13 problems

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 $(R,\mathfrak m)$ be an analytically unramified Gorenstein local ring of dimension $d\geq 3$, and let $I$ be a parameter ideal. The normal Hilbert coefficients…

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…