Birational geometry of linear blow-ups of projective spaces

Abstract

In this thesis we study the birational geometry of some linear blow-ups of projective spaces. Chapter 2 studies the blow-up $X$ of $\mathbb{P}^3$ at 6 points in very general position and the 15 lines through each pair of the points. We construct an infinite-order pseudo-automorphism $\phi_X$ of $X$, induced by the complete linear system of a degree 13 divisor of $X$. The effective cone of $X$ hasinfinitely many extremal rays. Therefore $X$ is not a Mori Dream Space. There is a unique anticanonical section of $X$ which is a Jacobian K3 Kummer surface $S$ of Picard number 17. The restriction of $\phi_X$ on $S$ is one of Keum's 192 infinite-order automorphisms of Jacobian K3 Kummer surfaces. To generalize, 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 and has 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 is not a Mori Dream Space, with infinitely many extremal effective divisors. In Chapter 3, builing on the work of Castravet-Tevelev, we computed the generators of the Cox ring of the blow-up of $\mathbb{P}^n$ along $r\geq n+3$ points $p_1,\ldots,p{r}$ on a rational normal curve $C\subset \mathbb{P}^n$ and along lines $\overline{p_1p_2},\overline{p_1p_3},\ldots,\overline{p_1p{r}}$. Using this result, we get a set of generators of the Cox ring of the blow-up of $\mathbb{P}^n$ at 1 line and n+1 points in general position. We conjectured that the Cox ring of the blow-up of $\mathbb{P}^n$ along $k$ lines and $n+3-2k$ points in general position is finitely generated and we checked this for small $n$. We proved that if this conjecture holds, then the blow-up of $\mathbb{P}^{n}$ along linear subvarieties $L_1$, $L_2,\ldots, L_m$ of codimension at least 2 in general position such that $\sum{i=1}^m (\dim L_i +1)=n+3$ is a Mori Dream Space. By Mukai's geometrization theorem, the latter statement implies an affirmative answer to Mukai's question of whether the algebra of invariants of $\mathbb{G}_a^2$ under the generalized Nagata actions is finitely generated.In the end, we give a proof of the Geometrization theorem following Mukai's idea.--Author's abstract