48 problems

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…

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 $\mathbb{G}_T$ 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.…

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…

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…

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 $n$.

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 $S$ be the homogeneous coordinate ring of $\mathbb{P}^n$, let $Z$ be a general set of $r$ points, and let $I(Z)$ be its vanishing ideal. Let $d$ be the Hilbert regularity of…

Let $S$ be the homogeneous coordinate ring of $\mathbb{P}^n$, let $Z$ be a general set of $r$ points, and let $I_{\langle d\rangle}=\langle I(Z)_d\rangle_S$. For positive integers…

Let $S$ be the homogeneous coordinate ring of $\mathbb{P}^n$, let $Z$ be a general set of $r$ points, and let $I(Z)$ be its vanishing ideal. For an integer $d$ that is the smallest…

Let $(R,\mathfrak{m})$ be a Cohen–Macaulay local ring with a canonical ideal $\mathcal{C}$. Let $x$ be an element of $R$ that is regular modulo $\mathcal{C}$, and suppose that…

Maximal Hilbert function conjecture. For every $s$ and $t$,

Maximal Hilbert function conjecture. For every $m,n\geq 2$, there exists an $m$-superfat point in $\mathbb{P}^2$ having maximal Hilbert function.

Let $A={\mathbb{C}}[x_1,\dots,x_n]$ be a polynomial ring, let $A_d$ denote its degree-$d$ component, and let $W\subseteq A_d$ be a $k$-dimensional subspace with $k\le d-1$ and…

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

The strong full Perazzo conjecture. The Hilbert vector $H$ is strongly minimal in $\mathcal{AG}(r,d)$.

The full Perazzo conjecture. The Hilbert vector $H$ is minimal in $\mathcal{AG}(r,d)$; equivalently, if $\hat{H}\preceq H$ is a comparable Artinian Gorenstein Hilbert vector, then…

Let $M$) be an upper triangular matrix of size $d+1$ with non-negative entries. For an upper triangular matrix, let $\operatorname{diag}(i,M)$ denote its $i$-th diagonal vector, an…

Let $v_d^{(n)}(\mathbb{P}^n)$ be the degree-$d$ Veronese variety, let $\mathfrak{L}(X)$ be its sequence of secant indices, and let ${\mathfrak{R}\mathfrak{L}}^G(X)$ and…