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.
After revision:  We show that under this more general notion, an arrangement of pseudocircles is embeddable into an orientable surface of genus~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.