31 problems

Let $Z=\sum_i m_iP_i$ be a fat point scheme in $\mathbb{P}^2$ for general points $P_i$, and suppose that $h^1(\mathcal I_Z(k))=0$. Let $\mu_k(Z)$ denote the multiplication map in d…

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

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 $Z_{s,m}$ 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.

Chudnovsky's conjecture.

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 $s\le 4$ 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 $Z a derivesubseteq \mathbb{P}^n$ be a fat point scheme, let $I:=IZ$ be its defining ideal, and let $\mathcal{M}=(x0,\dots,xn)$ be the graded maximal ideal. **Harbourne–Huneke con…

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 $S$ 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 $Z\subseteq\mathbb P^N$ 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…

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…

Conjecture on the transformed ideal.

Segre's version of the SHGH conjecture. For $X=m_1P_1+\cdots+m_rP_r$ with general points $P_i$, $X$ fails to impose the expected number of conditions on $V$ only if $[I_X]_j\ne0$ b…

Fix integers $d\ge 2$ and $n>2$. Let $S\subseteq {\mathbb P}^n$ be a finite collection of $s$ points, fix integers $m_1,\dots,m_s$, and set … Let $\operatorname{w}(Z)$ be the total…

Fatabbi–Lorenzini's generalized Segre bound. One has

Let ${\mathcal L}={\mathcal L}_{n,d}(m^s)$ be a homogeneous linear system of degree-$d$ hypersurfaces in ${\mathbb P}^n$ through $s$ general points, each with multiplicity $m$, and…