In previous work, we introduced a framework for proof nets of the multiplicative fragment of Linear Logic, where \emph{partially sequentialized} nets are allowed. In this paper we extend this result to include additives, using a definition of proof net, called \emph{J-proof net}, which is the typed version of an L-net of Faggian and Maurel. In J-proof nets, we can characterize nets with different degrees of sequentiality, by gradual insertion of sequentiality constraints (jumps). As a byproduct, we obtain a simple proof of the sequentialization theorem. Moreover, we provide a denotational semantics for J-proof nets which is an extension of the relational model, using pointed sets, and we prove that such a semantics is injective with respect to J-proof nets.