Heuristics for the Brauer-Manin obstruction for curves

We conjecture that if C is a curve of genus >1 over a number field k such that C(k) is empty, then a method of Scharaschkin (equivalent to the Brauer-Manin obstruction in the context of curves) supplies a proof that C(k) is empty. As evidence, we prove a corresponding statement in which C(F_v) is replaced by a random subset of the same size in J(F_v) for each residue field F_v at a place v of good reduction for C, and the orders of Jacobians over finite fields are assumed to be smooth (in the sense of having only small prime divisors) as often as random integers of the same size. If our conjecture holds, and if Shafarevich-Tate groups are finite, then there exists an algorithm to decide whether a curve over k has a k-point, and the Brauer-Manin obstruction to the Hasse principle for curves over the number fields is the only one.Comment: 7 page

The Grothendieck ring of varieties is not a domain

Let k be a field. Let K_0(V_k) denote the quotient of the free abelian group generated by the geometrically reduced varieties over k, modulo the relations of the form [X]=[X-Y]+[Y] whenever Y is a closed subvariety of X. Product of varieties makes K_0(V_k) into a ring. We prove that if the characteristic of k is zero, then K_0(V_k) is not a domain.Comment: 4 page

Smooth hypersurface sections containing a given subscheme over a finite field

We use the "closed point sieve" to prove a variant of a Bertini theorem over finite fields. Specifically, given a smooth quasi-projective subscheme X of P^n of dimension m over F_q, and a closed subscheme Z in P^n such that Z intersect X is smooth of dimension l, we compute the fraction of homogeneous polynomials vanishing on Z that cut out a smooth subvariety of X. The fraction is positive if m>2l.Comment: 7 pages. This paper appeared a few years ago. (I'm posting it in response to a request for the TeX file.