7 problems

Let $X$ be a smooth complex projective surface, and let $C$ be a reduced curve on $X$. Bounded Negativity Conjecture. For each such $X$, there exists a number $B_X\geq0$, depending…

Let $\mathbb{F}_e$ be the $e$-th Hirzebruch surface, with $C_e$ denoting its negative section. Let $P=\{p_1,\dots,p_r\}\subset \mathbb{F}_e$ be $r$ points in very general position,…

Let $\mathbb{F}_{e,r}$ denote the blow-up of the Hirzebruch surface $\mathbb{F}_e$ at $r$ very general points $p_1,\dots,p_r$. A curve is irreducible and reduced if it is an irredu…

Let $X$ be a composite of blow-ups of $\mathbb P^2$ at points $p_1,\ldots,p_n$ in very general position. A $(-1)$-rational curve is a rational curve with self-intersection $-1$. Th…

Let $C$ be a prime divisor on ${\rm Bl}_{s}(\mathbb{P}^{2})$. Negative-curve conjecture. If … then $C$ is a $(-1)$-curve. This conjecture describes the expected negative curves on…

Let $X_\mathcal K$ be the blowup associated with the Klein line configuration, let $H$ be the pullback of a line, and let $E_4$ and $E_3$ denote the exceptional divisors over the q…

Let $X=\operatorname{Bl}_r Y$ be the blow-up of a surface $Y$ at $r$ points. Let $C\subset X$ be a nonexceptional integral curve with $C^2<0$. List conjecture. There exist a positi…