Bertini theorems over finite fields
Abstract
Let X be a smooth quasiprojective subscheme of Pn of dimension m >= 0 over Fq. Then there exist homogeneous polynomials f over Fq for which the intersection of X and the hypersurface f=0 is smooth. In fact, the set of such f has a positive density, equal to zetaX(m+1)-1, where zetaX(s)=ZX(q-s) is the zeta function of X. An analogue for regular quasiprojective schemes over Z is proved, assuming the abc conjecture and another conjecture.
0
Turn this paper into a full lesson
ArcXiv compiles a staged curriculum from this paper: 8-12 lessons across beginner → advanced, synthesised section guides, visuals, flashcards, a quiz, exercises, and on-demand deep dives per section. Grounded in the abstract, never invented.