When is M 0,n (ℙ1,1) a Mori dream space?

The moduli space M ¯ 0, n ( ℙ 1 , 1 ) ${{\bar{M}}_{0,n}}\left( {{\mathbb{P}}^{1}},1 \right)$ of n-pointed stable maps is a Mori dream space whenever the moduli space M ¯ 0 , n + 3 of ( n + 3 ) ${{\bar{M}}_{0,n+3}}\; \text{of} \;(n+3)$ pointed rational curves is, and M ¯ 0 , n ( ℙ 1 , 1 ) ${{\bar{M}}_{0,n}}\left( {{\mathbb{P}}^{1}},1 \right)$ is a log Fano variety for n ≤ 5.

The Hilbert scheme of a pair of linear spaces

Let $$\mathcal {H}_{a,b}^n$$ H a , b n denote the component of the Hilbert scheme whose general point parameterizes an a-plane union a b-plane meeting transversely in $${\mathbf {P}}^n$$ P n . We show that $$\mathcal {H}_{a,b}^n$$ H a , b n is smooth and isomorphic to successive blow ups of $$\mathbf {Gr}(a,n) \times \mathbf {Gr}(b,n)$$ Gr ( a , n ) × Gr ( b , n ) or $$\text {Sym}^2 \mathbf {Gr}(a,n)$$ Sym 2 Gr ( a , n ) along certain incidence correspondences. We classify the subschemes parameterized by $$\mathcal {H}_{a,b}^n$$ H a , b n and show that this component has a unique Borel fixed point. We also study the birational geometry of this component. In particular, we describe the effective and nef cones of $$\mathcal {H}_{a,b}^n$$ H a , b n and determine when the component is Fano. Moreover, we show that $$\mathcal {H}_{a,b}^n$$ H a , b n is a Mori dream space for all values of a, b, n.

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

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.

Finite Generation of the Algebra of Type A Conformal Blocks via Birational Geometry

We study the birational geometry of the moduli space of parabolic bundles over a projective line, in the framework of Mori’s program. We show that the moduli space is a Mori dream space. As a consequence, we obtain the finite generation of the algebra of type A conformal blocks. Furthermore, we compute the H-representation of the effective cone that was previously obtained by Belkale. For each big divisor, the associated birational model is described in terms of moduli space of parabolic bundles.

A Remark on Rationally Connected Varieties and Mori Dream Spaces

In this short note, we show that a construction by Ottem provides an example of a rationally connected variety that is not birationally equivalent to a Mori dream space with terminal singularities.