81 problems

Eisenbud–Green–Harris conjecture. The ideal $I$ has the same Hilbert function as an ideal containing $x_1^{a_1},\dots,x_n^{a_n}$. This conjecture predicts that the Hilbert function…

Migliore–Miró-Roig conjecture. Such general equigenerated algebras have minimal Betti numbers among algebras with the Fröberg Hilbert function, subject to the constraints imposed b…

Almkvist's conjecture. For each $m$, the Hilbert function $H(m,n)$ is unimodal for all $n\geq 11$. Moreover, if $m$ is even, then $H(m,n)$ is unimodal for all $n$.

Rossi's conjecture. The Hilbert function of a one-dimensional Gorenstein local ring should be non-decreasing.

SI-sequence conjecture. The sequence $AV_{X,1}$ is an SI-sequence shifted by $1$. Its last nonzero term is

Let $X\subset\mathbf{P}^{r+1}$ be a nondegenerate reduced irreducible curve of genus $g$ and degree $n$, where $\pi_\alpha(n,r)=\frac{n^2}{2(r+\alpha)}+O(n)$ is the Eisenbud–Harris…

Let $(A,\mathfrak{m})$ be a local ring, with $\mathfrak{m}$ its maximal ideal, and write $e_i=e_i(\mathfrak{m})$ for the Hilbert coefficients and $\lambda(A/\mathfrak{m})$ for the…

Stanley's unimodality conjecture. The $h$-vector of $A$ is unimodal.

Let $\Gamma$ be the set of points in symmetric position considered in the preceding higher Castelnuovo conjecture, and let $\mathcal I_\Gamma$ be its ideal sheaf. Write…

Higher Castelnuovo conjecture. Then $\Gamma$ lies on a curve $D$ of degree at most $n+m-2$. This is related to the Eisenbud–Harris conjecture for curves: the cases $m=1$ and $m=2$…

The Lex Plus Powers conjecture for Hilbert functions. Then

The Lex Plus Powers conjecture. If $\mathcal H$ is $\mathbb A$-lpp valid and $L_{\mathcal H,\mathbb A}$ is the $\mathbb A$-lex plus powers ideal attaining $\mathcal H$, then…

Let ${\bf H}$ be a Hilbert function of the form … with $h_d<h_{d+2}$. Socle-element conjecture. (a) Any graded algebra of codimension $3$ with Hilbert function ${\bf H}$ has a socl…

Failure conjecture. Not all height four Gorenstein sequences satisfy the condition that $\Delta H_{\leq j/2}$ is an $O$-sequence.

Linear-configuration conjecture. The Hilbert function of $C_{\underline h}$ is $\underline h^{\min}$.

Minimal-Hilbert-function conjecture. Among these double schemes, there is a minimal Hilbert function, and it occurs when the support is $C_{t,r}$.

Let $X$ be an arithmetically Gorenstein scheme, and let $h_X$ denote its Hilbert function. A generalized stick figure is an arithmetically Gorenstein scheme of the type defined in…

Let $\mathbb{G}_T$ be the parameter space of graded quotients with Hilbert function $T$, and let $U_c$ and $U_e$ be cells in $\mathbb{G}_T$ of dimensions $c$ and $e$, respectively.…

Let $Z_{s,m}$ be a general union of $s$ flat fat points of multiplicity $m$ in $\mathbb P^2$. Assume $m\geq 4$ and $s\geq 2m-3$. Its Hilbert function should satisfy … A flat fat po…

Let $A=k[\![x,y]\!]/J$ be an AG $k$-algebra, let $I$ be an ideal in $A$, and let $H$ be a $2$-admissible $G(I)$-Gorenstein sequence. A symmetric decomposition is a decomposition of…

Let $A=k[ … /J$ be an AG $k$-algebra. Let $H=(h0,h1,ldots,hu)$ be a $2$-admissible sequence, where a sequence is partially strict unimodal if…

Minimality conjecture for full Perazzo algebras. The $h$-vectors of full Perazzo algebras are minimal among the $h$-vectors of all Artinian Gorenstein algebras with the same degree…

Polynomiality conjecture. We conjecture that $TH_V(k)$ is a polynomial of degree $\dim V$ for sufficiently large $k$. In the case $\dim V=1$, $degT(V)$ is the leading coefficient o…

Gotzmann threshold polynomiality conjecture. For all $n \ge 3$, $\tau_n(x_2^d)$ is a polynomial of degree $2^{n-2}$ in $d$ with dominant term

Let $I$ be a random monomial ideal in the Erdős–Rényi model considered in the paper, and let $S/I$ be its associated algebra with Hilbert function $h_{S/I}$. Based on simulations,…