Hostname: page-component-848d4c4894-2pzkn Total loading time: 0 Render date: 2024-05-24T18:52:59.469Z Has data issue: false hasContentIssue false

Graded rings of rank 2 Sarkisov links

Published online by Cambridge University Press:  11 January 2016

Gavin Brown
Affiliation:
School of Mathematics, Loughborough University, Leicestershire LE11 3TU, United KingdomG.D.Brown@lboro.ac.uk
Francesco Zucconi
Affiliation:
Dipartimento di Informatica e Matematica, Università di Udine, 33100 Udine, Italyzucconi@dimi.uniud.it
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We compute a class of Sarkisov links from Fano 3-folds embedded in weighted Grassmannians using explicit methods for describing graded rings associated to a variation of geometric invariant theory (GIT) quotient.

Type
Research Article
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2010

References

[ABR02] Altınok, S., Brown, G., and Reid, M., “Fano 3-folds, K3 surfaces and graded rings” in Topology and Geometry: Commemorating SISTAG, Contemp. Math. 314, Amer. Math. Soc., Providence, 2002, 2553.Google Scholar
[AKMW02] Abramovich, D., Karu, K., Matsuki, K., and Włodarczyk, J., Torification and factorization of birational maps, J. Amer. Math. Soc., 15 (2002), 531572.Google Scholar
[Alt98] Altınok, S., Graded rings corresponding to polarised K 3 surfaces and Q-Fano 3-folds, Ph.D. dissertation, University of Warwick, Coventry, 1998.Google Scholar
[BCZ04] Brown, G., Corti, A., and Zucconi, F., “Birational geometry of 3-fold Mori fibre spaces” in The Fano Conference, Univ. Torino, Turin, 2004, 235275.Google Scholar
[BDK+] Brown, G., Davis, S., Kasprzyk, A., Kerber, M., Sisask, O., and Tawn, S., The graded ring database webpage, grdb.lboro.ac.uk, 2004-2010.Google Scholar
[BKR] Brown, G., Kerber, M., and Reid, M., K3 surfaces and Fano 3-folds, Tom and Jerry, in preparation.Google Scholar
[Bro99] Brown, G., Flips arising as quotients of hypersurfaces, Math. Proc. Cambridge Philos. Soc., 127 (1999), 1331.Google Scholar
[Bro07] Brown, G., A database of polarized K3 surfaces, Experiment. Math., 16 (2007), 720.Google Scholar
[CM04] Corti, A. and Mella, M., Birational geometry of terminal quartic 3-folds, I, Amer. J. Math., 126 (2004), 739761.Google Scholar
[Cor96] Corti, A., Del Pezzo surfaces over Dedekind schemes, Ann. of Math. (2), 144 (1996), 641683.Google Scholar
[Cor00] Corti, A., “Singularities of linear systems and 3-fold birational geometry” in Explicit Birational Geometry of 3-Folds, London Math. Soc. Lecture Note Ser. 281, Cambridge Univ. Press, Cambridge, 2000, 259312.Google Scholar
[Cox95] Cox, D. A., The homogeneous coordinate ring of a toric variety, J. Algebraic Geom., 4 (1995), 1750.Google Scholar
[CPR00] Corti, A., Pukhlikov, A., and Reid, M., “Fano 3-fold hypersurfaces” in Explicit Birational Geometry of 3-Folds, London Math. Soc. Lecture Note Ser. 281, Cambridge Univ. Press, Cambridge, 2000, 175258.Google Scholar
[CR00] Corti, A. and Reid, M., eds., Explicit Birational Geometry of 3-Folds, London Math. Soc. Lecture Note Ser. 281, Cambridge Univ. Press, Cambridge, 2000.Google Scholar
[CR02] Corti, A. and Reid, M., “Weighted Grassmannians” in Algebraic Geometry, de Gruyter, Berlin, 2002, 141163.Google Scholar
[CS05] Corti, A. and Smith, I., Conifold transitions and Mori theory, Math. Res. Lett., 12 (2005), 767778.Google Scholar
[DH98] Dolgachev, I. V. and Hu, Y., Variation of geometric invariant theory quotients, Publ. Math. Inst. Hautes Études Sci. (1998), no. 87, 556.Google Scholar
