# Mori Dream Spaces and Blow-Ups

## TL;DR

This paper surveys Mori Dream Spaces, focusing on when blow-ups of toric varieties are Mori Dream Spaces, translating the problem into an interpolation question for toric surfaces, and providing new examples where blow-ups are not Mori Dream Spaces.

## Contribution

It connects Mori Dream Spaces with interpolation problems on toric surfaces and presents new instances where blow-ups fail to be Mori Dream Spaces.

## Key findings

- Gonzalez-Karu theorem provides new examples of non-Mori Dream Space blow-ups.
- Interpolation problems are key to understanding Mori Dream Space properties.
- Survey of the theory of Mori Dream Spaces and their blow-ups.

## Abstract

The goal of the present article is to survey the general theory of Mori Dream Spaces, with special regards to the question: When is the blow-up of toric variety at a general point a Mori Dream Space? We translate the question for toric surfaces of Picard number one into an interpolation problem involving points in the projective plane. An instance of such an interpolation problem is the Gonzalez-Karu theorem that gives new examples of weighted projective planes whose blow-up at a general point is not a Mori Dream Space.