61 problems

Let $(R,\mathfrak{m})$ be a regular local ring, and let $\mathfrak{p}$ and $\mathfrak{q}$ be prime ideals in $R$ such that … Here $\mathfrak{p}^{(n)}$ denotes the $n$th symbolic po…

SP-2 conjecture. The intersection of symbolic powers satisfies

Let $S$ be a polynomial ring and let $I$ be a monomial ideal in $S$. An ideal is Simis when its symbolic Rees algebra is generated by its elements of degree one; a monomial ideal h…

Conforti–Cornuéjols conjecture. All symbolic powers and ordinary powers of $I$ coincide if and only if $I$ has the packing property:

Let $I\subset K[x_1,\ldots,x_n]$ be a monomial ideal, and let $h$ be its big height. Let $\alpha(I)$ be the least degree of a nonzero element of $I$, and let … be its Waldschmidt c…

Let $R={\Bbbk}[x,y,a,b]$ be the polynomial ring from the preceding construction, let $S=R[z]$, and set $I=Q\cap(f,z)\subseteq S$. The source specifies that…

Let $I$ be a square-free monomial ideal. Eventual stability conjecture. Then, $\operatorname{assrad} I^s$ is constant for $s\gg 0$. This is the ordinary-power analogue of the event…

Linear quotients conjecture for symbolic powers. If $I(G)$ has a linear resolution, then $I(G)^{(k)}$ has linear quotients for every $k\geq 1$.

Componentwise linear symbolic power conjecture. If $I(G)$ has a linear resolution, then $I(G)^{(k)}$ is componentwise linear for every $k\geq 1$.

Demailly's conjecture. For every integer $s\geqslant 1$,

Converse conjecture. If $I$ is symbolic $F$-split, then

Let $G$ be a graph on $m$ vertices and let $\alpha=(\alpha_1,\dots,\alpha_m)\in(\mathbb Z^+)^m$, where $G^\alpha$ is obtained by duplicating the vertices of $G$ according to…

Restricted bipartite connectivity conjecture. The stable value of the depth of the symbolic powers of $I(G)$ is

Chudnovsky's conjecture.

Algebraic packing conjecture. One has

Let $G$ be a graph, let $I(G)$ denote its edge ideal, let $I(G)^{\{s\}}$ denote the squarefree part of its $s$-th symbolic power, and let ${\rm match}(G)$ denote the matching numbe…

Iarrobino's conjecture. For every $n\in\mathbb{N}$,

Let $I$ be a radical ideal of big height $c$ in a regular ring. For each positive integer $n$, let $I^{(n)}$ denote the $n$th symbolic power of $I$. Stable Harbourne conjecture. On…

Let $\mathbb{k}$ be a field, let $R=\mathbb{k}[x_1,\ldots,x_d]$, and let $I\subseteq R$ be a radical homogeneous ideal of bigheight $h$. Harbourne's conjecture. One has … for every…

Let $G$ be a simple graph, and let $I(G)$ be its edge ideal. For each integer $s\geq 1$, the symbolic power $I(G)^{(s)}$ and ordinary power $I(G)^s$ are ideals in the associated po…

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…

Suppose that $k$ is an algebraically closed field of characteristic $0$, and let $I$ be the defining ideal of a set of points $X\subseteq \mathbb{P}^n_k$. For an ideal $J$, let…

Let $I\subseteq k[x_0,\dots,x_n]$ be a homogeneous radical ideal of big height $h$, and let $\mathcal{M}=(x_0,\dots,x_n)$ be the graded maximal ideal. Stable Harbourne–Huneke conje…

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 $H=(V,E)$ be a hypergraph, and let $I(H)$ be its edge ideal. A minor of $H$ is obtained by deleting and contracting vertices. The hypergraph $H$ has the packing property if eve…