16 problems

Let $L=(L_{ij})$ be an $m\times n$ matrix whose entries have the form … with $a_{ijk}\in K$ for all $i,j,k$. Assume that, for each $i$, the forms $L_{i1},\dots,L_{in}$ are linearly…

Let $t_{ij}$ be distinct variables over a field $K$ with $1\leq i\leq m$ and $1\leq j\leq n$. Let $L=(L_{ij})$ be an $m\times n$ matrix with … where $a_{ijk}\in K$ for all $i,j,k$.…

Cubic generation conjecture. The prime ideal $Q_G$ is generated by the $3\times 3$-subdeterminants of any two-dimensional table obtained by flattening the $n$-dimensional table…

Let $n=qp$ and let $I=\langle f_0,f_1,\ldots,f_{p-1}\rangle\cap R^{S_n}$ be the elimination ideal, where the $f_i$ are the polynomials obtained by collecting the coefficients of th…

Let $\mathbf{k}=(k_0,\dots,k_n)$ and $\mathbf{r}=(r_1,\dots,r_n)$, let $\psi$ be a $k_0\times\cdots\times k_n$ tensor of indeterminates, and let $\psi^i$ denote its flattenings. Th…

Let $X$ be an $n\times n$ matrix of indeterminates, and let $I_n$ be the ideal generated by the $\tfrac{n}{2}\times\tfrac{n}{2}$ minors of $X$. For a nonzero polynomial $f\in I_n$,…

Let $X$ be a smooth projective variety of dimension $n$. Let $\kappa_k(X,L)$ be the $k$-th cactus variety, namely the closure of the union of the $k$-planes spanned by finite subsc…

Let $C$ be a smooth projective curve of genus $g$. For line bundles $A$ and $B$ on $C$ of sufficiently large degree with respect to $g$ and $k$, set $L=A\otimes B$. The $k$-th seca…

Let $n$ be a positive integer, let $\boldsymbol{\rm Mat}^{\mathrm{sym}}_{n\times n}$ be the space of symmetric $n\times n$ matrices, and let $I^{\mathrm{sym}}_w$ and…

Let $n$ be a positive integer, let $\boldsymbol{\rm Mat}^{\mathrm{sym}}_{n\times n}$ be the space of symmetric $n\times n$ matrices, and for…

Let $S$ be either a regular local ring with residue class field $K$ or a regular positively graded $K$-algebra. Let $I$ be a perfect ideal of grade two, and let $A$ be an Hilbert–B…

Let $I$ be the ideal considered in the alternating case, with an alternating basic entry sequence as in the preceding construction, and let $\mathcal{R}_R(I)$ denote its Rees algeb…

Let $R$ be the polynomial ring of entries of the degenerated generic matrix, let $J$ be the Jacobian ideal of its determinant, let $I$ be the ideal of submaximal minors, and let…

Assume $r=m-2$, and retain the notation for the degenerated generic matrix, the Jacobian ideal $J$, the colon ideal $J:I$, the submatrices $M_j,N_j$, and the determinants…

Let $k$ be the base field, let $R$ be the polynomial ring containing the entries of the generic $m\times m$ Hankel matrix $\mathcal{H}_m$, and let $J$ be the gradient ideal of…

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…