Mori Dream Spaces

Mori Dream Spaces are a class of projective varieties that have many nice geometric and algebraic properties. Examples of Mori Dream Spaces include flag varieties, projective toric varieties, spherical varieties and smooth Fano varieties. A smooth projective variety with finitely generated and free Picard group is a Mori Dream Space precisely when its Cox ring, also called its homogeneous coordinate ring, is finitely generated as an algebra over the base field. This fact provides a strong link between the birational geometry of a Mori Dream Space and the birational geometry of an ambient toric variety. The goal is to spend the semester learning about aspects that interest us.

Mori Dream Spaces were introduced in the seminal paper of Hu and Keel:

Yi Hu and Sean Keel, Mori dream spaces and GIT. Dedicated to William Fulton on the occasion of his 60th birthday. Michigan Math. J. 48 (2000), 331-348.

For more recent references, see:

Ivan Arzhantsev, Ulrich Derenthal, Jurgen Hausen, Antonio Laface, Cox rings. Cambridge Studies in Advanced Mathematics, 144. Cambridge University Press, Cambridge, 2015. viii+530 pp.
Jurgen Hausen, Three lectures on Cox rings. Torsors, etale homotopy and applications to rational points, 3-60, London Math. Soc. Lecture Note Ser., 405, Cambridge Univ. Press, Cambridge, 2013.
Cinzia Casagrande, Mori Dream Spaces and Fano varieties, expanded notes from a minicourse given at `Geometrie Algebrique et Geometrie Complexe', CIRM, Marseille, 12-16 March 2012.
Antonio Laface, Mauricio Velasco, A survey on Cox rings. Geom. Dedicata 139 (2009), 269-287.
David Cox, The homogeneous coordinate ring of a toric variety. J. Algebr. Geom., 4 (1995), no. 1, 17-50.

The timetable for the talks (plus lecture notes when available) is given below:

5 October 2016 Room 8W 2.20 Alastair Craw Organisational meeting and overview Lecture notes

12 October 2016 Room 8W 2.20 Alastair Craw Introduction to toric varieties Lecture notes

19 October 2016 Room 8W 2.20 Alastair Craw The Cox ring of a toric variety and quasi-torus actions Lecture notes

26 October 2016 Wolfson room Alastair Craw Good categorical quotients Lecture notes

2 November 2016 Wolfson room Alastair Craw Good geometric quotients and the secondary fan Lecture notes

9 November 2016 Wolfson room Alastair Craw The nef, movable and effective cones Lecture notes

16 November 2016 Room 8W 2.20 James Green Mori Dream Spaces and birational geometry Lecture notes

23 November 2016 Wolfson room Fran Burstall The Cox ring of a Flag variety Lecture notes
Kostant-Plücker relations

30 November 2016 Room 8W 2.20 Alastair Craw Hu-Keel's result on Mori Dream Spaces and Cox rings Lecture notes

14 December 2016 Room 6W 1.2 Claudio Onorati Which K3 surfaces are Mori Dream spaces? Lecture notes

5 October 2016	Room 8W 2.20	Alastair Craw	Organisational meeting and overview	Lecture notes
12 October 2016	Room 8W 2.20	Alastair Craw	Introduction to toric varieties	Lecture notes
19 October 2016	Room 8W 2.20	Alastair Craw	The Cox ring of a toric variety and quasi-torus actions	Lecture notes
26 October 2016	Wolfson room	Alastair Craw	Good categorical quotients	Lecture notes
2 November 2016	Wolfson room	Alastair Craw	Good geometric quotients and the secondary fan	Lecture notes
9 November 2016	Wolfson room	Alastair Craw	The nef, movable and effective cones	Lecture notes
16 November 2016	Room 8W 2.20	James Green	Mori Dream Spaces and birational geometry	Lecture notes
23 November 2016	Wolfson room	Fran Burstall	The Cox ring of a Flag variety	Lecture notes Kostant-Plücker relations
30 November 2016	Room 8W 2.20	Alastair Craw	Hu-Keel's result on Mori Dream Spaces and Cox rings	Lecture notes
14 December 2016	Room 6W 1.2	Claudio Onorati	Which K3 surfaces are Mori Dream spaces?	Lecture notes