Shamil Asgarli (UBC)

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.

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.