8 problems

Let $X$ be a projective manifold of dimension $n$, and let $L$ be a strictly nef line bundle, meaning that $L\cdot C>0$ for every irreducible curve $C\subset X$. Serrano's conjectu…

Let $X$ be an $n$-dimensional nonsingular projective algebraic variety over the complex numbers, and let $\Delta=\sum_{i\in I}\Delta_i$ be a reduced simple normal crossing divisor,…

Let $X$ be a projective variety of dimension $n$, and let $D$ be a strictly nef divisor on $X$, meaning that $D\cdot C>0$ for every irreducible curve $C$ in $X$. Serrano's conjectu…

Nagata's conjecture. If $n\geq 10$ and $t>\sqrt{n}$, then $A_t$ is ample. In particular,

Log version of Serrano's conjecture. The $\mathbb Q$-Cartier divisor $K_X+\Delta+tL$ is ample for $t>2n$.

Ampleness conjecture. Any strictly nef divisor on $X$ is ample.

Let $X$ be a projective manifold of dimension $n$, and let $L$ be a strictly nef divisor, meaning that $L\cdot C>0$ for every irreducible curve $C$ on $X$. Adjoint strict-nefness c…

Debarre's ampleness conjecture. There is a positive lower bound $d\gg1$, depending on $N$ and $c$, such that whenever $d_1,\dots,d_c\geqslant d$, the cotangent bundle $\Omega_X$ is…