M22 Is of General Type

We recall that it is a famous result due to Harris, Mumford and Eisenbud thatMg is of general type for g ≥ 24 (cf. [HM], [H1], [EH]). On the other hand, a classical theorem due to Severi saysMg is unirational for g ≤ 10 (see [AC1] for a modern exposition). Similar results for higher g were proved in the 1980’s: Sernesi showed that M12 is unirational (cf. [Se]), then Chang and Ran settled the unirationality of M11 and M13 and gave a different proof of Sernesi’s result in genus 12 (cf. [CR1]). One also knows that M15 andM16 have negative Kodaira dimension (cf. [CR2]). The case ofM14 remained open for a long time until recently when Verra proved that M14 is also unirational (cf. [V]). The highest genus not entirely covered by [EH] is g = 23. One knows that the Kodaira dimension ofM23 is ≥ 2 (cf. [F1]).