10 problems

Let $X$ be a Segre variety, and let $I(\tau(X))$ denote the homogeneous ideal of its tangential variety. The relevant Schur-module factors are expressed using exterior powers…

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…

Griffiths–Harris conjecture. If $Y_{Z,z}$ is the disjoint union of two lines, then

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…

Projective-equivalence conjecture. The subsets $S\in X$ of cardinality $6$ such that $\pi_i(S)$ and $\pi_j(S)$ are projectively equivalent, for some fixed $i\neq j$, fill a dense o…

Let $K$ be the coefficient field, let $R=K[x_{ijk}:(i,j,k)fin [2]\times[2t]\times[2t]]$, and let $I(2t,2t)^{\{t\}}$ be the defining ideal of the $t$-secant variety of the Segre var…

Let $\sigma_2((\mathbb{P}^1)^{\times 5})$ denote the second secant variety of the product of five projective lines. Five-factor secant aG conjecture. The variety … is arithmeticall…

Let $n_1,\dots,n_d$ and $r$ be positive integers, and let … denote the Zariski closure of tensors of format $n_1\times\dots\times n_d$ and rank $r$. The aCM conjecture. All secant…

Segre–split secant conjecture. If $m\geq 1$, then $\sigma_s(\operatorname{Seg}(\mathbb P^m\times\operatorname{Split}_d(\mathbb P^n))$ is nondefective for all cases except those out…

Safety-bound conjecture. The following statements hold: