The enumerative geometry of cubic hypersurfaces: point and line conditions. (English) Zbl 07832897
Summary: The set of smooth cubic hypersurfaces in \(\mathbb{P}^n\) is an open subset of a projective space. A compactification of the latter which allows to count the number of smooth cubic hypersurfaces tangent to a prescribed number of lines and passing through a given number of points is termed a 1-complete variety of cubic hypersurfaces, in analogy with the space of complete quadrics. Imitating the work of Aluffi for plane cubic curves, we construct such a space in arbitrary dimensions by a sequence of five blow-ups. The counting problem is then reduced to the computation of five total Chern classes. In the end, we derive the desired numbers in the case of cubic surfaces.
MSC:
| 14-XX | Algebraic geometry |
References:
| [1] | Aluffi, P., The enumerative geometry of plane cubics. I. Smooth cubics, Trans. Am. Math. Soc., 317, 2, 501-539, 1990 · Zbl 0703.14035 |
| [2] | Aluffi, P., The enumerative geometry of plane cubics. II. Nodal and cuspidal cubics, Math. Ann., 289, 4, 543-572, 1991 · Zbl 0712.14011 · doi:10.1007/BF01446588 |
| [3] | Aluffi, P., Some characteristic numbers for nodal and cuspidal plane curves of any degree, Manuscr. Math., 72, 4, 425-444, 1991 · Zbl 0749.14035 · doi:10.1007/BF02568288 |
| [4] | Bruns, W.; Herzog, J., Cohen-Macaulay rings. Cambridge Studies in Advanced Mathematics, 1993, Cambridge: Cambridge University Press, Cambridge · Zbl 0788.13005 |
| [5] | Brodmann, M.; Schenzel, P., Arithmetic properties of projective varieties of almost minimal degree, J. Algebraic Geom., 16, 2, 347-400, 2007 · Zbl 1126.14054 · doi:10.1090/S1056-3911-06-00442-5 |
| [6] | Breiding, P., Timme, S.: Homotopycontinuation. jl: a package for homotopy continuation in julia. In International Congress on Mathematical Software, pp 458-465. Springer, (2018) · Zbl 1396.14003 |
| [7] | De Concini, C., Procesi, C.: Complete symmetric varieties. II. Intersection theory. In Algebraic groups and related topics, volume 6 of Adv. Stud. Pure Math., pp 481-513. Amsterdam, (1985) · Zbl 0596.14041 |
| [8] | Eisenbud, D.; Harris, J., 3264 and All That: A Second Course in Algebraic Geometry, 2016, Cambridge: Cambridge University Press, Cambridge · Zbl 1341.14001 · doi:10.1017/CBO9781139062046 |
| [9] | Eisenbud, D.: Commutative Algebra, with a View Toward Algebraic Geometry. Graduate Texts in Mathematics, Springer-Verlag, New York (1995) · Zbl 0819.13001 |
| [10] | Fulton, W., Harris, J.: Representation theory, volume 129 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1991. A first course, Readings in Mathematics · Zbl 0744.22001 |
| [11] | Fulton, W., Kleiman, S., MacPherson, R.: About the enumeration of contacts. In Algebraic geometry—open problems (Ravello, 1982), volume 997 of Lecture Notes in Math., pp 156-196. Springer, Berlin, (1983) · Zbl 0529.14030 |
| [12] | Fulton, W.: Intersection theory, volume 2 of Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics. Springer-Verlag, Berlin, second edition, (1998) · Zbl 0885.14002 |
| [13] | Gelfand, I.M., Kapranov, M.M., Zelevinsky, A.V.: Discriminants, resultants, and multidimensional determinants. Theory & Applications. Birkhäuser Boston Inc, Boston, MA, Mathematics (1994) · Zbl 0827.14036 |
| [14] | Hartshorne, R.: Algebraic geometry. Graduate Texts in Mathematics, No. 52. Springer-Verlag, New York-Heidelberg, (1977) · Zbl 0367.14001 |
| [15] | Harris, J.: Algebraic geometry. A first course, volume 133 of Graduate Texts in Mathematics. Springer-Verlag, New York, (1995) · Zbl 0852.55001 |
| [16] | Kleiman, S., Speiser, R.: Enumerative geometry of cuspidal plane cubics. In Proceedings of the Vancouver Conference in Algebraic Geometry 6, 227-268 (1984) · Zbl 0601.14045 |
| [17] | Lee, W.; Park, E.; Schenzel, P., On the classification of non-normal cubic hypersurfaces, J. Pure Appl. Algebra, 215, 8, 2034-2042, 2011 · Zbl 1225.14044 · doi:10.1016/j.jpaa.2010.12.007 |
| [18] | Maillard, Synère-Nicolas.: Recherches des caractéristiques des systèmes élémentaires de courbes planes du troisième ordre. PhD thesis, Imprimerie Cusset, (1871) |
| [19] | Manivel, L., Michałek, M., Monin, L., Seynnaeve, T., Vodička, M.: Complete quadrics: Schubert calculus for gaussian models and semidefinite programming, 2020. To appear in: Journal of the European Mathematical Society |
| [20] | Michalek, M.; Monin, L.; Wisniewski, JA, Maximum likelihood degree, complete quadrics, and \(\mathbb{C}^*\)-action, SIAM J. Appl. Algebra Geometry, 5, 1, 60-85, 2021 · Zbl 1461.62228 · doi:10.1137/20M1335960 |
| [21] | Schubert, H.: Kalkül der abzählenden Geometrie. BG Teubner, (1879) · JFM 11.0460.01 |
| [22] | Segre, B., The Non-singular Cubic Surfaces, 1942, Oxford: Oxford University Press, Oxford · JFM 68.0358.01 |
| [23] | Semple, JG, On complete quadrics (i), J. Lond. Math. Soc., 1, 4, 258-267, 1948 · Zbl 0034.08701 · doi:10.1112/jlms/s1-23.4.258 |
| [24] | Slavov, K., The moduli space of hypersurfaces whose singular locus has high dimension, Math. Z., 279, 1, 139-162, 2015 · Zbl 1310.14038 · doi:10.1007/s00209-014-1360-0 |
| [25] | Sturmfels, B., The Hurwitz form of a projective variety, J. Symbolic Comput., 79, 1, 186-196, 2017 · Zbl 1359.14043 · doi:10.1016/j.jsc.2016.08.012 |
| [26] | Sukarto, E.: Orbit and orbit closure containments for cubic surfaces (2020) |
| [27] | Tseng, D., Collections of hypersurfaces containing a curve, Int. Math. Res. Not., 13, 3927-3977, 2020 · Zbl 1454.14112 · doi:10.1093/imrn/rny120 |
| [28] | Vainsencher, I.: Schubert calculus for complete quadrics. In Enumerative Geometry and Classical Algebraic Geometry, pp 199-235. Springer (1982) · Zbl 0501.14032 |
| [29] | Vakil, R., The characteristic numbers of quartic plane curves, Canad. J. Math., 51, 5, 1089-1120, 1999 · Zbl 0967.14034 · doi:10.4153/CJM-1999-048-1 |
| [30] | van Gastel, L. J..: Characteristic numbers of plane curves. In Enumerative Algebraic Geometry: Proceedings of the 1989 Zeuthen Symposium, volume 123, page 259. American Mathematical Soc (1991) · Zbl 0753.14046 |
| [31] | Zeuthen, H. G.: Sur la détermination des caractéristiques de surfaces du second ordre. Nouvelles annales de mathématiques: journal des candidats aux écoles polytechnique et normale, 2e série, 7:385-403, (1868) · JFM 01.0236.05 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
