions of fat points.

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 $t,s,m\geq1$ and $n_1,\ldots,n_t\geq1$. Work over $\mathbb C$ and put \[Y=\mathbb P^{n_1-1}\times\cdots\times\mathbb P^{n_t-1},\qquad S=\mathbb C[x_{i,j}:1\leq i\leq t,1\leq j\leq n_i]. \] Give $x_{i,j}$ multidegree equal to the $i$th standard basis vector. For generic distinct points $P_1,\ldots,P_s\in Y$, let $\mathfrak p_j$ be their multihomogeneous prime ideals and set \[I_X=\bigcap_{j=1}^s\mathfrak p_j^m,\qquad X=mP_1+\cdots+mP_s. \] Write $\mathbb N_0=\{0,1,\ldots\}$ and $H_X(\boldsymbol a)=\dim_{\mathbb C}(S/I_X)_{\boldsymbol a}$ for $\boldsymbol a\in\mathbb N_0^t$.

Statement recorded in the supplied source.

Calculate $H_X(\boldsymbol a)$ for all the parameters above and every multidegree $\boldsymbol a$. [S2]

Short English statement

Determine every multigraded Hilbert-function value for equally thickened generic points in a product of projective spaces.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 1500015 (AMR-014-0015). 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), §3.1, Problem J. https://arxiv.org/abs/1801.01692.