Moduli of cubic surfaces and their anticanonical divisors. (English) Zbl 1441.14153

Summary: We consider the moduli space of log smooth pairs formed by a cubic surface and an anticanonical divisor. We describe all compactifications of this moduli space which are constructed using geometric invariant theory and the anticanonical polarization. The construction depends on a weight on the divisor. For smaller weights the stable pairs consist of mildly singular surfaces and very singular divisors. Conversely, a larger weight allows more singular surfaces, but it restricts the singularities on the divisor. The one-dimensional space of stability conditions decomposes in a wall-chamber structure. We describe all the walls and relate their value to the worst singularities appearing in the compactification locus. Furthermore, we give a complete characterization of stable and polystable pairs in terms of their singularities for each of the compactifications considered.


14L24 Geometric invariant theory
14J10 Families, moduli, classification: algebraic theory
14J26 Rational and ruled surfaces
14J45 Fano varieties
14Q10 Computational aspects of algebraic surfaces


Full Text: DOI arXiv


[1] Allcock, D.; Carlson, JA; Toledo, D., The complex hyperbolic geometry of the moduli space of cubic surfaces, J. Algebr. Geom., 11, 659-724, (2002) · Zbl 1080.14532
[2] Arnold, VI, Local normal forms of functions, Invent. Math., 35, 87-109, (1976) · Zbl 0336.57022
[3] Arnold, VI, Critical points of smooth functions, and their normal forms, Usp. Mat. Nauk, 30, 3-65, (1975)
[4] Bruce, J.; Wall, C., On the classification of cubic surfaces, J. London Math. Soc. (2), 2, 245-256, (1979) · Zbl 0393.14007
[5] Ding, WY; Tian, G., Kähler-Einstein metrics and the generalized Futaki invariant, Invent. Math., 110, 315-335, (1992) · Zbl 0779.53044
[6] Dolgachev, I.: Lectures on invariant theory, London Mathematical Society Lecture Note Series, vol. 296. Cambridge University Press, Cambridge (2003) · Zbl 1023.13006
[7] Dolgachev, IV; Hu, Y., Variation of geometric invariant theory quotients, Inst. Hautes Études Sci. Publ. Math., 87, 5-56, (1998) · Zbl 1001.14018
[8] Du Plessis, A.; Wall, C., Hypersurfaces in \(\mathbb{P}^n\mathbb{C}\) with one-parameter symmetry groups, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci, 4652, 2515-2541, (2000) · Zbl 0980.32011
[9] Gallardo, P., Martinez-Garcia, J.: Variations of GIT quotients package v0.6.13. https://doi.org/10.15125/BATH-00458 (2017)
[10] 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
[11] Gallardo, P., Martinez-Garcia, J., Spotti, C.: Applications of the moduli continuity method to log K-stable pairs. arXiv preprint arXiv:1811.00088 (2018)
[12] Hacking, P.; Keel, S.; Tevelev, J., Stable pair, tropical, and log canonical compactifications of moduli spaces of del Pezzo surfaces, Invent. Math., 178, 173-227, (2009) · Zbl 1205.14012
[13] Hacking, P.; Prokhorov, Y., Smoothable del Pezzo surfaces with quotient singularities, Compos. Math., 146, 169-192, (2010) · Zbl 1194.14054
[14] Hilbert, D., Über die vollen invariantensysteme, Math. Ann., 42, 313-373, (1893) · JFM 25.0173.01
[15] Ishii, S.; etal., Moduli space of polarized del Pezzo surfaces and its compactification, Tokyo J. Math, 5, 289-297, (1982) · Zbl 0534.14018
[16] Laza, R., Deformations of singularities and variation of GIT quotients, Trans. Amer. Math. Soc., 361, 2109-2161, (2009) · Zbl 1174.14004
[17] Laza, R., Pearlstein, G., Zhang, Z.: On the moduli space of pairs consisting of a cubic threefold and a hyperplane. Adv. Math. (2019) (to appear) (arXiv:1710.08056) · Zbl 1411.14042
[18] Looijenga, E.: Root systems and elliptic curves. Invent. Math. 38(1), 17-32 (1976/77) · Zbl 0358.17016
[19] Mukai, S.: An introduction to invariants and moduli, Cambridge Studies in Advanced Mathematics, vol. 81. Cambridge University Press, Cambridge (2003)
[20] Naruki, I., Cross ratio variety as a moduli space of cubic surfaces, Proc. London Math. Soc., 3, 1-30, (1982) · Zbl 0508.14005
[21] Odaka, Y.; Spotti, C.; Sun, S., Compact moduli spaces of del Pezzo surfaces and Kähler-Einstein metrics, J. Diff. Geom., 102, 127-172, (2016) · Zbl 1344.58008
[22] Pinkham, H., Deformations of normal surface singularities with \({\mathbb{C}}^*\) action, Math. Ann., 232, 65-84, (1978) · Zbl 0351.14004
[23] Shustin, E.; Tyomkin, I., Versal deformation of algebraic hypersurfaces with isolated singularities, Math. Ann., 313, 297-314, (1999) · Zbl 0951.14022
[24] Thaddeus, M., Geometric invariant theory and flips, J. Amer. Math. Soc., 9, 691-723, (1996) · Zbl 0874.14042
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.