Birational geometry of blow-ups of projective spaces along points and lines

TL;DR

This paper studies the birational geometry of blow-ups of projective spaces at points and lines, revealing infinite extremal rays in their effective cones and constructing automorphisms related to K3 surfaces, showing they are not Mori Dream Spaces.

Contribution

It constructs infinite-order pseudo-automorphisms on blow-ups of projective spaces and demonstrates the non-Mori Dream Space property in these cases, linking to K3 surface automorphisms.

Findings

Effective cone has infinitely many extremal rays.

Constructs an infinite-order pseudo-automorphism induced by a degree 13 divisor.

Blow-ups of projective spaces at points and lines are not Mori Dream Spaces.

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 $X$, induced by the complete linear system of a divisor of degree 13. 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 of Jacobian K3 Kummer surfaces. In general, 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 Mori Dream, with infinitely many extremal effective divisors. 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.