The defect of Fano $3$-folds

Author:
Anne-Sophie Kaloghiros

Journal:
J. Algebraic Geom. **20** (2011), 127-149

DOI:
https://doi.org/10.1090/S1056-3911-09-00531-1

Published electronically:
October 7, 2009

Erratum:
J. Algebraic Geom. **21** (2012), 397-399.

MathSciNet review:
2729277

Full-text PDF

Abstract |
References |
Additional Information

Abstract: This paper studies the rank of the divisor class group of terminal Gorenstein Fano $3$-folds. If $Y$ is not $\mathbb {Q}$-factorial, there is a small modification of $Y$ with a second extremal ray; Cutkosky, following Mori, gave an explicit geometric description of contractions of extremal rays on terminal Gorenstein $3$-folds. I introduce the category of weak-star Fanos, which allows one to run the Minimal Model Program (MMP) in the category of Gorenstein weak Fano $3$-folds. If $Y$ does not contain a plane, the rank of its divisor class group can be bounded by running an MMP on a weak-star Fano small modification of $Y$. These methods yield more precise bounds on the rank of $\operatorname {Cl} Y$ depending on the Weil divisors lying on $Y$. I then study in detail quartic $3$-folds that contain a plane and give a general bound on the rank of the divisor class group of quartic $3$-folds. Finally, I indicate how to bound the rank of the divisor class group of higher genus terminal Gorenstein Fano $3$-folds with Picard rank $1$ that contain a plane.

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References
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*Math. Ann.*, 272(2):257–265, 1985. MR **796252 (86j:14005)**
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*Pacific J. Math.*, 226(1):65–81, 2006. MR **2247856 (2007e:14024)**
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*Adv. in Math.*, 47(2):107–230, 1983. MR **690465 (85e:14058)**
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*Ann. of Math. (2)*, 144(3):641–683, 1996. MR **1426888 (98e:14037)**
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*Math. Ann.*, 280(3):521–525, 1988. MR **936328 (89k:14070)**
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*Manuscripta Math.*, 104(3):325–331, 2001. MR **1828878 (2002g:14056)**
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Additional Information

**Anne-Sophie Kaloghiros**

Affiliation:
Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, United Kingdom

MR Author ID:
912655

ORCID:
0000-0002-8305-8229

Email:
A.S.Kaloghiros@dpmms.cam.ac.uk

Received by editor(s):
August 5, 2008

Received by editor(s) in revised form:
February 24, 2009

Published electronically:
October 7, 2009

Additional Notes:
This work was partially supported by Trinity Hall, Cambridge