On the moduli space of pairs consisting of a cubic threefold and a hyperplane. (English) Zbl 1411.14042

A Hodge structure is said to be of \(K3\) type if it is effective, of weight \(2\), and has \(h^{2,0}=1\). Such structures arise occasionally as the middle cohomology of a projective variety and are always interesting. It is natural to look at weighted hypersurfaces of this kind. There are famously 95 families of \(K3\) surfaces that arise in this way [M. Reid, in: Journees de geometrie algebrique, Angers/France 1979, 273–310 (1980; Zbl 0451.14014)]. For fourfolds, one needs to consider degree \(d=\frac{1}{2} (w_1+\cdots+w_5)\). The first thing that the authors do here (or the last, since it is relegated to an appendix) is to classify the families of such hypersurfaces that contain a Fermat member \(\sum x_i^{n_i}=0\), which amounts to writing 2 as a sum of integer reciprocals. There are seventeen such families, but for fourteen of them the last two weights are the same, which essentially means that they were on Reid’s list (the other 81 do not contain Fermat hypersurfaces). The first of the three new cases is the family of cubic fourfolds in \(\mathbb{P}^5\). The others are of degree \(6\) with weights \(1,2,2,2,2,3\), which is the case studied here, and degree \(12\) with weights \(3, 3, 4, 4, 4, 6\).
Such a degree \(6\) hypersurface \(Z\) is a double cover of \(\mathbb{P}^4\) branched along a smooth cubic threefold \(X\) and a hyperplane \(H\), and this is the reason for studying that moduli problem. The first step, though, is to go back to \(Z\) and find a birational modification \(Y\) of \(Z\) that is a smooth cubic fourfold with an Eckardt point \(p\): by this we mean that \(Y\) contains a hyperplane section that is a cone on a cubic surface with vertex \(p\).
The cubic surface is the blow-up of six points in \(\mathbb{P}^2\) and the exceptional curves, together with the pullback of a general line, give seven classes in \(H^4(Y)\cap H^{2,2}(Y)\). The lattice they generate in the primitive cohomology is show to be isometric to \(E_6(2)\). There is a period map associated with its orthogonal complement \(T\), much as for lattice-polarised \(K3\) surfaces, and the authors establish all the expected properties, including a Torelli theorem. To understand \(\operatorname{O}(T)\) they identify \(T\) (which is isometric with \(2U\oplus 3D_4\)) with the Milnor lattice of the singularity \(\Sigma O_{16}\), the suspension of the cone on a cubic surface, and use the results of W. Ebeling [Invent. Math. 77, 85–99 (1984; Zbl 0527.14031)].
Inside the period domain (or the moduli space, which is a quotient of it by a suitable orthogonal group) the authors identify two important Heegner divisors, both arising from reflections. The vectors giving rise to them have square \(2\) and divisor \(1\), respectively square \(4\) and divisor \(2\) (the latter has 36 components). They correspond to nodal cubics and to the case where \(H\) is tangent to \(X\). By embedding \(T\) in \(II_{2,26}\) and using quasi-pullback of the Borcherds form they get a relation in the Picard group of the moduli space. They extend the period map and show, following a route described by Looijenga, that the Baily-Borel compactification agrees with the GIT moduli space for polarisation of slope \(\frac{1}{3}\).


