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Smoothness in pencils of hypersurfaces over finite fields

Shamil Asgarli (UBC)
Tuesday, June 7, 2022 - 2:30pm
PDL C-38
Seminar 2:30-3:20
Title:Smoothness in pencils of hypersurfaces over finite fields
Abstract: 
It is a principle in classical algebraic geometry that smooth
hypersurfaces of a given degree are "dense" in the space of all
hypersurfaces of that degree. This is literally true in the Zariski
topology when the ground field is algebraically closed. When the ground
field is a finite field, this is still true by the Lang-Weil theorem if the
"density" is appropriately defined. In this work, we study abundance of
smooth hypersurfaces from a different point of view: when do smooth
hypersurfaces of a given degree over a finite field exist across linear
families? More concretely, here is a typical question we investigate: Let q
be a fixed prime power, and let n and d be positive integers. Can we find
two hypersurfaces F and G of degree d in P^n defined over F_q such that
each F_q-member of the pencil <F, G> is a smooth hypersurface? We show that
the answer is yes when q is sufficiently large with respect to n and d.
This is joint work with Dragos Ghioca.
Event Type: 
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