[Dol03] Dolgachev, I., Lectures on Invariant Theory, London Math. Soc. Lecture Note Ser. 296, Cambridge Univ. Press, Cambridge, 2003.Google Scholar
[FS04] Fujino, O. and Sato, H., Introduction to the toric Mori theory, Michigan Math. J., 52 (2004), 649665.Google Scholar
[Gri01] Grinenko, M. M., On fibrations into del Pezzo surfaces, Mat. Zametki, 69 (2001), 550565.Google Scholar
[HK00] Hu, Y. and Keel, S., Mori dream spaces and GIT, Michigan Math. J., 48 (2000), 331348.Google Scholar
[IF00] Iano-Fletcher, A. R., “Working with weighted complete intersections” in Explicit Birational Geometry of 3-Folds, London Math. Soc. Lecture Note Ser. 281, Cambridge Univ. Press, Cambridge, 2000, 101173.Google Scholar
[Kaw88] Kawamata, Y., Crepant blowing-up of 3-dimensional canonical singularities and its application to degenerations of surfaces, Ann. of Math. (2), 127 (1988), 93163.Google Scholar
[Kaw96] Kawamata, Y., “Divisorial contractions to 3-dimensional terminal quotient singularities” in Higher-Dimensional Complex Varieties (Trento, Italy, 1994), de Gruyter, Berlin, 1996, 241246.Google Scholar
[Kaw02] Kawakita, M., Divisorial contractions in dimension three which contract divisors to compound A1 points, Compos. Math., 133 (2002), 95116.Google Scholar
[Kaw05] Kawakita, M., Three-fold divisorial contractions to singularities of higher indices, Duke Math. J., 130 (2005), 57126.Google Scholar
[KM98] Kollár, J. and Mori, S., Birational Geometry of Algebraic Varieties, Cambridge Tracts in Math., 134, Cambridge Univ. Press, Cambridge, 1998.Google Scholar
[Kol92] Kollár, J., ed., Flips and Abundance for Algebraic Threefolds, Soc. Math. France, Paris, 1992.Google Scholar
[Kol97] Kollár, J., Polynomials with integral coefficients, equivalent to a given polynomial, Electron. Res. Announc. Math. Soc., 3 (1997), 1727.Google Scholar
[MM82] Mori, S. and Mukai, S., Classification of Fano 3-folds with B2 ≥ 2, Manuscripta Math., 36 (1981/1982), 147162.Google Scholar
[Pap06] Papadakis, S. A., Type II unprojection, J. Algebraic Geom., 15 (2006), 399414.Google Scholar
[Rei83] Reid, M., “Decomposition of toric morphisms” in Arithmetic and Geometry, Vol. II, Progr. Math., 36, Birkh¨auser Boston, Boston, 1983, 395418.Google Scholar
[Rei87] Reid, M., “Young person’s guide to canonical singularities” in Algebraic Geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), Proc. Sympos. Pure Math., 46, Amer. Math. Soc., Providence, 1987, 345414.Google Scholar
[Rei92] Reid, M., What is a flip? unpublished manuscript of Utah seminar, 1992.Google Scholar
[Rei97] Reid, M., “Chapters on algebraic surfaces” in Complex Algebraic Geometry (Park City, Utah, 1993), IAS/Park City Math. Ser., 3, Amer. Math. Soc., Providence, 1997, 3159.Google Scholar
[Rei00] Reid, M., “Graded rings and birational geometry” in Proceedings of Algebraic Geometry Symposium (Kinosaki), 2000, 172.Google Scholar
[Rei02] Reid, M., Quasi-Gorenstein unprojection, unpublished manuscript, 2002.Google Scholar
[Ryd02] Ryder, D., Elliptic and K3 fibrations birational to Fano 3-fold weighted hyper-surfaces, Ph.D. dissertation, University of Warwick, Coventry, 2002.Google Scholar
[Ryd06] Ryder, D., Classification of elliptic and K3 fibrations birational to some Q- Fano 3-folds, J. Math. Sci. Univ. Tokyo, 13 (2006), 1342.Google Scholar
[Tak06] Takagi, H., Classification of primary Q-Fano threefolds with anti-canonical Du Val K3 surfaces, I, J. Algebraic Geom., 15 (2006), 3185.Google Scholar
[Tha96] Thaddeus, M., Geometric invariant theory and flips, J. Amer. Math. Soc., 9 (1996), 691723.Google Scholar