Quadric cones on the boundary of the Mori cone for very general blowups of the plane

In this paper we show the existence of cones over a 8-dimensional rational sphere at the boundary of the Mori cone of the blow-up of the plane at \(s\ge 13\) $s\ge 13$ very general points. This gives evidence for De Fernex’s strong \(\Delta \) $\mathrm{\Delta }$ -conjecture, which is known to imply Nagata’s conjecture. This also implies the existence of a multitude of good and wonderful rays as defined in Ciliberto et al. (Clay Math Proc 18:185–203, 2013).