14J15 Moduli, classification: analytic theory; relations with modular forms
14D07 Variation of Hodge structures (algebro-geometric aspects)
14D22 Fine and coarse moduli spaces
14J10 Families, moduli, classification: algebraic theory
14J17 Singularities of surfaces or higher-dimensional varieties
14J35 \(4\)-folds


Full Text: DOI arXiv


[1] Allcock, D., The moduli space of cubic threefolds, J. Algebraic Geom., 12, 2, 201-223, (2003), MR 1949641 · Zbl 1080.14531
[2] Allcock, D.; Carlson, J. A.; Toledo, D., The complex hyperbolic geometry of the moduli space of cubic surfaces, J. Algebraic Geom., 11, 4, 659-724, (2002), MR 1910264 · Zbl 1080.14532
[3] Allcock, D.; Carlson, J. A.; Toledo, D., The moduli space of cubic threefolds as a ball quotient, Mem. Amer. Math. Soc., vol. 209, (2011), no. 985, xii+70, MR 2789835
[4] Arnold, V. I.; Guseĭn-Zade, S. M.; Varchenko, A. N., Singularities of differentiable maps. vol. I: the classification of critical points, caustics and wave fronts, Monographs in Mathematics, vol. 82, (1985), Birkhäuser Boston, Inc. Boston, MA, Translated from the Russian by Ian Porteous and Mark Reynolds, MR 777682 · Zbl 0554.58001
[5] Arnold, V. I.; Guseĭn-Zade, S. M.; Varchenko, A. N., Singularities of differentiable maps. vol. II: monodromy and asymptotics of integrals, Monographs in Mathematics, vol. 83, (1988), Birkhäuser Boston, Inc. Boston, MA, Translated from the Russian by Hugh Porteous, Translation revised by the authors and James Montaldi, MR 966191
[6] Beauville, A., Le groupe de monodromie des familles universelles d’hypersurfaces et d’intersections complètes, (Complex Analysis and Algebraic Geometry, Göttingen, 1985, Lecture Notes in Math., vol. 1194, (1986), Springer Berlin), 8-18, MR 855873
[7] Beauville, A., Determinantal hypersurfaces, Michigan Math. J., 48, 39-64, (2000), Dedicated to William Fulton on the occasion of his 60th birthday, MR 1786479 · Zbl 1076.14534
[8] Beauville, A.; Donagi, R., La variété des droites d’une hypersurface cubique de dimension 4, C. R. Acad. Sci. Paris Sér. I Math., 301, 14, 703-706, (1985), MR 818549 · Zbl 0602.14041
[9] Benoist, O., Degrés d’homogénéité de l’ensemble des intersections complètes singulières, Ann. Inst. Fourier (Grenoble), 62, 3, 1189-1214, (2012), MR 3013820 · Zbl 1254.14061
[10] Bergeron, N.; Li, Z.; Millson, J.; Moeglin, C., The Noether-Lefschetz conjecture and generalizations, Invent. Math., 208, 2, 501-552, (2017), MR 3639598 · Zbl 1368.14051
[11] Borcherds, R. E., Automorphic forms on \(\operatorname{O}_{s + 2, 2}(\mathbf{R})\) and infinite products, Invent. Math., 120, 1, 161-213, (1995), MR 1323986
[12] Bourbaki, N., Lie groups and Lie algebras. chapters 4-6, Elements of Mathematics (Berlin), (2002), Springer-Verlag Berlin, Translated from the 1968 French original by A. Pressley, MR 1890629 · Zbl 0983.17001
[13] Bruinier, J. H., On the rank of Picard groups of modular varieties attached to orthogonal groups, Compos. Math., 133, 1, 49-63, (2002), MR 1918289 · Zbl 1036.11018
[14] Carlson, J. A.; Griffiths, P. A., Infinitesimal variations of Hodge structure and the global Torelli problem, (Journées de Géometrie Algébrique d’Angers, Juillet 1979/Algebraic Geometry, Angers, 1979, (1980), Sijthoff & Noordhoff, Alphen aan den Rijn—Germantown, Md.), 51-76, MR 605336
[15] Carlson, J.; Müller-Stach, S.; Peters, C., Period mappings and period domains, Cambridge Studies in Advanced Mathematics, vol. 85, (2003), Cambridge University Press Cambridge, MR 2012297 · Zbl 1030.14004
