9 problems

De Jong–Debarre conjecture. If $d\leq n$, then

Let $n\geq 5$, let $X\subset\mathbb{P}^n$ be a non-singular hypersurface of degree $d\geq 3$, and set … An $m$-plane means a linear projective subspace of dimension $m$ contained i…

Let $a(t) = a_0 + \cdots + a_m t^m \in \mathbf{Z}_{\geq 0}[t]$ with $a_0 \neq 0$ and $r \in \mathbf{Z}_{\geq 1}$. A $(p;a)$-tic hypersurface is a hypersurface defined by the profil…

Fix $n\geq 2$ and $m=2n$. Let $X_{m-1}\subset\mathbb{P}^{m-1}$ be the hypersurface defined by $e_{m-1}(x_1,\dotsc,x_m)=0$, and let $F_{n-1}(X_{m-1})$ denote the Fano scheme of proj…

Let $\mathrm{\textbf{F}}_{k}(\mathrm{SD}_n^r)$ denote the Fano scheme of $k$-planes contained in the space of symmetric $n\times n$ matrices of rank at most $r$. Let $\kappa(s)$ be…

One-Splitting conjecture. Assume $r\geq 2$ and $d\geq 3$. The Fano scheme $\mathbf{F}_k(X_{r,d})$ is one-split if and only if

Let $P_{n,n}^n$ be the permanental hypersurface of $n\times n$ matrices, and let $\mathbf{F}_1(P_{n,n}^n)$ be its Fano scheme of lines. Reduced local-complete-intersection conjectu…

Let $D_{n,n}^n$ and $P_{n,n}^n$ be respectively the determinantal and permanental hypersurfaces of $n\times n$ matrices, and let $\mathbf{F}_1(-)$ denote the corresponding Fano sch…

Let $K$ be an algebraically closed field of characteristic zero, let $X^{n-1}\subset \mathbb P^n$ be a hypersurface of degree $n-\lambda$, and let $B\subset \mathbb F(X)$ be an irr…