4 problems

Let $D=D_1+\dots+D_n$ be a reduced curve with simple normal crossings on a smooth algebraic surface $V$, and let $\Gamma$ be the dual graph of $D$. For each component, set…

Let $S$ be a polynomial ring and let $I\subseteq S$ be a quadratic ideal. Assume that $S/I$ is Cohen–Macaulay. The dual graph of $I$ has vertex set $\operatorname{Min}(I)$, with an…

The quadratic Cohen–Macaulay Hirsch conjecture. If $S/I$ is Cohen–Macaulay, then $I$ is Hirsch.

Degree-two Cohen–Macaulay Hirsch conjecture. If $S/I$ is Cohen–Macaulay, then $I$ is Hirsch.