Abstract
We characterize the image of the period map for cubic fourfolds with at worst simple singularities as the complement of an arrangement of hyperplanes in the period space. It follows then that the geometric invariant theory (GIT) compactification of the moduli space of cubic fourfolds is isomorphic to the Looijenga compactification associated to this arrangement. This paper builds on and is a natural continuation of our previous work on the GIT compactification of the moduli space of cubic fourfolds.
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@incollection {clemensschmid, MRKEY = {756848},
AUTHOR = {Morrison, David R.},
TITLE = {The {C}lemens-{S}chmid exact sequence and applications},
BOOKTITLE = {Topics in Transcendental Algebraic Geometry},
VENUE={{P}rinceton, {N}.{J}., 1981/1982},
SERIES = {Ann. of Math. Stud.},
NUMBER = {106},
PAGES = {101--119},
PUBLISHER = {Princeton Univ. Press},
ADDRESS = {Princeton, NJ},
YEAR = {1984},
MRCLASS = {14D05 (32G10 32G20)},
MRNUMBER = {756848},
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FJOURNAL = {L'Enseignement Mathématique. Revue Internationale. IIe Série},
VOLUME = {23},
YEAR = {1977},
NUMBER = {1-2},
PAGES = {39--110},
ISSN = {0013-8584},
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@book {GIT, MRKEY = {1304906},
AUTHOR = {Mumford, David and Fogarty, J. and Kirwan, F.},
TITLE = {Geometric Invariant Theory},
SERIES = {Ergeb. Math. Grenzgeb.},
NUMBER = {34},
EDITION = {Third},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {1994},
PAGES = {xiv+292},
ISBN = {3-540-56963-4},
MRCLASS = {14D25 (58E05 58F05)},
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TITLE={Integral symmetric bilinear forms and some of their applications},
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@book {peterssteenbrink, MRKEY = {2393625},
AUTHOR = {Peters, Chris A. M. and Steenbrink, Joseph H. M.},
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SERIES = {Ergeb. Math. Grenzgeb.},
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JOURNAL = {C. R. Acad. Sci. Paris Sér. {\rm A-B}},
VOLUME = {284},
YEAR = {1977},
NUMBER = {11},
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Math. Phys. Chem.},
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MRCLASS = {14D15 (14C30 32G11)},
MRNUMBER = {84g:14011},
MRREVIEWER = {Author's review},
URL = {http://www.numdam.org/item?id=CM_1980__42_3_315_0},
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NUMBER = {3},
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[wallsextic] C. T. C. Wall, "Sextic curves and quartic surfaces with higher singularities," , preprint , 1998.
@techreport{wallsextic,
author={Wall, C. T. C.},
TITLE={Sextic curves and quartic surfaces with higher singularities},
TYPE={ preprint},
YEAR={1998},
NOTE={31 pp},
}
Full Article