Before revision:  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 areembeddable 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\emph{ .
After revision:  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+ 4 are. Moreover, this bound is tight. We actually have similar results for a much general notion of arrangement, which we call an\emph{ .