## Taiwanese Journal of Mathematics

### BIRATIONAL MAPS OF $3$-FOLDS

Jungkai Chen

#### Abstract

We show that a $3$-fold terminal flip or divisorial contractioncan be factored into a sequence of  flops, blow-downs to a smoothcurve in a smooth $3$-fold or divisorial contractions to  pointswith minimal discrepancies.

#### Article information

Source
Taiwanese J. Math., Volume 19, Number 6 (2015), 1619-1642.

Dates
First available in Project Euclid: 4 July 2017

https://projecteuclid.org/euclid.twjm/1499133731

Digital Object Identifier
doi:10.11650/tjm.19.2015.5337

Mathematical Reviews number (MathSciNet)
MR3434269

Zentralblatt MATH identifier
1357.14021

Subjects
Primary: 14E30: Minimal model program (Mori theory, extremal rays)

#### Citation

Chen, Jungkai. BIRATIONAL MAPS OF $3$-FOLDS. Taiwanese J. Math. 19 (2015), no. 6, 1619--1642. doi:10.11650/tjm.19.2015.5337. https://projecteuclid.org/euclid.twjm/1499133731

#### References

• G. Brown, Flips arising as quotients of hypersurfaces, Math. Proc. Cambridge Philos. Soc. 127(1) (1999), 13–31.
• J. A. Chen, Factoring threefold divisorial contractions to points, Ann Scoula Ecole Norm. Sup Pisa, (5) 13(2) (2014), 435–463.
• J. A. Chen and M. Chen, Explicit birational geometry of threefolds of general type, I, Ann. Sci. Éc. Norm. Supér 43 (2010), 365–394.
• ––––, Explicit birational geometry of threefolds of general type, II, J. of Diff. Geom. 86 (2010), 237–271.
• J. A. Chen and C. D. Hacon, Factoring $3$-fold flips and divisorial contractions to curves, J. reine angew. Math. 657 (2011), 173–197.
• A. Corti, Factoring birational maps of $3$-folds after Sarkisov, Journal of Algebraic Geometry 4 (1995), 223–254.
• S. D. Cutkosky, Elementary contractions of Gorenstein $3$-folds, Math. Ann. 280(3) (1988), 521–525.
• T. Hayakawa, Blowing ups of $3$-dimensional terminal singularities, Publ. Res. Inst. Math. Sci. 35(3) (1999), 515–570.
• ––––, Blowing ups of $3$-dimensional terminal singularities. II, Publ. Res. Inst. Math. Sci. 36(3) (2000), 423–456.
• ––––, Divisorial contractions to $3$-dimensional terminal singularities with discrepancy one, J. Math. Soc. Japan 57(3) (2005), 651–668.
• M. Kawakita, Divisorial contractions in dimension three which contract divisors to smooth points, Invent. Math. 145(1) (2001), 105–119.
• ––––, Divisorial contractions in dimension three which contract divisors to compound $A_1$ points, Compo. Math. 133 (2002), 95–116.
• ––––, General, elephants, of, three-fold, divisorial, contractions, J., Amer., Math., Soc. 16(2), (2002), 331–362.
• ––––, Three-fold divisorial contractions to singularities of higher indices, Duke Math. J. 130(1) (2005), 57–126.
• ––––, Supplement to classification of three-fold divisorial contractions, Nagoya Math. J. 206 (2012), 67–73.
• Y. Kawamata, Divisorial contractions to $3$-dimensional terminal quotient singularities, Higher-dimensional complex varieties (Trento, 1994), 241–246, de Gruyter, Berlin, 1996.
• J. Kollár, Flops, Nagoya Math. J. 113 (1989), 15–36.
• J. Kollár and S. Mori, Classification of three-dimensional flips, J. Amer. Math. Soc. 5(3) (1992), 533–703.
• D. Markushevich, Minimal discrepancy for a terminal cDV singularity is $1$, J. Math. Sci. Tokyo 3 (1996), 445–456.
• S. Mori, Threefolds whose canonical bundles are not numerically effective, Ann. Math. 116 (1982), 133–176.
• ––––, On $3$-dimensional terminal singularities, Nagoya Math. J. 98 (1985), 43–66.
• S. Mori and Y. Prokhorov, $3$-fold extremal contractions of type (IA), Kyoto J. Math. 51(2) (2011), 393–438.
• ––––, Threefold Extremal Contractions of Types (IC) and (IIB), Proceedings of the Edinburgh Mathematical Society 57 (2014), no. 1, 231–252.
• M. Reid, Decomposition of toric morphisms, Arithmetic and geometry, Vol. II, 395–418, Progr. Math. 36, Birkhaüser Boston, Boston, MA, 1983.
• N. Tziolas, Terminal $3$-fold divisorial contractions of a surface to a curve. I, Compositio Math. 139(3) (2003), 239–261.
• ––––, Three dimensional divisorial extremal neighborhoods, Math. Ann. 333(2) (2005), 315–354.
• ––––, $\mathbb Q$-Gorenstein deformations of nonnormal surfaces, Amer. J. Math. 131(1) (2009), 171–193.