14 problems

Let $M$ be the coordinate ring of the Veronese, under the hypotheses of Theorem, and let $\beta_{i,i+1}$ denote the corresponding graded Betti numbers. Erman–Martinova conjecture.…

Positive geometry conjecture. The closure

Let $\mathbf P^n$ be projective $n$-space, embedded by $\mathcal{O}_{\mathbf P^n}(d)$, and let $\mathrm{K}_{p,1}(\mathbf P^n,\mathcal{O}_{\mathbf P^n}(d))$ denote the Koszul cohomo…

Let $\mathbf P^n$ be projective $n$-space, embedded by $\mathcal{O}_{\mathbf P^n}(d)$, and let $\mathrm{K}_{p,q}(\mathbf P^n,\mathcal{O}_{\mathbf P^n}(d))$ denote the corresponding…

The three-dimensional restriction conjecture. The algebra $\operatorname{A}(X_{n,d}^G)$ is quadratic if and only if, for all $i<j<k$, the algebra…

Complete intersection classification conjecture. Such a subvariety exists if and only if it is one of the following:

Maximal-loss conjecture. In each projection step, one loses the maximal number of quadrics, namely the codimension of the projected variety. Equivalently, successive general-point…

Planar Veronese secant-index conjecture. The maximal number of points in the finite reduced intersection of $r$ linearly independent degree-$d$ curves is $b_{r-1}$. The statement p…

Let $v_d^{(n)}(\mathbb{P}^n)$ be the degree-$d$ Veronese variety, let $\mathfrak{L}(X)$ be its sequence of secant indices, and let ${\mathfrak{R}\mathfrak{L}}^G(X)$ and…

Precise nonvanishing range conjecture. In the situation of that theorem, one has

Let $s,d,n$ be positive integers, let $\nu_d\mathbb{P}^n$ be the degree-$d$ Veronese re-embedding of projective space, and write $\sigma_s(\nu_d\mathbb{P}^n)$ for its $s$-secant va…

Let $X=v_d(\mathbb P^n)$ be the $d$th Veronese embedding of projective $n$-space, and define … Here $g_j(X)$ denotes the $j$th component of the gap vector of $X$, and…

Let $K$ be a field, let $n,d,ke\mathbb{Z}$ with $k,n\geq 2$ and $d\geq 2k-2$, and let $Cat(t,d-t;n)$ denote the $t$-th generic catalecticant. Write $I_k(Cat(t,d-t;n))$ for the idea…

Let $v_k(\mathbb{P}^d)$ be the $k$-uple Veronese embedding of $\mathbb{P}^d$, with $d\geq2$ and $k\geq3$, and let $\Sigma$ be its secant variety. Write $N_{p,q}$ for the indicated…