17 problems

Let $X^n\subseteq\mathbb{P}^{n+a}$ be a smooth variety, and let $\delta_{*}$ denote the dual defect of $X$. Dual-defect lower-bound conjecture. If $\delta_{*}>0$, then … The conjec…

Let $X\subset\mathbb{P}^N$ be as in the paper's standing setup, assume that $\operatorname{def}(X)>0$, and suppose that…

Let $P$ be an integral polytope. A polytope is called a defect polytope when its associated projective toric variety has degenerate dual variety, equivalently when the invariant…

Degree conjecture. If the matroid $M(L)$ is connected, then $\nabla(L^{-1})$ is a hypersurface of degree

Dual-defectivity conjecture. The reciprocal linear space $L^{-1}$ is dual defective, meaning

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 $Z\subsetneq \mathbb{P}^N$ be an irreducible and linearly non-degenerate projective variety. If the dual variety $\check{Z}\subset \check{\mathbb{P}}^N$ is a hypersurface with…

Let $X\subseteq\mathbb P^n$ be a smooth hypersurface of degree $d\geq 3$, and let $X^\vee$ be its dual hypersurface. A Frobenius form is a defining equation of the form … for linea…

Landman's duality defect conjecture. If

Abuaf's dimension conjecture. One has

Let a weighted projective space be a projective toric variety, and call it defective when its dual variety is not a hypersurface. A cone over a weighted projective space is a weigh…

The dual-defective-to-LQEL conjecture. $X$ is a local quadratic entry locus manifold (LQELM).

Let $X=\mathbb{P}^{k_1}\times\dots\times\mathbb{P}^{k_r}$ be a Segre embedding, and let $X^*$ be its dual variety. When $X^*$ is a hypersurface, its defining equation up to scale i…

Dickenstein–Nill dual-defectivity conjecture. If

Let $X\subset\mathbb{P}^n$ be a smooth, irreducible, non-degenerate projective variety. For each $r\in\{0,\ldots,n\}$, define … where $p(q^{-1}(H^{\bot}))$ is the contact locus of…

Let $X \subset \mathbb{P}_{\mathbb{C}}^n$ be a non-degenerate, irreducible projective variety, and let $X^* \subset \mathbb{P}_{\mathbb{C}}^N$ be its projective dual. For…

A smooth irreducible non-degenerate variety $X\subset\mathbb{P}^N$ is dual defective if its dual variety has positive defect, and it has cyclic Picard group when…