Birational Geometry of Blow-ups of Projective Spaces Along Points and Lines

Abstract

Abstract Consider the blow-up $X$ of ${\mathbb{P}}^3$ at $6$ points in very general position and the $15$ lines through the $6$ points. We construct an infinite-order pseudo-automorphism $\phi _X$ on $X$. The effective cone of $X$ has infinitely many extremal rays and, hence, $X$ is not a Mori Dream Space. The threefold $X$ has a unique anticanonical section, which is a Jacobian K3 Kummer surface $S$ of Picard number 17. The restriction of $\phi _X$ on $S$ realizes one of Keum’s 192 infinite-order automorphisms. We show the blow-up of ${\mathbb{P}}^n$ ($n\geq 3$) at $(n+3)$ very general points and certain $9$ lines through them is not a Mori Dream Space. As an application, for $n\geq 7$, the blow-up of $\overline{M}_{0,n}$ at a very general point has infinitely many extremal effective divisors.