26 problems

Conjecture on the components of $\mathcal{SC}_r$. For all $r\in[N+1]$:

Let $AD^{2s,2n}_{\binom{2n}{2}-m}$ denote the skew-symmetric determinantal locus in even dimension with the indicated rank parameter. For any fixed $m>0$, the even skew-symmetric d…

Let $D^{s,n}_{n^2-m}$ denote the determinantal locus of general $n\times n$ matrices of the indicated rank parameter. For any fixed $m>0$, the general determinantal-locus conjectur…

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 $W(\underline{0};\underline{d})$ 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 $C$ 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…

Galois-width conjecture.

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

The partition characterization of relaxed SLMFs. Suppose $\#\Omega=r(m+n-r)$ and every vertex of $G_\Omega$ has degree at least $r+1$. Then $\Omega$ is a relaxed $(r,r,m)$-SLMF if…

Let $M$ be the matrix whose determinantal rank loci are denoted by $V_k$, and let $N_k$ denote the locus of cubic surfaces with $k$ singularities. For each $k$, consider the minors…

Let $V_k$ be the locus of cubic surfaces for which the matrix $M$ has rank at most $20-k$, and let $N_2$ denote the locus of binodal cubic surfaces. The $19\times 19$ minors of $M$…

Let $ell \leq m$ be positive integers and let $t$ be an integer satisfying $1<t<ell$. For the determinantal code, write $hat{w}_r(t;ell,m)$ for the weight indexed by $r$. Weight-ord…

Let $X\subset \mathbb{P}^n$ be a general linear standard determinantal scheme of codimension $c\ge 2$ defined by the maximal minors of a $t\times (t+c-1)$ matrix with linear entrie…

Radicality conjecture.

Let $V$ be a parameter space of smooth projective varieties, let $Y/V$ be the corresponding family, and let the Kodaira-Spencer map be the map referred to as. Let…

Let $\Bbbk$ be an algebraically closed field of characteristic zero, let $S$ be a polynomial ring over $\Bbbk$, and let $I\subseteq S$ be a prime ideal containing no linear form. S…

The strong maximal rank conjecture. For a general $[C]\in\mathcal{M}_g$, the locus $\mathfrak{Quad}_{g,d}^r(C)$ has the expected determinantal dimension

Let $C$ be a curve of genus $g$, and let $L$ be a line bundle that factors as … For an integer $k$, write $\operatorname{Sec}^k(C)$ for the $k$-th secant variety of the image of…

Tropical-basis conjecture. The $(r+1)\times(r+1)$ minors of a $d\times n$ matrix of variables are a tropical basis if and only if $r\leq 2$ or $r\geq \operatorname{min}(d,n)-2$.

The dimension conjecture for good determinantal schemes. Under these assumptions,