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
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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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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

American Mathematical Society