Maximal almost disjoint families, determinacy, and forcing

We study the notion of J-MAD families where J is a Borel ideal on omega. We show that if J is any finite or countably iterated Fubini product of the ideal of finite sets Fin, then there are no analytic infinite J-MAD families, and assuming Projective Determinacy and Dependent Choice there are no infinite projective J-MAD families; and under the full Axiom of Determinacy + V = L(R) or under AD(+) there are no infinite J-mad families. Similar results are obtained in Solovay's model. These results apply in particular to the ideal Fin, which corresponds to the classical notion of MAD families, as well as to the ideal Fin circle times Fin. The proofs combine ideas from invariant descriptive set theory and forcing.