On the birational geometry of spaces of complete forms II: skew-forms

Moduli spaces of complete skew-forms are compactifications of spaces of skew-symmetric linear maps of maximal rank on a fixed vector space, where the added boundary divisor is simple normal crossing. In this paper we compute their effective, nef and movable cones, the generators of their Cox rings, and for those spaces having Picard rank two we give an explicit presentation of the Cox ring. Furthermore, we give a complete description of both the Mori chamber and stable base locus decompositions of the effective cone of some spaces of complete skew-forms having Picard rank at most four.