  By a theorem of Wahl, the canonically embedded curves which are hyperplane
section of K3 surfaces are distinguished by the non-surjectivity of their Wahl
map. In this paper we address the problem of distinguishing hyperplane sections
of abelian surfaces. The somewhat surprising result is that the Wahl map of
such curves is (tendentially) surjective, but their second Wahl map has corank
at least 2 (in fact a more precise result is proved).
