9 problems

Let $K$ be a field, let $S=K[x_1,\ldots,x_N]$ be standard graded, and let $I\subseteq S$ be a homogeneous ideal containing a regular sequence of degrees…

Conjecture on the v-number of linear powers. If $I$ has linear powers, then

Assume that ${\Bbbk}$ has characteristic $2$, let $R={\Bbbk}[x,y,a,b]$, set $Q=(x^3,y^3)$ and $f=xya-(x^2+y^2)b$, and let $n=2^s$ for some $s\in\mathbb N$. Characteristic-two upper…

Let $S$ be a standard graded polynomial ring and let $I\subset S$ be a graded ideal with linear powers, meaning that every power $I^k$ has a linear free resolution. Let $\alpha(I)$…

Asymptotic regularity conjecture. Since the function $\operatorname{reg} I^{n-1}/I^n$ can be any numerical asymptotically linear function $f(n)\geq dn-1$ with slope $d$, the same s…

Defect-sequence characterization conjecture. The sequence $\{e_n\}$ is the defect sequence of the function $\operatorname{reg} I^n$ if and only if it is weakly decreasing when…

Let $I$ be a graded ideal, let $\mathfrak m$ denote the homogeneous maximal ideal, and let $c_I$ and $W_{c_I}$ be the quantities defined in the paper. Then the regularity stabiliza…

Let $R$ be a polynomial ring over an infinite field, and let $I$ be a graded ideal of $R$. The completely $\mathfrak m$-full/componentwise linearity conjecture. The ideal $I$ is co…

Let $R=K[x_1,\ldots,x_n]$ be a standard graded polynomial ring with homogeneous maximal ideal $\mathfrak m$. An ideal $I$ is completely $\mathfrak m$-full if each ideal obtained fr…