6 problems

Fano ampleness conjecture. If $Y$ is Fano with $\rho(Y)=1$, then ${\cal E}$ is ample, and is spanned in most cases. This is presented as the guideline for the paper’s section on co…

Let $M$ be a Fano manifold of Picard number one and let $\mathcal{E}$ be an ample vector bundle of rank $r$ on $M$. Let $L$, the extremal ray $R_1$, and a minimal rational curve…

Let $X$ be a smooth projective variety, let $\mathscr{E}$ be an ample vector bundle on $X$, and let $T_X$ denote the tangent bundle of $X$. Andreatta–Wiśniewski's conjecture. If ……

Let $X$ be a variety and let $Y\subset X$ be a smooth subvariety. Its normal bundle $N_Y$ is a vector bundle, and it is ample when $\mathscr O(1)$ is ample on $\mathbb P(N_Y)$. Har…

Interior-of-the-Mori-cone conjecture. The class $[C]$ lies in the interior of

Algebraic-dimension conjecture. One has