60 problems

Let $d\geq 2$ and let $Z$ be a set of $\binom{d+2}{2}-1$ general points in $\mathbb{P}^3$. The $d$-Weddle scheme is the locus of points $P$ for which the projection $\pi_P(Z)$ lies…

Let $R$ be a commutative ring, let $n$ be a positive integer, and let $A$ be an $n\times n$ symmetric matrix over $R$. For each $S\subseteq [n]$, write $A_S$ for the principal mino…

Let $A$ be an $n\times n$ matrix over a field, and for $S\subseteq[n]$ let $A_S$ be the determinant of the principal submatrix indexed by $S$. The principal minor map sends $A$ to…

Let $d=(d_K)_{K\in{[m]\choose n}}$ be the tuple of maximal minors of a generic $m\times n$ matrix. For each $\alpha\in\mathbb{Z}$, let $F_{\alpha}$ be the $\mathcal{D}$-module asso…

Let $\tau$ be a binary tree on binary random variables, and let $I_f$ be the ideal of phylogenetic invariants of its general Markov model. Each edge of $\tau$ induces a split of th…

Radicality conjecture. The ideal above is radical. This stronger assertion includes cases involving certain unions of splitting-type strata. The source records the case $r=2$ as as…

Let $p\in \operatorname{Spec} A$ be a closed point corresponding to a vector bundle with a fixed splitting type in the versal deformation of…

Consider the versal deformation of a rank $r$ vector bundle of the form ${\mathcal O}^{r-1}\oplus {\mathcal O}(d)$ on ${\mathbb P}^1$. For each splitting type, the construction giv…

Eisenbud–Popescu conjecture. The matrix $M$ is $1$-generic if and only if the syzygy ideals of all last syzygies coincide with the ideal of the determinantal variety:

Let $W(\underline{0};\underline{d})$ be the locus of good determinantal schemes in $\mathbb{P}^{n+c}$ of codimension $c$ given by the maximal minors of a $t\times(t+c-1)$ matrix wh…

Let $a_0\le a_1\le\cdots\le a_{t+c-2}$ and $b_1\le\cdots\le b_t$ be integers. Set … and, for $i=0,1,\ldots,c-3$, set … Assume that $a_{i-\min([c/2]+1,t)}\ge b_i$ for…

Let $x=(x_1,\dots,x_n)$, let $T$ be a nonzero polynomial of order $m$, and let $P_T$ be the Padé matrix associated with the map sending $Q\in\mathbb{C}[x]_{\leq e}$ to the restrict…

Let $C$ be a non-degenerate integral curve in the weighted projective space $\mathbf{P}(1,1,m,n)$, where $m$ and $n$ are positive integers. Write $n\bmod m$ for the remainder on di…

Let $f\in \mathbb C[x_{1},\dots,x_{n}]\setminus \mathbb C$ be a non-constant polynomial, let $d\in\mathbb Z_{\geq 1}$, and let $Z^{\mathrm{top},(d)}_{(f)}(s)$ denote the twisted to…

Shellability and Hilbert-series conjecture. The abstract simplicial complex corresponding to $\tilde{I^{m,n}_{r,k}}$ is shellable, and the Hilbert series of the principal component…

Determinantal radical conjecture. These relations generate the radical of the variety defined by the $|Y|\times |Y|$ minors of $\mathbf{A}$.

Eisenbud's normality conjecture. $1$-generic subvarieties, or the slightly more general class intended by Eisenbud, should be normal.

Galois-width conjecture.

Let $q$ be odd, let $t$ be odd, and for $0\leq k\leq m$ and $\delta\in\mathbb F_q$ define … where $G=\operatorname{diag}(1,1,\dots,1,\delta,0,\dots,0)$ has rank $k$. Minimum-weight…

Let $N$ be the matrix-size parameter, let $[N+1]=\{1,\ldots,N+1\}$, and let $S=S^2\mathbb{C}^{N+1}$ be the subspace of symmetric $(N+1)\times(N+1)$ matrices. For $r\in[N+1]$, defin…

Let $Q$ be the quiver and $I$ the relations defining the representation space in the preceding construction, and let…

Let $R$ be a Gorenstein local ring and let $I$ be a perfect ideal of height $3$ admitting a minimal free resolution of Dynkin format. Such formats include the formats associated wi…

Let $X$ be the space of symmetric $n\times n$ matrices with coordinate ring $S$, and let $\overline{O}_p$ denote the determinantal variety of symmetric matrices of rank at most $p$…

For $*\in\{\emptyset,\wedge,S\}$, let $c_{sm}^{\tau_{n,k}^{* \circ}}$, $Con(\tau^*_{n,k})$, and $Ch(\tau^*_{n,k})$ denote the Chern–Schwartz–MacPherson class, conormal class, and c…

Let $\tau_{n,k}^{\circ}$, $\tau_{n,k}^{\wedge \circ}$, and $\tau_{n,k}^{S \circ}$ denote the corresponding generic determinantal, skew-symmetric determinantal, and symmetric determ…