[16] Casalaina-Martin, S.; Laza, R., The moduli space of cubic threefolds via degenerations of the intermediate Jacobian, J. Reine Angew. Math., 633, 29-65, (2009), MR 2561195 · Zbl 1248.14041
[17] Casalaina-Martin, S.; Jensen, D.; Laza, R., The geometry of the ball quotient model of the moduli space of genus four curves, (Compact Moduli Spaces and Vector Bundles, Contemp. Math., vol. 564, (2012), Amer. Math. Soc. Providence, RI), 107-136, MR 2895186 · Zbl 1260.14032
[18] Cools, F.; Coppens, M., Star points on smooth hypersurfaces, J. Algebra, 323, 1, 261-286, (2010), MR 2564838 · Zbl 1193.14053
[19] Cynk, S.; van Straten, D., Infinitesimal deformations of double covers of smooth algebraic varieties, Math. Nachr., 279, 7, 716-726, (2006), MR 2226407 · Zbl 1101.14006
[20] Dolgachev, I. V., Weighted projective varieties, (Group Actions and Vector Fields, Vancouver, B.C., 1981, Lecture Notes in Math., vol. 956, (1982), Springer Berlin), 34-71, MR 704986
[21] Dolgachev, I. V., Mirror symmetry for lattice polarized K3 surfaces, J. Math. Sci., 81, 3, 2599-2630, (1996), Algebraic Geometry, vol. 4, MR 1420220 · Zbl 0890.14024
[22] Dolgachev, I. V., Classical algebraic geometry. A modern view, (2012), Cambridge University Press Cambridge, MR 2964027 · Zbl 1252.14001
[23] Dolgachev, I. V.; Hu, Y., Variation of geometric invariant theory quotients, Publ. Math. Inst. Hautes Études Sci., 87, 5-56, (1998), With an appendix by Nicolas Ressayre, MR 1659282 · Zbl 1001.14018
[24] Donagi, R.; Tu, L. W., Generic Torelli for weighted hypersurfaces, Math. Ann., 276, 3, 399-413, (1987), MR 875336 · Zbl 0588.14003
[25] Ebeling, W., An arithmetic characterisation of the symmetric monodromy groups of singularities, Invent. Math., 77, 1, 85-99, (1984), MR 751132 · Zbl 0527.14031
[26] Fulton, W., Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, vol. 2, (1998), Springer-Verlag Berlin, MR 1644323 · Zbl 0885.14002
[27] Gallardo, P.; Martinez-Garcia, J., Variations of geometric invariant quotients for pairs, a computational approach, Proc. Amer. Math. Soc., 146, 2395-2408, (2018) · Zbl 1391.14089
[28] (Griffiths, P., Topics in Transcendental Algebraic Geometry, Annals of Mathematics Studies, vol. 106, (1984), Princeton University Press Princeton, NJ), MR 756842 · Zbl 0528.00004
[29] Gritsenko, V. A.; Hulek, K.; Sankaran, G. K., The Kodaira dimension of the moduli of K3 surfaces, Invent. Math., 169, 3, 519-567, (2007), MR 2336040 · Zbl 1128.14027
[30] Gritsenko, V. A.; Hulek, K.; Sankaran, G. K., Abelianisation of orthogonal groups and the fundamental group of modular varieties, J. Algebra, 322, 2, 463-478, (2009), MR 2529099 · Zbl 1173.14027
[31] Hassett, B., Special cubic fourfolds, Compos. Math., 120, 1, 1-23, (2000), MR 1738215 · Zbl 0956.14031
[32] Kondō, S., On the Kodaira dimension of the moduli space of K3 surfaces. II, Compos. Math., 116, 2, 111-117, (1999), MR 1686793 · Zbl 0948.14007
[33] Laza, R., Deformations of singularities and variation of GIT quotients, Trans. Amer. Math. Soc., 361, 4, 2109-2161, (2009), MR 2465831 · Zbl 1174.14004
[34] Laza, R., The moduli space of cubic fourfolds, J. Algebraic Geom., 18, 3, 511-545, (2009), MR 2496456 · Zbl 1169.14026
[35] Laza, R., The moduli space of cubic fourfolds via the period map, Ann. of Math. (2), 172, 1, 673-711, (2010), MR 2680429 · Zbl 1201.14026
[36] Laza, R., Maximally algebraic potentially irrational cubic fourfolds, (2018)
[37] Laza, R.; O’Grady, K., Birational geometry of the moduli space of quartic K3 surfaces, (2016)
