# Mori dream spaces and blow-ups

@article{Castravet2018MoriDS, title={Mori dream spaces and blow-ups}, author={Ana-Maria Castravet}, journal={Algebraic Geometry: Salt Lake City 2015}, year={2018} }

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…

## 16 Citations

Blow-Ups of Toric Surfaces and the Mori Dream Space Property

- MathematicsExp. Math.
- 2021

A conjecture is state which generalizes a theorem of González and Karu which states that the blow-ups of toric surfaces of Picard number one at the identity point of the torus are Mori dream spaces.

New Examples and Non-examples of Mori Dream Spaces when Blowing up Toric Surfaces

- Mathematics
- 2017

We study the question of whether the blow-ups of toric surfaces of Picard number one at the identity point of the torus are Mori Dream Spaces. For some of these toric surfaces, the question whether…

Mori Dream Spaces and blow-ups of weighted projective spaces

- MathematicsJournal of Pure and Applied Algebra
- 2019

A Lefschetz-type theorem for Mori dream spaces.

- Mathematics
- 2020

Let $Z$ be a smooth Mori dream space of dimension at least four whose extremal contractions are of fiber type of relative dimension at least two and let $X$ be a smooth ample divisor in $Z$. We show…

Remarks on nef and movable cones of hypersurfaces in Mori dream spaces

- MathematicsJournal of Pure and Applied Algebra
- 2022

Negative curves in blowups of weighted projective planes

- Mathematics
- 2020

We study the Mori dream space property for blowups at a general point of weighted projective planes or, more generally, of toric surfaces with Picard number one. Such a variety is a Mori dream space…

Examples of Non-finitely Generated Cox Rings

- MathematicsCanadian Mathematical Bulletin
- 2019

Abstract We bring examples of toric varieties blown up at a point in the torus that do not have finitely generated Cox rings. These examples are generalizations of our earlier work, where toric…

Curves generating extremal rays in blowups of weighted projective planes

- MathematicsJournal of the London Mathematical Society
- 2020

We consider blowups at a general point of weighted projective planes and, more generally, of toric surfaces with Picard number 1. We give a unifying construction of negative curves on these blowups…

Non-existence of negative curves

- Mathematics
- 2021

Let X be a projective toric surface of Picard number one blown up at a general point. We bring an infinite family of examples of such Xwhose Kleiman-Mori cone of curves is not closed: there is no…

On base loci of higher fundamental forms of toric varieties

- MathematicsJournal of Pure and Applied Algebra
- 2020

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