Title: Arrangements of pseudocircles on surfaces,
Abstract: A pseudocircle is a simple closed curve on some surface. Arrangements of
pseudocircles were introduced by Grünbaum, who defined them as collections of
pseudocircles that pairwise intersect in exactly two points, at which they
cross. There are several variations on this notion in the literature, one of
which requires that no three pseudocircles have a point in common. Working
under this definition, Ortner proved that an arrangement of pseudocircles is
embeddable into the sphere if and only if all of its subarrangements of size at
most $4$ are embeddable into the sphere. Ortner asked if an analogous result
held for embeddability into a compact orientable surface $\Sigma_g$ of genus
$g>0$. In this paper we answer this question, under an even more general
definition of an arrangement, in which the pseudocircles in the collection are
not required to intersect each other, or that the intersections are crossings:
it suffices to have one pseudocircle that intersects all other pseudocircles in
the collection. We show that under this more general notion, an arrangement of
pseudocircles is embeddable into $\Sigma_g$ if and only if all of its
subarrangements of size at most $4g+5$ are embeddable into $\Sigma_g$, and that
this can be improved to $4g+4$ under the concept of an arrangement used by
Ortner. Our framework also allows us to generalize this result to arrangements
of other objects, such as arcs.