The moduli space of cubic threefolds
Author:
Daniel Allcock
Journal:
J. Algebraic Geom. 12 (2003), 201-223
DOI:
https://doi.org/10.1090/S1056-3911-02-00313-2
Published electronically:
November 18, 2002
MathSciNet review:
1949641
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Abstract |
References |
Additional Information
Abstract: We describe the moduli space of cubic hypersurfaces in $\mathbb {C}P^{4}$ in the sense of geometric invariant theory. That is, we characterize the stable and semistable hypersurfaces in terms of their singularities, and determine the equivalence classes of semistable hypersurfaces under the equivalence relation of their orbit-closures meeting.
allcock:ch4-cubic-moduli D. Allcock, J. Carlson, and D. Toledo. The complex hyperbolic geometry of the moduli space of cubic surfaces. J. Alg. Geom. 11 (2002), 659–724.
arnold-gusein-zade-varchenko:vol-1 V. I. Arnol$’$d, S. Gusein-Zade, and A. Varchenko. Singularities of Differentiable Maps, vol 1. Birkhäuser, 1985.
avritzer-miranda:stability-pencils-quadrics-in-P4 D. Avritzer and R. Miranda. Stability of pencils of quadrics in ${P}^4$. Bol. Soc. Mat. Mexicana (3), 5(2):281–300, 1999.
birkes:orbits-linear-algebraic-groups D. Birkes. Orbits of linear algebraic groups. Ann. Math., 93:459–475, 1971.
bruce-wall:classification-of-cubic-surfaces J. W. Bruce and C. T. C. Wall. On the classification of cubic surfaces. J. London Math. Soc. (2), 19:245–256, 1979.
collino:fundamental-group-of-fano-surface A. Collino. The fundamental group of the Fano surface I, II. In A. Conte, editor, Algebraic Threefolds, volume 947 of LNM, pages 209–18, 219–220. Springer, 1981.
granlund:GMP T. Granlund. The GNU Multiple Precision arithmetic library. Free Software Foundation, Cambridge, MA, 1996. Available at http://www.gnu.org.
hilbert:die-vollen-invarianetensysteme D. Hilbert. Über die vollen Invariantensysteme. Math. Ann., 36:473–534, 1890. English translation by Michael Ackerman, Hilbert’s Invariant Theory Papers, Mathsci Press, 1978.
miranda:stability-pencils-cubic-curves R. Miranda. On the stability of pencils of cubic curves. American J. Math., 102(6):1177–1202, 1980.
mumford:GIT D. Mumford, J. Fogarty, and F. Kirwan. Geometric Invariant Theory, 3rd ed. Springer-Verlag, 1994.
newstead:intro-moduli-spaces-and-orbit-problems P. E. Newstead. Introduction to moduli spaces and orbit problems. Tata Inst. Lecture Notes. Springer-Verlag, 1978.
shah:complete-moduli-space-K3-surfaces-degree-2 J. Shah. A complete moduli space for K3 surfaces of degree 2. Ann. Math., 112:485–510, 1980.
shah:degenerations-K3-surfaces-degree-4 J. Shah. Degenerations of K3 surfaces of degree 4. T.A.M.S., 263:271–308, 1981.
allcock:ch4-cubic-moduli D. Allcock, J. Carlson, and D. Toledo. The complex hyperbolic geometry of the moduli space of cubic surfaces. J. Alg. Geom. 11 (2002), 659–724.
arnold-gusein-zade-varchenko:vol-1 V. I. Arnol$’$d, S. Gusein-Zade, and A. Varchenko. Singularities of Differentiable Maps, vol 1. Birkhäuser, 1985.
avritzer-miranda:stability-pencils-quadrics-in-P4 D. Avritzer and R. Miranda. Stability of pencils of quadrics in ${P}^4$. Bol. Soc. Mat. Mexicana (3), 5(2):281–300, 1999.
birkes:orbits-linear-algebraic-groups D. Birkes. Orbits of linear algebraic groups. Ann. Math., 93:459–475, 1971.
bruce-wall:classification-of-cubic-surfaces J. W. Bruce and C. T. C. Wall. On the classification of cubic surfaces. J. London Math. Soc. (2), 19:245–256, 1979.
collino:fundamental-group-of-fano-surface A. Collino. The fundamental group of the Fano surface I, II. In A. Conte, editor, Algebraic Threefolds, volume 947 of LNM, pages 209–18, 219–220. Springer, 1981.
granlund:GMP T. Granlund. The GNU Multiple Precision arithmetic library. Free Software Foundation, Cambridge, MA, 1996. Available at http://www.gnu.org.
hilbert:die-vollen-invarianetensysteme D. Hilbert. Über die vollen Invariantensysteme. Math. Ann., 36:473–534, 1890. English translation by Michael Ackerman, Hilbert’s Invariant Theory Papers, Mathsci Press, 1978.
miranda:stability-pencils-cubic-curves R. Miranda. On the stability of pencils of cubic curves. American J. Math., 102(6):1177–1202, 1980.
mumford:GIT D. Mumford, J. Fogarty, and F. Kirwan. Geometric Invariant Theory, 3rd ed. Springer-Verlag, 1994.
newstead:intro-moduli-spaces-and-orbit-problems P. E. Newstead. Introduction to moduli spaces and orbit problems. Tata Inst. Lecture Notes. Springer-Verlag, 1978.
shah:complete-moduli-space-K3-surfaces-degree-2 J. Shah. A complete moduli space for K3 surfaces of degree 2. Ann. Math., 112:485–510, 1980.
shah:degenerations-K3-surfaces-degree-4 J. Shah. Degenerations of K3 surfaces of degree 4. T.A.M.S., 263:271–308, 1981.
Additional Information
Daniel Allcock
Affiliation:
Department of Mathematics, University of Texas at Austin, Austin, Texas 78712
MR Author ID:
620316
Email:
allcock@math.utexas.edu
Received by editor(s):
September 7, 2000
Published electronically:
November 18, 2002
Additional Notes:
Partly supported by National Science Foundation grant DMS-0070930
