Mori dreamness of blowups of weighted projective planes

Abstract

We consider the blowup X(a, b, c) of a weighted projective space $${\mathbb {P}}(a,b,c)$$ P ( a , b , c ) at a general nonsingular point. We give a sufficient condition for a curve to be a negative curve on X(a, b, c) in terms of $$\chi ({\mathcal {O}}_X(C))$$ χ ( O X ( C ) ) . This can be applied to find the effective cone of X(a, b, c) and can serve as a starting point to prove the Mori dreamness of blowups of many weighted projective planes. We confirm the Mori dreamness of some X(a, b, c) as examples of our method.