4 problems

Let $R=K[x_1,\ldots,x_n]$, and let $I$ be a squarefree monomial ideal. Write $\operatorname{ara} I$ for its arithmetical rank and let $\tau(I)$ be the maximum height of a minimal p…

Let $R=K[x_1,\ldots,x_n]$, let $J$ be a pure-dimensional ideal of $R$ of height $t$, and let $I$ be a pure-dimensional monomial ideal. Write $\operatorname{ara} J$ for the arithmet…

Let $R=K[x_1,\ldots,x_n]$ and let $I$ be a full-supported matroidal ideal of degree $d$, meaning that $I$ is generated by square-free monomials of degree $d$ whose variables collec…

Let $R=K[x_1,\dots,x_n]$ be a polynomial ring, and let $I\subseteq R$ be a matroidal ideal of degree $d$. The arithmetical rank conjecture. Then … This conjecture proposes an exact…