[38] Laza, R.; Saccà, G.; Voisin, C., A hyper-Kähler compactification of the intermediate Jacobian fibration associated to a cubic fourfold, Acta Math., 218, 1, 55-135, (2017) · Zbl 1409.14053
[39] Lehn, C.; Lehn, M.; Sorger, C.; van Straten, D., Twisted cubics on cubic fourfolds, J. Reine Angew. Math., 731, 87-128, (2017), MR 3709061 · Zbl 1376.53096
[40] Looijenga, E., New compactifications of locally symmetric varieties, (Proceedings of the 1984 Vancouver Conference in Algebraic Geometry, CMS Conf. Proc., vol. 6, (1986), Amer. Math. Soc. Providence, RI), 341-364, MR 846027
[41] Looijenga, E., Compactifications defined by arrangements. I. the ball quotient case, Duke Math. J., 118, 1, 151-187, (2003), MR 1978885 · Zbl 1052.14036
[42] Looijenga, E., Compactifications defined by arrangements. II. locally symmetric varieties of type IV, Duke Math. J., 119, 3, 527-588, (2003), MR 2003125 · Zbl 1079.14045
[43] Looijenga, E., The period map for cubic fourfolds, Invent. Math., 177, 1, 213-233, (2009), MR 2507640 · Zbl 1177.32010
[44] Looijenga, E.; Swierstra, R., The period map for cubic threefolds, Compos. Math., 143, 4, 1037-1049, (2007), MR 2339838 · Zbl 1120.14007
[45] Looijenga, E.; Swierstra, R., On period maps that are open embeddings, J. Reine Angew. Math., 617, 169-192, (2008), MR 2400994 · Zbl 1161.14007
[46] Mukai, S., An introduction to invariants and moduli, Cambridge Studies in Advanced Mathematics, vol. 81, (2003), Cambridge University Press Cambridge, Translated from the 1998 and 2000 Japanese editions by W.M. Oxbury, MR 2004218
[47] Nikulin, V. V., Integer symmetric bilinear forms and some of their geometric applications, Izv. Akad. Nauk SSSR Ser. Mat., 43, 1, 111-177, (1979), 238, MR 525944 · Zbl 0408.10011
[48] Pinkham, H. C., Deformations of algebraic varieties with \(G_m\) action, Astérisque, vol. 20, (1974), Société Mathématique de France Paris, MR 0376672 · Zbl 0304.14006
[49] Reid, M., Canonical 3-folds, (Journées de Géometrie Algébrique d’Angers, Juillet 1979/Algebraic Geometry, Angers, 1979, (1980), Sijthoff & Noordhoff, Alphen aan den Rijn—Germantown, Md.), 273-310, MR 605348
[50] Reid, M., Chapters on algebraic surfaces, (Complex Algebraic Geometry, Park City, UT, 1993, IAS/Park City Math. Ser., vol. 3, (1997), Amer. Math. Soc. Providence, RI), 3-159, MR 1442522 · Zbl 0910.14016
[51] Saitō, M.-H., Weak global Torelli theorem for certain weighted projective hypersurfaces, Duke Math. J., 53, 1, 67-111, (1986), MR 835796 · Zbl 0606.14031
[52] Shah, J., A complete moduli space for K3 surfaces of degree 2, Ann. of Math. (2), 112, 3, 485-510, (1980), MR 595204 · Zbl 0412.14016
[53] Tanimoto, S.; Várilly-Alvarado, A., Kodaira dimension of moduli of special cubic fourfolds, (2015), in press
[54] Thaddeus, M., Geometric invariant theory and flips, J. Amer. Math. Soc., 9, 3, 691-723, (1996), MR 1333296 · Zbl 0874.14042
[55] Voisin, C., Théorème de Torelli pour LES cubiques de \(\mathbf{P}^5\), Invent. Math., 86, 3, 577-601, (1986), MR 860684 · Zbl 0622.14009
[56] Voisin, C., Hodge theory and complex algebraic geometry. I, Cambridge Studies in Advanced Mathematics, vol. 76, (2002), Cambridge University Press Cambridge, Translated from the French original by Leila Schneps, MR 1967689 · Zbl 1005.14002
[57] Yonemura, T., Hypersurface simple K3 singularities, Tohoku Math. J. (2), 42, 3, 351-380, (1990), MR 1066667 · Zbl 0733.14017
[58] Yu, C.; Zheng, Z., Moduli spaces of symmetric cubic fourfolds and locally symmetric varieties, (2018)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.