Vittarioideae worthy of subgeneric rank, while "Rheopteris" had not yet been discovered. "Haplopteris" he explicitly synonymized with "Vittaria" in 1914. Carl Christensen used the name "Vittarioideae" in Verdoorn's "Manual of Pteridology" in 1938, but did not include a description, leaving it nomenclaturally invalid. Ren-Chang Ching raised Vittariaceae to the rank of a family in 1940. The first well-sampled molecular phylogenetic study of the vittarioids was based on the chloroplast gene rbcL. In this study, it was found that the type species of "Monogramma" is embedded in "Haplopteris"; the segregation of "Vaginularia" from "Monogramma" was also supported, as members of "Vaginularia" formed
Surface of general type describe these schemes explicitly, and there are few pairs of Chern numbers for which this has been done (except when the scheme is empty). There are some indications that these schemes are in general too complicated to write down explicitly: the known upper bounds for the number of components are very large, some components can be non-reduced everywhere, components may have many different dimensions, and the few pieces that have been studied explicitly tend to look rather complicated. The study of which pairs of Chern numbers can occur for a surface of general type is known as "" and there