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… 
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References

SHOWING 1-10 OF 58 REFERENCES
On images of Mori dream spaces
The purpose of this paper is to study the geometry of images of morphisms from Mori dream spaces. First we prove that a variety which admits a surjective morphism from a Mori dream space is again a
A Lefschetz hyperplane theorem for Mori dream spaces
Let X be a smooth Mori dream space of dimension ≥ 4. We show that, if X satisfies a suitable GIT condition which we call small unstable locus, then every smooth ample divisor Y of X is also a Mori
Mori dream spaces of Calabi-Yau type and the log canonicity of the Cox rings
We prove that a Mori dream space over a field of characteristic zero is of Calabi-Yau type if and only if its Cox ring has at worst log canonical singularities. By slightly modifying the arguments we
On blowing up the weighted projective plane
We investigate the blow-up of a weighted projective plane at a general point. We provide criteria and algorithms for testing if the result is a Mori dream surface and we compute the Cox ring in
Birational geometry of hypersurfaces in products of projective spaces
We study the birational properties of hypersurfaces in products of projective spaces. In the case of hypersurfaces in $${\mathbb {P}}^m \times {\mathbb {P}}^n$$Pm×Pn, we describe their nef, movable
Hilbert's 14th problem and Cox rings
Our main result is the description of generators of the total coordinate ring of the blow-up of $\mathbb{P}^n$ in any number of points that lie on a rational normal curve. As a corollary we show that
Cox rings and pseudoeffective cones of projectivized toric vector bundles
We study projectivizations of a special class of toric vector bundles that includes cotangent bundles, whose associated Klyachko filtrations are particularly simple. For these projectivized bundles,
Explicit log Fano structures on blow‐ups of projective spaces
In this paper, we determine which blow‐ups X of Pn at general points are log Fano, that is, when there exists an effective Q ‐divisor Δ such that −(KX+Δ) is ample and the pair (X,Δ) is klt. For these
Hilbert’s 14th problem over finite fields and a conjecture on the cone of curves
  • B. Totaro
  • Mathematics
    Compositio Mathematica
  • 2008
Abstract We give the first examples over finite fields of rings of invariants that are not finitely generated. (The examples work over arbitrary fields, for example the rational numbers.) The group
...
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