## Osaka Journal of Mathematics

### Mori Dream Spaces extremal contractions of K3 surfaces

Alice Garbagnati

#### Abstract

We will give a criterion to assure that an extremal contraction of a K3 surface which is not a Mori Dream Space produces a singular surface which is a Mori Dream Space. We list the possible Néron--Severi groups of K3 surfaces with this property and an extra geometric condition such that the Picard number is greater than or equal to 10. We give a detailed description of two geometric examples for which the Picard number of the K3 surface is 3, i.e. the minimal possible in order to have the required property. Moreover we observe that there are infinitely many examples of K3 surfaces with the required property and Picard number equal to 3.

#### Article information

Source
Osaka J. Math. Volume 54, Number 3 (2017), 409-433.

Dates
First available in Project Euclid: 7 August 2017

https://projecteuclid.org/euclid.ojm/1502092821

Subjects
Primary: 14J28: $K3$ surfaces and Enriques surfaces
Secondary: 14E30: Minimal model program (Mori theory, extremal rays)

#### Citation

Garbagnati, Alice. Mori Dream Spaces extremal contractions of K3 surfaces. Osaka J. Math. 54 (2017), no. 3, 409--433. https://projecteuclid.org/euclid.ojm/1502092821.

#### References

• M. Artebani, J. Hausen and A. Laface: On Cox rings of K3-surfaces Compos. Math. 146 (2010), 964–998.
• F. Galluzzi, G. Lombardo and C. Peters: Automorphs of indefinite binary quadratic forms and K3 surfaces with Picard number 2, Rend. Semin. Mat. Univ. Politec. Torino, 68 (2010), 57–77.
• R. Lazarsfeld: Positivity in algebraic geometry. I, Classical setting: line bundles and linear series. Vol. 48, Springer-Verlag, Berlin, 2004.
• Y. Hu and S. Keel: Mori Dream Spaces and GIT, Michigan Math. J. 48 (2000), 331–348.
• S. Kondo: Algebraic K3 surfaces with finite automorphism groups, Nagoya Math. J. 116 (1989), 1–15.
• S.J. Kovács: The cone of curves of a K3 surface. Math. Ann. 300 (1994), 681–691.
• K. Oguiso: On automorphisms of the punctual Hilbert schemes of $K3$ surfaces, Eur. J. Math. 2 (2016), 246–261.
• B. Saint-Donat: Projective models of K3 surfaces, Amer. J. Math. 96 (1974), 602–639.