The moduli space of cubic threefolds as a ball quotient

Abstract

The moduli space of cubic threefolds in CP 4 , with some minor birational modifications, is the Baily-Borel compactification of the quotient of the complex 10-ball by a discrete group. We describe both the birational modifications and the discrete group explicitly.In the second-to-last step we used the relation σ −2 +σ 2 = σ+σ −1 = −1. That ·|· is a C-valued Hermitian form now follows from α|β = β| α ; to prove this it suffices to check that the imaginary part of (2.2) changes sign when α and β are exchanged, i.e.,

Braid groups and Hodge theory

“…[DK], [Ko], [ACT1], [ACT2], [Al2], [Lo2]. The action of the mappingclass group of a closed surface on the homology of its finite abelian covers is studied in [Lo1].…”

The moduli space of cubic fourfolds via the period map

“…This builds on earlier results of Voisin [48] and Hassett [21], and was inspired by the recent work of Allcock-Carlson-Toledo [1], [2] and Looijenga-Swierstra [31] on the moduli space of cubic threefolds.…”

“…Since the case is quite similar (see 4.1.2, especially Remark 4.7), there is no difficulty in adapting the arguments to the case . In particular we note that the semi-stable reduction in this case is essentially contained in Allcock-Carlson-Toledo [2,5].…”

“…In our analysis, the Type IV fourfolds play the same role as the triple conic for plane sextics (see [43]) or the chordal cubic for cubic threefolds (see [2], [31]). …”

“…As hinted in the previous paragraph, the essential fact that makes the proofs of Section 3 work is that the singularities of Type II and III fourfolds are double suspensions of special surface singularities, the so-called insignificant limit singularities of Mumford and Shah. In Section 4, we analyze the degenerations to Type IV fourfolds, completing the second step of our proof. A partial analysis of the degenerations to Type IV fourfolds was done by Hassett [21, 4.4] and Allcock-Carlson-Toledo [2,5]. We complete the analysis by proceeding as follows.…”