Abstract. Given a family of varieties X → P n over a number field, we determine conditions under which there is a Brauer-Manin obstruction to weak approximation for 100% of the fibres which are everywhere locally soluble.

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We discuss the Brauer-Manin obstruction on del Pezzo surfaces of degree 4. We outline a detailed algorithm for computing the obstruction and provide associated programs in MAGMA. This is illustrated with the computation of an example with an irreducible cubic factor in the singular locus of the defining pencil of quadrics (in contrast to previous examples, which had at worst quadratic irreducible factors). We exploit the relationship with the Tate-Shafarevich group to give new types of examples of III [2], for families of curves of genus 2 of the form y2 = f(x), where f(x) is a quintic containing an irreducible cubic factor.

The Brauer-Manin obstruction is a concept which has been very effective in finding counter-examples to the Hasse principle, that is, sets of polynomial equations which have solutions in every completion of the rational numbers but have no rational solutions. The standard way of calculating the Brauer-Manin obstruction involves listing all the p-adic solutions to some accuracy, at finitely many primes p; this is a process which may be timeconsuming. The result described in this paper shows that, at some primes, we do not need to list all p-adic solutions, but only those lying over a closed subset; and, at other primes, we need only to list solutions modulo p.

We show that, on a sufficiently general diagonal quartic surface, there is no Brauer-Manin obstruction to the existence of rational points.