7 problems

Let $(R,\mathfrak{m})$ be a local complete intersection ring of dimension $3$. Let … be the punctured spectrum of $R$. Gabber's conjecture. The group $\operatorname{Pic}(U_R)$ is t…

Let $X$ be the smooth variety and let $f_1,\dots,f_r\in \mathscr{O}_X(X)$ be a regular sequence. Set $g=f_1\cdots f_r$ and let $U$ denote the open subset on which the relevant rest…

Let $X$ be a non-singular variety, and let $Y_1,\dots,Y_l$ be local complete intersection subvarieties whose intersection decomposes as … where each $Z_j$ is a local complete inter…

Let $Z$ be a reduced local complete intersection of codimension $r$. Let $p(Z)$ denote its singularity level, and let $\widetilde{\alpha}(Z)$ be the negative of the largest root of…

Let $\gamma$ be any $(p,2)$-clan with $p\geq 2$, and let $Y_{\gamma}$ be its orbit closure. lci–Gorenstein equivalence conjecture. $Y_{\gamma}$ is a local complete intersection if…

Let $\gamma$ be a clan, and let $Y_{\gamma}$ be its orbit closure. General lci pattern-avoidance conjecture. Ordinary pattern avoidance should characterize whether $Y_{\gamma}$ is…

Niu's generic-linkage conjecture. The variety $Y$ is also a local complete intersection with log canonical singularities.