14 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…

Demailly's conjecture. The inequality

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

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 $H$ be a hypergraph with edge ideal $I=I(H)$. Write $\alpha(I)$ for the least degree of a nonzero homogeneous element of $I$, and let $\operatorname{ht}(I)$ denote the height o…

Let $I$ be a monomial ideal. Define its initial degree by $\alpha(I)$, its naive Waldschmidt constant by $\widetilde{\alpha}(I)$, and its big-height by … The naive Chudnovsky bound…

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…

Equality conjecture. Equality holds for all sufficiently large $s$:

Conjectural sharpness of the upper bound. The upper bound is conjectured to be the actual bound for $r\geqslant 2^N$.

Let $X_\mathcal K$ be the blowup associated with the Klein line configuration, with $H$ the pullback of a line and $E_4,E_3$ the exceptional divisors over its quadruple and triple…

Let $I$ be a monomial ideal, and let $\operatorname{big-height}(I)$ denote the maximum of the heights of the associated primes $P\in\operatorname{Ass}(I)$. Write…

Nagata–Iarrobino conjecture. If $I$ is the ideal of $r\gg0$ generic points of $\mathbb{A}^{n}$, then

Let $K$ be an algebraically closed field, let $I$ be the radical ideal of a finite set of points in $\mathbb{P}^{N}$, and let $\alpha(I)$ denote the least degree of a nonzero form…

Let $K[\mathbb{P}^{N}]$ be the homogeneous coordinate ring, let $I$ be the ideal of a finite set of points in $\mathbb{P}^{N}$, and let $\alpha(I)$ denote its initial degree. Evolu…