  We study the Mori Dream Space (MDS) property for blowups of weighted
projective planes at a general point and, more generally, blowups of toric
surfaces defined by a rational plane triangle. The birational geometry of these
varieties is largely governed by the existence of a negative curve in them,
different from the exceptional curve of the blowup.
  We consider a parameter space of all rational triangles, and within this
space we study how the negative curves and the MDS property vary. One goal of
the article is to catalogue all known negative curves and show their location
in the parameter space. In addition to the previously known examples we
construct two new families of negative curves. One of them is, to our
knowledge, the first infinite family of special negative curves.
  The second goal of the article is to show that the knowledge of negative
curves in the parameter space often determines the MDS property. We show that
in many cases this is the only underlying mechanism responsible for the MDS
property.
