Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911

   
 
 

 

Mirror quintics, discrete symmetries and Shioda maps


Authors: Gilberto Bini, Bert van Geemen and Tyler L. Kelly
Journal: J. Algebraic Geom. 21 (2012), 401-412
DOI: https://doi.org/10.1090/S1056-3911-2011-00544-4
Published electronically: May 11, 2011
MathSciNet review: 2914798
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Abstract | References | Additional Information

Abstract: In a recent paper, Doran, Greene and Judes considered one parameter families of quintic threefolds with finite symmetry groups. A surprising result was that each of these six families has the same Picard-Fuchs equation associated to the holomorphic $ 3$-form. In this paper we give an easy argument, involving the family of Mirror Quintics, which implies this result. Using a construction due to Shioda, we also relate certain quotients of these one-parameter families to the family of Mirror Quintics. Our constructions generalize to degree $ n$ Calabi-Yau varieties in $ (n-1)$-dimensional projective space.


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Additional Information

Gilberto Bini
Affiliation: Dipartimento di Matematica, Università di Milano, Via Saldini 50, I-20133 Milano, Italia
Email: gilberto.bini@unimi.it

Bert van Geemen
Affiliation: Dipartimento di Matematica, Università di Milano, Via Saldini 50, I-20133 Milano, Italia
Email: lambertus.vangeemen@unimi.it

Tyler L. Kelly
Affiliation: Department of Mathematics, University of Pennsylvania, David Rittenhouse Lab., 209 South 33rd Street, Philadelphia, Pennsylvania 19104-6395
Email: tykelly@math.upenn.edu

DOI: https://doi.org/10.1090/S1056-3911-2011-00544-4
Received by editor(s): May 19, 2009
Received by editor(s) in revised form: July 17, 2009
Published electronically: May 11, 2011

American Mathematical Society