87 problems

Let $I$ be an ideal of points in $\mathbb{P}^{n-1}$, where $ngeq 2$, and let $I^{(s)}$ denote its $s$-th symbolic power. Write … for the least degree of a nonzero element of $I$. C…

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

Let $I$ be the defining ideal of $s$ reduced points in ${\mathbb P}^N$, let $\mathfrak m$ denote the homogeneous maximal ideal, and let $h$ be the parameter appearing in the contai…

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 graph with edge ideal $I(G)$, and let $I(G)^{(t)}$ denote its $t$-th symbolic power. The Castelnuovo–Mumford regularity of an ideal is denoted by $\operatorname{reg}$.…

Demailly's conjecture. The inequality

Eisenbud–Mazur conjecture. The containment

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…

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 $S$ 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…

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

Jacobian-ideal conjecture. The generators $g_1,\ldots,g_s$ can be chosen so that

Let $I$ be a homogeneous ideal, and let $I^{[k]}$ denote the generalized power defined in the cited source. Let $\rho(J)$ denote the resurgence of an ideal $J$. Unbounded resurgenc…

Stable Harbourne conjecture. For all $n \gg 0$,

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 $R=k[x,y,z]$ be the polynomial ring in three variables over a field $k$, and let $P$ be the prime ideal defining the space monomial curve $(t^a,t^b,t^c)$ in $\mathbb A^3_k$. Fo…

Harbourne–Huneke degree-bound conjecture. For every $r>0$,

Let $(R,\mathfrak m)$ be a Cohen–Macaulay local ring and let $P$ be a prime ideal of height $g\geq 2$. Let $J=(z_1,\dots,z_g)$ be an ideal generated by a regular sequence contained…

SP-2 conjecture. The intersection of symbolic powers satisfies

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 $6\mid n$ and let $E$ be any elliptic curve in $\mathbb{P}^2$. Apply the Elliptified Böröczky construction to the points of $E[n]$, and let $\mathcal{B}_{En}$ denote the triple…

Let $S=K[x_1,\dots,x_n]$ be a polynomial ring over a field $K$, and let $I\subset S$ be a polymatroidal ideal. For $k\ge 1$, write $I^{(k)}$ for the $k$th symbolic power of $I$. Po…

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 homogeneous ideal in the polynomial ring $R={\Bbbk}[x_1,\dots,x_r]$, and let $I^{(n)}$ denote its $n$th symbolic power. Herzog–Hoa–Trung question. Does the limit … exi…

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…