Bertini theorems over finite fields
Copyright policy )Bertini theorems over finite fields
Let X be a smooth quasiprojective subscheme of P^n of dimension m >= 0 over F_q. Then there exist homogeneous polynomials f over F_q 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 zeta_X(m+1)^{-1}, where zeta_X(s)=Z_X(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.
22 pages, Latex 2e
- University of California, Berkeley United States
Mathematics - Number Theory, 14J70 (Primary) 11M38, 11M41, 14G40, 14N05 (Secondary), Bertini theorem, Finite ground fields in algebraic geometry, Mathematics - Algebraic Geometry, Projective techniques in algebraic geometry, Arithmetic ground fields for surfaces or higher-dimensional varieties, Varieties over finite and local fields, FOS: Mathematics, Number Theory (math.NT), Arithmetic varieties and schemes; Arakelov theory; heights, finite field, Algebraic Geometry (math.AG), arithmetic scheme
Mathematics - Number Theory, 14J70 (Primary) 11M38, 11M41, 14G40, 14N05 (Secondary), Bertini theorem, Finite ground fields in algebraic geometry, Mathematics - Algebraic Geometry, Projective techniques in algebraic geometry, Arithmetic ground fields for surfaces or higher-dimensional varieties, Varieties over finite and local fields, FOS: Mathematics, Number Theory (math.NT), Arithmetic varieties and schemes; Arakelov theory; heights, finite field, Algebraic Geometry (math.AG), arithmetic scheme
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