16 problems

Let $\mathcal{A}$ be a central arrangement in $\mathbb{K}^{n+1}$. Write $\operatorname{rgin}(J(\mathcal{A}))$ for the generic initial ideal of its Jacobian ideal, and set … A minim…

Jacobian-ideal conjecture. The generators $g_1,\ldots,g_s$ can be chosen so that

Let $f^{(r)}$ be a sub-Hankel determinant, and let $J$ denote its full Jacobian ideal, generated by the partial derivatives of $f^{(r)}$. An ideal is of linear type if its Rees alg…

Let $k$ be a field, let $R=k[x_1,\ldots,x_n]$, and let $\mathcal A$ be a central hyperplane arrangement with $m$ hyperplanes, defining linear forms whose product is $f$. Let $J_f$…

Let $k$ be a field, let $R=k[x_1,\ldots,x_n]$, and let $\mathcal A$ be a central hyperplane arrangement with defining linear forms whose product is $f$. Let $I\subseteq R$ be the i…

Let $k$ be a field of characteristic zero and let … Let $f\in R$ define an isolated hypersurface singularity, with Jacobian ideal $j(f)$ and Tjurina ideal $tj(f)$. Hassanzadeh–Neja…

Ramirez's conjecture. The ideal of maximal minors of $K$ is

Let $\mathbb K$ be a field with $\operatorname{char}(\mathbb K)\nmid m$, let $R=\mathbb K[x_1,\ldots,x_n]$, and let $f\in R$ be the defining polynomial of a central hyperplane arra…

Let $X=V(f)\subseteq \mathbb{A}_k^n$ be an affine hypersurface with one-dimensional singularities. For a minimal prime $\mathfrak q$ of the Jacobian ideal $I(f)$, let…

Let $hh_1^{r_1}\cdots h_s^{r_s}=0$ be the equation of the unique non-reduced curve, where ${h=0}$ is reduced, $\operatorname{deg}(h_i)=m_i\geq 1$, and $r_i\geq 2$. Let $D_k$ be a m…

Carlson–Griffiths conjecture. If $V(f)$ does not possess singularities which are large in relation to the degree, then $f$ is determined by $E(f)$, up to a nonzero scalar multiple.

Unimodality conjecture. The sequence of dimensions $\dim SD(J_f)_k$ is unimodal, namely

Let $D:f=0$ be a hypersurface in $\mathbb{P}^n$ with only isolated singularities, let $J=J_f$ be its Jacobian ideal, and write $ct(D)$ and $st(D)$ for the coincidence and stability…

Let $D$ be a divisor in a complex manifold $S$, defined locally at a point $p$ by a reduced $h \in \mathcal{O}_{S,p}$. Suppose that the Jacobian ideal $J_h$ is radical, equidimensi…

Let $D \subseteq S$ be a divisor in a complex manifold $S$, defined at a point $p$ by $h \in \mathcal{O}_{S,p}$. Assume that its Jacobian ideal $J_h$ is radical, equidimensional, a…

Let $J$ be an ideal generated by quadrics in a polynomial ring $S$, and let $r={\rm ht}(J)\geq 2$. Let $\Theta$ be the Jacobian matrix of $J$, let $I_r(\Theta)$ denote its ideal of…