142 problems

Let $X$ be a variety, let $\mathcal{L}$ be a line bundle on $X$, and let $\mathbb{X}\subseteq X$ be a zero-dimensional scheme. The scheme $\mathbb{X}$ imposes independent condition…

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…

Secant-dimension conjecture. For each integer $r\geq 1$, we have

Lower-weight conjecture. The only nonzero lower weight is

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 embedded by a sufficiently positive line bundle $L$, and let $\Sigma_k$ denote its $k^{\text{th}}$ secant variety. Property $N_{k+2,p}$ means t…

Let $f$ be a degree-$d$ homogeneous polynomial, and let $\wr{f}$ and $\bwr{f}$ denote its Waring rank and border Waring rank, respectively. The Waring rank is the minimum number of…

Let $S$ be the polynomial ring in $n$ variables over the complex numbers. For positive integers $k,d,n$, let $\operatorname{rk}_k^\circ(kd,n)$ denote the $k$-rank of a general form…

Arrondo–Bernardi conjecture. The secant variety $\sigma_s(\operatorname{Split}_d(\mathbb P^n))$ is nondefective unless $d=2$ and $2\leq s\leq\frac{n}{2}$.

Garcia–Stillman–Sturmfels conjecture. The ideal of $\sigma_2(\operatorname{Seg}(\mathbb{P}V_1\times\cdots\times\mathbb{P}V_n))$ is generated by the $3\times 3$ minors of all flatte…

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…

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…

Let $\Delta$ be a full triangulation of a configuration $\mathcal{A}$ of maximal rank $d$, and let $C\subseteq\mathcal{A}$ be called $r$-partitionable if it is the disjoint union o…

GSS conjecture. The ideal $I(\sigma_2(X))$ is generated by the $3\times3$ minors of flattenings; equivalently, $\sigma_2(X)$ is the scheme-theoretic intersection of the varieties…

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…

GSS conjecture. Equality holds when $r=2$:

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 $n=\sum_{j=1}^k n_j$ 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 $T$ be a general tensor of format $(a+1,b+1,n+2)$, where $n=a+b$ and $a,b\geq2$. Three-factor defective-span conjecture. … except in the cases $(3,3,6)$, $(3,4,7)$, and…

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.