5 problems

Let $X$ be a surface and let $A$ be an ample line bundle on $X$. For an integer $p\geq 0$ and an integer $\ell$, set $L=K_X+\ell A$. Mukai's conjecture. The line bundle $L$ satisfi…

Let $X$ be a smooth projective variety of dimension $n$, let $A$ be an ample divisor, and let $p\geq 0$. For an integer $r$, consider the adjoint line bundle…

Let $\mathcal{F}$ be a regular foliation of rank $p$ on a smooth projective variety $X$, and let $L$ be an ample line bundle on $X$. Foliated Fujita conjectures. The line bundle ……

Let $X$ be a smooth projective variety and let $L$ be a line bundle on $X$. For every proper subvariety $Z\varsubsetneq X$, write $(L^{\dim Z}.Z)$ for the intersection number of…

Work over an algebraically closed field $k$ of characteristic zero. Let $X$ be a smooth projective variety of dimension $n$, let $x\in X$ be a point, and let $L$ be an ample line b…