Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911

   
 

 

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.


References [Enhancements On Off] (What's this?)


Additional Information

Anne-Sophie Kaloghiros
Affiliation: Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, United Kingdom
Email: A.S.Kaloghiros@dpmms.cam.ac.uk

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

Journal of Algebraic Geometry
The Journal of Algebraic Geometry
is sponsored by the Department of Mathematical Sciences
of Tsinghua University
and is distributed by the American Mathematical Society
for University Press, Inc.
Online ISSN 1534-7486; Print ISSN 1056-3911
© 2017 University Press, Inc.
Comments: jag-query@ams.org
AMS Website