92 problems

Let $N\geq 7$, let $\mathbb S_N^+$ be the Spinor variety, let $\sigma_2(\mathbb S_N^+)$ be its secant variety of lines, and let $\Sigma_{2,N}$ be the distance--$2$ orbit. The singu…

Let $C$ be a nonhyperelliptic curve of genus $g\geq 9$. Let $E\to C$ be a vector bundle of rank $r$ and slope at most $g-2$ such that the tautological line bundle…

Let $\mathbb G(r,n)$ denote the Grassmannian of $r$-planes in projective $n$-space, and let $h$ be a positive integer. A projective variety is called $h$-defective when its $h$-sec…

Let $X$ be a smooth curve embedded by a line bundle $L$, and let $Sec^{k-1}X$ denote its $(k-1)$-st secant variety. Secant-variety degree-bound conjecture. If $\deg(L)\geq 2g+2k$,…

Let $X$ be a smooth variety, let $L$ be an ample line bundle on $X$, and fix an integer $k\geq 1$. For sufficiently large $n$, the line bundle $L^n$ defines an embedding of $X$; wr…

Let $X$ be a smooth curve and let $L$ be a very ample line bundle defining an embedding of $X$. For each integer $k$, let $Sec^kX$ denote the $k$-th secant variety. Eisenbud–Koh–St…

Work over an algebraically closed field of characteristic $0$. Let $C\subset\mathbb{P}^n$ be a linearly normal embedding of a smooth curve of genus $g$ by a line bundle $L$, and le…

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 $n=\sum_{j=1}^k n_j$ and consider a $(k+1)$-tensor space $V$ of format $(n_1+1,\dots,n_k+1,n+2)$, with $T\in V$ a general tensor. Higher-factor defective-span conjecture. If…

Let $V=\mathbb{C}^{n_1+1}\otimes\dots\otimes\mathbb{C}^{n_k+1}\otimes\mathbb{C}^{n+2}$, where $n=\sum_{j=1}^k n_j$, and let $T\in V$ be a general tensor. Let $\alpha_T$ be the coho…

Oeding's conjecture. All secant varieties of Segre products of projective spaces are arithmetically Cohen–Macaulay; equivalently, $\widehat{\sigma}_r$ is Cohen–Macaulay for every…

Minor flattening conjecture. The displayed equality holds.

Tangency flattening conjecture. The displayed equality holds.

Let $SV_{\mathbf{d},\mathbf{n}}$ be a Segre-Veronese variety, let $\sigma_{\mathbf{r}}(SV_{\mathbf{d},\mathbf{n}})$ denote its secant variety, and let a Hadamard product mean the c…

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 $V$ be a finite-dimensional vector space, let $f\in S^3V$, and let $E\subseteq V^*$ be a linear subspace. Write $f|_E$ for the restriction of $f$ to $E$, and let…

General secant non-containment conjecture. If

Secant non-containment conjecture. If

Let $V$) be a finite-dimensional vector space, and let $\Upsilon^{i,d-i}$ denote the corresponding catalecticant map; write $\operatorname{Minors}_r(\Upsilon^{i,d-i})$ for the idea…

Let $\mathcal{H}=\mathbbm{C}^{d_1}\otimes\cdots\otimes\mathbbm{C}^{d_n}$ be the Hilbert space of multipartite quantum states. The expected tensor rank is the smallest $k$ for which…

Asymptotic expected-dimension conjecture. There exists a threshold $d53fdd53f(d53d)$ such that, whenever $r>d53f$, the neurovariety $d4b1_{d53d,r}$ attains the expected dimension.

Let $\mathcal{M}^\mathrm{IG}_d$ and $\mathcal{M}^{\Gamma}_d$ denote the moment varieties associated with the inverse Gaussian and gamma distributions, respectively. For $k\geq 2$,…

Let $C\subseteq \mathbf{P}^d$ be a rational normal curve of degree $d$, and let … be its $k$-th secant variety, defined as the closure of the union of $(k+1)$-secant $k$-planes to…

Let $X\subset\mathbb{P}^{N}$ be an irreducible and non-degenerate variety with non-degenerate Gauss map. The variety $X$ is $h$-identifiable if a general fiber of its $(h+1)$-secan…