49 problems

Let $X$ be a variety, let $\mathcal{L}$ be a line bundle on $X$, and let $\mathbb{X}\subseteq X$ be a zero-dimensional scheme. The scheme $\mathbb{X}$ imposes independent condition…

Hirschowitz–Harbourne conjecture. The system $L$ is special if and only if there exists a $-1$-curve $C\subset\mathbb{P}^{2}$ such that

Let $Z\subseteq\mathbb P^N$ be a fat point subscheme, let $R=K[x_0,\ldots,x_N]$ be the homogeneous coordinate ring, and let $M=(x_0,\ldots,x_N)$. Huneke–Harbourne containment conje…

SHGH conjecture. Either

Self-intersection–genus conjecture. One has

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 $(r,d)$ be integers satisfying … and $(r,d)\notin\{(7,2),(8,2),(9,2),(8,3),(9,3)\}$. Let $l(d,\delta,\mu^r)$ denote the dimension of the space of homogeneous degree-$\delta$ po…

Let $(r,d)$ be integers satisfying … and $(r,d)\notin\{(7,2),(8,2),(9,3)\}$. Let $l(d,\delta,\mu^r)$ denote the dimension of the space of homogeneous degree-$\delta$ polynomials on…

Let $Z$ be a scheme of fat points in general position in the plane, with ideal sheaf ${\cal J}_Z$. For an integer $d$, consider the cohomology and sections of ${\cal J}_Z(d)$. Harb…

Let ${\mathbb X}=\{P_1,\ldots,P_s\}\subseteq {\mathbb P}^2={\mathbb P}^2_k$ be a set of $s$ distinct points over an algebraically closed field $k$ of characteristic $0$, and let…

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…

Non-square grid conjecture. The Artinian algebra $A$ fails to have the weak Lefschetz property.

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.

Smoothability conjecture. $Z$ is smoothable for all $m,n\in\mathbb{N}$.

Let $I\subset k[\mathbf{x}]$ be a zero-dimensional ideal, where $\mathbf{x}=(x_1,\ldots,x_n)$, and let $h$ be a positive integer. For a positive integer $H$, consider the chain … a…

Let $s\le 4$ and let $m_i$ be the multiplicities of $s$ general fat points. For each $i$, take a contact star configuration formed by $m_i+1$ lines tangent to the same conic. The c…

Let $R={\mathbb C}[x,y,z]$, let $Y=\{P_1,\ldots,P_r\}$ be a general set of points in $\mathbb{P}^2$, and let $I_Y$ be its defining ideal. An irreducible homogeneous polynomial…

Let $Z\subset {\mathbb{P}}^N$ be a fat point scheme. For a homogeneous ideal $J$, write $\alpha(J)$ for its least nonzero degree, and write $\widehat{\alpha}(I(Z))$ for the Waldsch…

Let $\mathbb K$ be a field of characteristic $0$, let $R=\mathbb K[x_0,\ldots,x_N]$, and let $I$ be the defining ideal of a finite set of points in $\mathbb P_{\mathbb K}^N$. Stabl…

Let $S$ be the projectivization of the Atiyah bundle over an elliptic curve, equipped with the polarization considered in the paper, and let the special case mean the linear system…

Let $S$ be the blow-up of $s$ general points of $\mathbb P^2$, let $F$ be a divisor on $S$, and let an exceptional curve mean a smooth rational curve $E$ with $E^2=-1$. SHGH conjec…

Let $Z=p1+cdots+ps eqmathbb P^2$ be a union of general points, where either $s$ is a square or $s8$, and let $I(mZ)$ denote the degree-$m$ fat-point ideal. SHGH conjecture. The…

Principal-curve ideal conjecture. The ideal $(f,g,q)$ is perfect, that is, unmixed. Moreover, it admits a linear syzygy if and only if either $s=3$ or $s=5$, or else $s>5$ and…