32 problems

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 $X$ 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 $I_{n,k}$ be the determinantal ideal and let $preceq$ be the term order defined in the paper. An initial ideal is called spherical when it satisfies the stated Stanley–Reisner…

Let $R=\mathbb{K}[a_1,a_2,a_3,a_4,a_5,b_1,b_2,b_3,b_4,b_5]$, and let … … and … For a matrix $M$, write $I_t(M)$ for the ideal generated by its $t\times t$ minors. Eisenbud's conjec…

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 $i,j,k$.…

Let $X$ and $Y$ be generic $n$ by $n$ matrices, let $J$ be the ideal generated by the off-diagonal entries of $XY-YX$, and consider determinants of matrices whose columns are the d…

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 $R$ be a ring in which $2$ is invertible, and let $\varphi$ be a symmetric $n\times n$ matrix of rank $r$. For each $i\leq r$, let $I_i$ be the ideal generated by the…

Let $X$ be a projective variety, let $L$ define an embedding $X\subseteq\mathbb{P}^r$, and let $\Sigma_k(X,L)$ denote the $k$-th secant variety. Suppose that $L=L_1\otimes L_2$, wh…

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 $R=K[X]$ and let $\frac{}{}\prec$ be the monomial order defined by Soll and Welker. Let $I_t$ be the determinantal ideal and let $\Delta$ be the simplicial complex defined by t…

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 $\operatorname{RL}$ be the property defined by … Here, being $(4,n-2,n,1)$-generic is the genericity property for the relevant determinantal ideals. Genericity conjecture. For…

Let $J_\Delta$ be an $r$-determinantal facet ideal generated by minors indexed by the facets of a simplicial complex $\Delta$, where $r=n$ for a generic $n\times m$ matrix. The ide…

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 $M_j$ and $N_j$ be the submatrices defined in the preceding notation, let $I_j(M_j)$ and $I_j(N_j)$ denote their ideals of $j$-minors, and suppose that $r=m-2$. For…

Let $R$ be the polynomial ring of entries of the degenerated generic matrix, let $J$ be the ideal generated by the partial derivatives of its determinant, and let $m$ and $r$ be as…

Let $R$ be the polynomial ring of entries of the degenerated generic matrix, let $J$ be the ideal generated by the partial derivatives of its determinant, and let $r$ and $m$ be as…