Before revision:  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.
After revision:  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 , and asked if an analogous result held for embeddability into a compact orientable surface \Sigma_g of genus g>0.