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Speaker:
Alex Degtyarev
Affiliation:
Bilkent Univ. / MPIM
Date:
Tue, 04/08/2015 - 16:30 - 18:00
Location:
MPIM Lecture Hall
Parent event:
Summer Tropical Seminar In 1981, T. Shioda showed that the straight lines contained in a Fermat surface of degree prime to 6 generate the Picard group of the surface *over the rationals*, and he conjectured that the same lines also generate the Picard group *over the integers*. This conjecture was settled in the affirmative in 2013, when it was also extended to certain---but not all---more general Delsarte surfaces. (The dependence of the discrepancy between the two groups on the surface is the subject of an ongoing investigation.) Another generalization is the higher dimensional version of the conjecture, i.e., whether the projective spaces of middle dimension contained in a Fermat variety generate---over the integers---the group of algebraic classes in the middle homology of the variety. I will discuss an algebraic reduction of this problem (similar to the one that worked well in the case of surfaces), the numerical evidence, and the open questions that arise. This part is a joint project with Ichiro Shimada (Hiroshima University).
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