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The moduli space of cubic fourfolds. (English) Zbl 1169.14026
Geometric Invariant Theory [cf. D. Mumford, J. Fogarty and F. Kirwan, Geometric invariant theory. Springer-Verlag, 3rd edition (1993; Zbl 0797.14004)] provides a powerful tool for constructing compact moduli spaces for algebro-geometric objects. In this paper, this technique is applied to cubic hypersurfaces in projective $$5$$-space. The main result is the characterization of GIT-stable cubic fourfolds: Cubic fourfolds with only isolated simple singularities (i.e. singularities of A-D-E type) are GIT-stable, and this condition characterizes GIT-stable cubics, with some minor exceptions. Moreover, the irreducible components of the complement of the locus of cubic fourfolds with isolated simple singularities inside the compactified moduli space are explicitly described. In a sequel to this paper [R. Laza, The moduli space of cubic fourfolds via the period map. preprint, arXiv:0705.0949 (2007)], these results will be used to investigate the image of the period map for cubic fourfolds.

##### MSC:
 14J10 Families, moduli, classification: algebraic theory 14J35 $$4$$-folds 14D20 Algebraic moduli problems, moduli of vector bundles
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