  The aim of this paper is to prove the two-variable anticyclotomic Iwasawa
main conjecture for Hida families and a definite version of the horizontal
non-vanishing conjecture, which are formulated in Longo-Vigni. Our approach is
based on the two-variable anticyclotomic control theorem for Selmer groups for
Hida families and the relation between the two-variable anticyclotomic
$L$-function for Hida families built out of $p$-adic families of Gross points
on definite Shimura curves studied in Castella-Longo and Castella-Kim-Longo and
the self-dual twist of the specialisation to the anticyclotomic line of the
three-variable $p$-adic $L$-function of Skinner-Urban.
