8 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 $f:X\to Y$ be a finite, degree $d$, morphism from a reduced pure $n$-dimensional projective variety $X$ onto an $n$-dimensional projective manifold $Y$, with $X$ having at wors…

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…

Kovács' conjecture. The variety $X$ is either projective space or the $p$-dimensional quadric hypersurface.

Extension conjecture. The same conclusion should hold without the hypothesis that the Kähler forms $\omega_i$ are rational: $s_{\lambda}(\omega_1,\ldots,\omega_e)$ should have the…

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 ……

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

Algebraic-dimension conjecture. One has