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The Moduli Space of Cubic Threefolds Daniel Allcock

Summary: The Moduli Space of Cubic Threefolds
Daniel Allcock
February 6, 2001
We describe the moduli space of cubic hypersurfaces in C P 4 in the sense of geometric invariant
theory. That is, we characterize the stable and semistable hypersurfaces in terms of their singu-
larities, and determine the equivalence classes of semistable hypersurfaces under the equivalence
relation of their orbit-closures meeting.
x1. Introduction
Mumford's geometric invariant theory provides a construction of complete moduli spaces of families
of varieties. In this paper we apply his methods to obtain a concrete description of the moduli
space of cubic hypersurfaces in C P 4 . More precisely, we work out which cubic threefolds are
stable, which are semistable, which of the semistable orbits are minimal, and which semistable
threefolds degenerate to which minimal orbits, all in terms of the singularities of the threefolds.
Many authors have treated stability and semistability in other settings. Hilbert treated point-sets
in the projective line, plane curves of degree  6 and cubic surfaces [8]. Shah provided much more
detailed information about sextic plane curves [12] and analyzed quartic surfaces [13]. Mumford
and Tate treated point-sets in projective spaces of arbitrary dimension [10, chap. 3], Miranda
treated pencils of cubics in P 2 [9], and Avritzer and Miranda have recently treated pencils of
quadrics in P 4 [3]. For further references, see [10]. After writing this paper we learned that Collino


Source: Allcock, Daniel - Department of Mathematics, University of Texas at Austin


Collections: Mathematics