7 problems

Let $X\subset\mathbb P^n$ be a projective variety of dimension $d\geq 1$, with $\operatorname{reg}(X)=m$ and Hilbert polynomial $P_X$. Let $\delta$ satisfy…

Let $\mathbb X$ be a general set of $s={d+1\choose 2}+k$ points in $\mathbb P^2$, where $0<k<d+1$, and let $I_{\mathbb X}$ be its defining ideal. The ideal generation conjecture st…

Weakened Minimal Resolution Conjecture. The set $X$ has a generic Hilbert matrix as in the source definition, and, for every fixed $(i,j)>(0,0)$, one has $\beta_{1,(i,j)}>0$ if and…

Let ${\mathbb X}=\{P_1,\dots,P_s\}\subseteq\mathbb{P}^N$ be a sufficiently large set of general points, and let $\alpha_m({\mathbb X})$ denote the least degree of a hypersurface pa…

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…

Let $K[\mathbb{P}^{N}]$ be the homogeneous coordinate ring, let $M$ be its homogeneous maximal ideal, and let $I$ be the ideal of a finite set of points $p_i\in\mathbb{P}^{N}$. Evo…

Let $I\subseteq K[\mathbb{P}^{2}]$ be the homogeneous radical ideal of a finite set of points, and let $\alpha(I)$ denote its initial degree. Plane containment conjecture. Whenever…