  We investigate the geometry of the graphs of nonseparating curves for
surfaces of finite positive genus with potentially infinitely many punctures.
This graph has infinite diameter and is known to be Gromov hyperbolic by work
of the author. We study finite covers between such surfaces and show that lifts
of nonseparating curves to the nonseparating curve graph of the cover span
quasiconvex subgraphs which are infinite diameter and not coarsely equal to the
nonseparating curve graph of the cover. In the finite type case, we also
reprove a theorem of Hamenst\"{a}dt identifying the Gromov boundary with the
space of ending laminations on full genus subsurfaces. We introduce several
tools based around the analysis of bicorn curves and laminations which may be
of independent interest for studying the geometry of nonseparating curve graphs
of infinite type surfaces and their boundaries.
