The singular locus of hypersurface sections containing a closed subscheme over finite fields

2017/04/26 by Wutz, Franziska
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1704.08108

Abstract

We prove that there exist hypersurfaces that contain a given closed subscheme Z of the projective space over a finite field and intersect a given smooth scheme X off of Z smoothly, if the intersection V = Z ∩ X is smooth. Furthermore, we can give a bound on the dimension of the singular locus of the hypersurface section and prescribe finitely many local conditions on the hypersurface. This is an analogue of a Bertini theorem of Bloch over finite fields and is proved using Poonen's closed point sieve. We also show a similar theorem for the case where V is not smooth.

Related

• Bertini theorems for hypersurface sections containing a subscheme over finite fields
• Bertini Type Theorems
• Transverse linear subspaces to hypersurfaces over finite fields
• Bertini theorems for singular schemes and nearby cycles in mixed characteristic
• The probability that a complete intersection is smooth