  By a hyperelliptic curve over Q, we mean a smooth, geometrically irreducible,
complete curve C over Q equipped with a fixed map of degree 2 to P^1 defined
over Q. Thus any hyperelliptic curve C over Q of genus g can be embedded in
weighted projective space P(1,1,g+1) via an equation of the form
  C : z^2 = f(x,y) = f_0 x^n + f_1 x^{n-1} y + ... + f_n y^n where n=2g+2, the
coefficients f_i lie in Z, and f factors into distinct linear factors over
Q-bar. Define the height H(C) of C by H(C):=max{|f_i|}, and order all
hyperelliptic curves over Q of genus g by height. Then we prove that, as g
tends to infinity:
  1) a density approaching 100% of hyperelliptic curves of genus g have no
rational points;
  2) a density approaching 100% of those hyperelliptic curves of genus g that
have points everywhere locally fail the Hasse principle; and
  3) a density approaching 100% of hyperelliptic curves of genus g have empty
Brauer set, i.e., have a Brauer-Manin obstruction to having a rational point.
  We also prove positive proportion results of this type for individual genera,
including g = 1.
