A look into the Severi varieties of curves in higher codimension

We show some non-emptiness results for the Severi varieties of nodal curves with fixed geometric genus in $\textbf{P}^n, n > 2$. For $n = 3$, we also fix a vector bundle $E$ of rank 2 and look at the variety $V_\delta(E)$ parameterizing sections of $E$ whose 0-locus is nodal, with fixed geometric genus. We establish some basic facts about $V_\delta(E)$ and prove some (almost sharp) non-obstructedness results for these varieties.