  Given a weight two modular form f with associated p-adic Galois
representation V_f, for certain quadratic imaginary fields K one can construct
canonical classes in the Galois cohomology of V_f by taking the Kummer images
of Heegner points on the modular abelian variety attached to f. We show that
these classes can be interpolated as f varies in a Hida family, and construct
an Euler system of big Heegner points for Hida's universal ordinary deformation
of V_f. We show that the specialization of this big Euler system to any form in
the Hida family is nontrivial, extending results of Cornut and Vatsal from
modular forms of weight two and trivial character to all ordinary modular
forms, and propose a horizontal nonvanishing conjecture for these cohomology
classes. The horizontal nonvanishing conjecture implies, via the theory of
Euler systems, a conjecture of Greenberg on the generic ranks of Selmer groups
in Hida families.
