6 problems

Let $X$ be a variety, let $i$ be a nonnegative integer, and write $\underline{\Omega}^{i}_{X}$ for the $i$-th graded piece of the Du Bois complex of $X$. For a sheaf or complex, le…

Let ${\Gamma}$ be a connected non-bipartite graph, let $R$ be the polynomial ring associated to its vertices, and let $I$ be the edge ideal of ${\Gamma}$. Depth-drop conjecture. If…

Let ${\Gamma}$ be a connected graph, let $R$ be the polynomial ring associated to its vertices, let $I$ be the edge ideal of ${\Gamma}$, and let ${\mathfrak m}$ be the homogeneous…

Let $W(C_m)$ be the graph obtained by attaching a whisker to each vertex of the cycle $C_m$, let $I(W(C_m))$ be its edge ideal in the corresponding polynomial ring…

Let $C_n$ be a cycle of length $n$, let $I(C_n)$ be its edge ideal in the corresponding polynomial ring $R=\mathbb{K}[x\mid x\in V(C_n)]$, and let $\nu(C_n)$ denote its matching nu…

Depth criterion conjecture. One has