15 problems

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…

Negative curve conjecture. Every negative curve on $X_n$ is a $(-1)$-rational curve.

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,…

SHGH conjecture. Every negative curve on $Y$ is a rational $(-1)$-curve.

Let $X'$ be the blowup in characteristic $2$ described in the source, and let $C$ be its negative curve with self-intersection zero, defined by the polynomial $xi$ from the precedi…

Uncovered-region conjecture. Most of the $st$-plane that is not already covered by diamonds contains no negative curve.

Let $a,b,c$ be positive pairwise coprime integers, let $\mathbb P(a,b,c)$ be the corresponding weighted projective plane, and let $\pi\colon X(a,b,c)\to\mathbb P(a,b,c)$ be the blo…

Let $X$ be the blowup at a general point $e$ of the weighted projective plane $\mathbb{P}(9,10,13)$: … A negative curve on $X$ is an irreducible curve whose class generates the non…

Let $X\to\mathbb{P}^2$ be the blow-up of $r\geq 0$ general points of $\mathbb{P}^2$. A negative curve on $X$ is an irreducible curve with negative self-intersection, and a $(-1)$-c…

Let $f:X\to\mathbb{P}^2$ be the blow-up of the projective plane in general points $P_1,\ldots,P_s$. A negative curve is a reduced and irreducible curve with negative self-intersect…

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 $X$ be a smooth projective surface. A negative curve is a curve $C$ on $X$ with $C^2<0$. Bounded negativity conjecture. There is a nonnegative integer $b_X\geq 0$ such that for…

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…