14 problems

Dynkin-diagram conjecture. The ideal is in the linkage class of a complete intersection if and only if its associated graph is a Dynkin diagram.

Pokrovskiy's conjecture. There exists an integer $f(k)$ such that every $2k$-strong tournament with minimum semi-degree at least $f(k)$ is $k$-linked.

Let $(R,\operatorname{\mathfrak m})$ be the local ring considered in the paper, let $I_k$ be the $k$-th iterated link, and let $\delta(I_k)$ denote the kernel of the natural surjec…

The iterated-link reduction-number conjecture. For every $k=1,\ldots,s$, one has

Girão–Popielarz–Snyder's conjecture. A minimum out-degree of $O(k)$ is already sufficient for the relevant tournament linkage conclusion.

Fix integers $c\geq 3$, $d\geq 1$, and $t\geq 1$. For a perfect ideal, the codimension, deviation, and type are respectively $c$, $d$, and $t$; associate to the triplet the diagram…

Let $I$ be a licci ideal of codimension $c$, and write its decoration as … Suppose that $J$ is an ideal geometrically linked to $I$, with $\kappa(I)\leq\kappa(J)$. The doubling con…

Thomassen's weak linkage conjecture. If $G$ is $k$-edge-connected, then there are $k$ pairwise edge-disjoint paths $P_1,\dots,P_k$ in $G$, with $P_i$ joining $s_i$ and $t_i$ for ea…

Girã o, Snyder, and Popielarz's conjecture. There exists a constant $C>0$ such that every $(2k+1)$-connected tournament with minimum out-degree at least $Ck$ is $k$-linked.

Gir2o–Snyder–Popielarz conjecture. There exists a constant $C>0$ such that every $(2k+1)$-connected tournament with minimum out-degree at least $Ck$ is $k$-linked.

Let $R$ be a commutative local or graded ring, and let $M$ be a perfect module with a minimal free resolution of one of the formats … A module is in the BR-linkage class of a Buchs…

Let $R$ be a Gorenstein local ring with infinite residue field, let $\mathfrak{m}$ be its maximal ideal, and let $I$ be a perfect ideal of height $3$. Let $W(d_1)$ denote the space…

Let $R$ be a Gorenstein local ring and let $I$ be a perfect ideal of height $3$ admitting a minimal free resolution of Dynkin format. Such formats include the formats associated wi…

Let $\mathfrak{I}\subseteq Q$ be a grade $3$ perfect ideal of class $\mathbf{G}$ that is not Gorenstein. Write $m_{\mathfrak{I}}$ for its first deviation, $n_{\mathfrak{I}}$ for it…