16 problems

Let $S/R$ be an extension of commutative rings, and let … When $S=\overline{R}$, the integral closure of $R$ in its total ring of fractions $\operatorname{Q}(R)$, write $R^*=R_S^*$…

Normalization conjecture for tropical curves. There is an embedding of $C$ with an ideal $I$ having the property that the integral closure of $B$ in its field of fractions is obtai…

Let $K$ be a field of arbitrary characteristic and let … Let $f\in R$ define an isolated hypersurface singularity, let $j(f)$ be its Jacobian ideal, and write $j(f):f$ for the idea…

Let $\mathcal{O}_2=k[x_1,x_2]$ and let $\mathcal{I}\triangleleft\mathcal{O}_2$ be a finite-codimensional, integrally closed ideal. Let $\mathcal{O}$ denote the ambient module in th…

Let $k[x_1,x_2]$ be the polynomial ring in two variables, and let $\mathcal{I}\triangleleft k[x_1,x_2]$ be a finite-codimensional, integrally closed ideal. Write…

Let $R$ be a commutative ring with no nilpotent elements, and let $\overline{R}$ denote its integral closure. The quotient $\overline{R}/R$ is called $h$-reduced when it has the pr…

Let $\mathbb{k}$ be an algebraically closed field, and let $R$ and $S$ be normal domains finitely generated over $\mathbb{k}$. Set … For a rational exponent, write…

Squarefree monomial specialization conjecture. The preceding result continues to hold for squarefree monomial ideals without the condition that the generating sets of variables in…

Catenarity conjecture for integrally closed domains. Every integrally closed domain is catenary.

Strong Rees property characterization conjecture. Then $\dim A=2$, and either $A$ is a regular local ring or

Archimedean characterization conjecture. $D$ is completely integrally closed if and only if $D$ is Archimedean.

Power-series complete integral-closure conjecture. $D[[X]]$ is integrally closed if and only if $D[[X]]$ is completely integrally closed, and this holds if and only if $D$ is compl…

Polynomial-extension conjecture. The Briançon–Skoda number satisfies

Let $S=\mathbb{K}[x_1,\dots,x_n]$ and let $I\subset S$ be an integrally closed monomial ideal. If $\ell(I)$ denotes the analytic spread of $I$, then Seyed Fakhari's conjecture. … T…

Let $R$, $\mathfrak{a}$ and $J$ be as in the relevant corollary, with positive integers $a_1,\ldots,a_d,b_1,\ldots,b_d$ supplied by that setup. Suppose that … Put…

Suppose that $\mathfrak{a}$ and $J$ are $\mathfrak{m}$-primary ideals in a $d$-dimensional Noetherian local or Noetherian graded $k$-algebra $(R,\mathfrak{m})$, where $k$ is a fiel…