This thesis concerns the hyperkähler eightfold constructed by Lehn, Lehn, Sorgen, and van Straten, built from twisted cubics on a cubic fourfold. We study its period, its birational properties and we describe some geometric features.

This thesis analyses the singularities of toroidal compactifications. Motivated by a result of Shepherd-Barron about the first Voronoi compactification of the moduli space of principally polarised abelian varieties, the object taken into consideration consists of the perfect cone (also known as first Voroni) compactification of arithmetic quotients of type IV domains. These are of importance in the context of algebraic geometry because they are used to construct moduli spaces of polarised K3 surfaces and are strongly related to moduli spaces of hyperkähler varieties of higher dimension. The local analysis of singularities of a toroidal compactification reduces to that of finite quotients of toric varieties. The main result of this thesis gives a description of the singularities of the perfect cone compactification of the moduli space of pseudo-polarised K3 surfaces of square-free degree.