2 problems

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The $(-1)$ -curves conjecture for blow-ups of the plane
Let $X$ be the blow-up of ${\mathbb P}^2$ at $r$ very general points. Let $N_1(X)$ be the space of numerical curve classes, let $Q\subset N_1(X)$ be the quadric cone of classes…
Conjectureopenblow-ups mori-cone plane-geometryarXiv posted 2 days ago
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The negative-curve conjecture for blow-ups of the plane
Negative-curve conjecture. Every integral curve $C$ on $X$ with negative self-intersection is a $(-1)$ -curve of $X$ , meaning that it is smooth and rational and satisfies $C^2=-1$ .
Conjectureopenbirational-geometry blow-ups mori-conearXiv posted 2 days ago

The (−1)(-1)(−1)-curves conjecture for blow-ups of the plane