The 2.0 release of the Universal Dependency treebanks demonstrates the effectiveness of the UD scheme to cope with very diverse languages. The next step would be to get more of syntactic analysis , and the " enhanced dependencies " sketched in the UD 2.0 guidelines is a promising attempt in that direction. In this work we propose to go further and enrich the enhanced dependency scheme along two axis: extending the cases of recovered arguments of non-finite verbs, and neutralizing syntactic alternations. Doing so leads to both richer and more uniform structures, while remaining at the syntactic level, and thus rather neutral with respect to the type of semantic representation that can be further obtained. We implemented this proposal in two UD treebanks of French, using deterministic graph-rewriting rules. Evaluation on a 200 sentence gold standard shows that deep syntactic graphs can be obtained from surface syntax annotations with a high accuracy. Among all arguments of verbs in the gold standard, 13.91% are impacted by syntactic alternation normalization, and 18.93% are additional deep edges.