Nets of conics and associated Artinian algebras of length 7. Translation and update of the 1977 version by J. Emsalem and A. Iarrobino. (English) Zbl 1511.14014
The present paper is devoted to nets of conics and associated Artinian algebras of length \(7\) and it constitutes a translation and an update of the typescript written by Emsalem and Iarrobino in 1977. In the first part of the paper the authors classify the orbits of nets of conics under the action of the projective linear group and they determine the specialisations of these orbits by using geometric and algebraic methods. Then the authors show that Artinian algebras of the Hilbert function \(H=(1,3,3,0)\) determined by nets can be smoothed, i.e. deformed to a direct sum of fields, and that algebras of the Hilbert function \(H=(1,r,2,0)\) determined by pencils of quadrics can also be smoothed.
As a reader, I can strongly recommend this paper both as a source of useful references and as a source devoted to the classification of nets of conics (which is difficult to find in one piece in the literature).
As a reader, I can strongly recommend this paper both as a source of useful references and as a source devoted to the classification of nets of conics (which is difficult to find in one piece in the literature).
Reviewer: Piotr Pokora (Kraków)
MSC:
| 14C21 | Pencils, nets, webs in algebraic geometry |
| 14-02 | Research exposition (monographs, survey articles) pertaining to algebraic geometry |
| 13E10 | Commutative Artinian rings and modules, finite-dimensional algebras |
Keywords:
nets of conics; Artinian algebra; deformation; discriminant; Hessian; Hilbert function; isomorphism class; normal form; parametrization; pencils of conics; planar conics; planar cubic; quadric; smoothable; ternary cubicSoftware:
Book3264ExamplesReferences:
| [1] | Abdallah, N., Altafi, N., Iarrobino, A., Secealanu, A., Yaméogo, J.: Lefschetz properties for some codimension three Artinian Gorenstein algebras (2022). arXiv:2203.01258 |
| [2] | Abdallah, N., Emsalem, J., Iarrobino, A., Yaméogo, J.: Limits of graded Gorenstein algebras of Hilbert function \((1,3^k\!,1))\). arXiv:2302.00287 |
| [3] | Alda, V., Les réseaux de coniques, Czechoslovak Math. J., 7, 82, 48-56 (1957) · Zbl 0089.16802 · doi:10.21136/CMJ.1957.100230 |
| [4] | Anema, A.; Top, J.; Tuijp, A., Hesse pencils and 3-torsion structures, SIGMA Symmetry Integrability Geom. Methods Appl., 14, 102 (2018) · Zbl 1403.14030 |
| [5] | Araujo, C., Castravet, A.-M., Cheltsov, I., Fujita, K., Kaloghiros, A.-S., Martinez-Garcia, J., Shramov, C., Süss, H., Viswanathan, N.: The Calabi Problem for Fano Threefolds. MPIM preprint (2021) (to appear, London Mathematical Society Lecture Series, Cambridge University Press). https://www.maths.ed.ac.uk/cheltsov/pdf/Fanos.pdf |
| [6] | Artebani, M.; Dolgachev, I., The Hesse pencil of plane cubic curves, Enseign. Math., 55, 3-4, 235-273 (2009) · Zbl 1192.14024 · doi:10.4171/LEM/55-3-3 |
| [7] | Baldisserri, N., Nets of conics in a plane over a field of characteristic 2, Rend. Mat., 5, 3-4, 355-365 (1985) · Zbl 0649.51008 |
| [8] | Barth, W., Moduli of vector bundles on the projective plane, Invent. Math., 42, 63-91 (1977) · Zbl 0386.14005 · doi:10.1007/BF01389784 |
| [9] | Bertone, C.; Cioffi, F.; Roggero, M., Smoothable Gorenstein points via marked schemes and double-generic initial ideals, Exp. Math., 31, 1, 120-137 (2022) · Zbl 1489.13035 · doi:10.1080/10586458.2019.1592034 |
| [10] | Bernardi, A.; Gimigliano, A.; Idà, M., Computing symmetric rank for symmetric tensors, J. Symbolic Comput., 46, 1, 34-53 (2011) · Zbl 1211.14057 · doi:10.1016/j.jsc.2010.08.001 |
| [11] | Bhargava, M.; Ho, W., Coregular spaces and genus one curves, Cambridge J. Math., 4, 1, 1-119 (2016) · Zbl 1342.14074 · doi:10.4310/CJM.2016.v4.n1.a1 |
| [12] | Bhosle, UN, Nets of quadrics and vector bundles on a double plane, Math. Z., 192, 1, 29-43 (1986) · Zbl 0619.14025 · doi:10.1007/BF01162017 |
| [13] | Bielmans, P.: Fanography, a tool to visually study the geography of Fano three-folds. https://fanography.pythonanywhere.com |
| [14] | Blind, B.: La théorie des invariants des formes quadratiques ternaires revisité (2008). arXiv:0805.4135 |
| [15] | Bourbaki, N., Elements of Mathematics, Commutative Algebra (1972), Paris: Hermann, Paris · Zbl 0279.13001 |
| [16] | Boughon, P.; Nathan, J.; Samuel, P., Courbes planes en caractéristique 2, Bull. Soc. Math. France, 83, 275-278 (1955) · Zbl 0065.36403 · doi:10.24033/bsmf.1463 |
| [17] | Briançon, J., Description de \(Hilb^n C[x, y]\), Invent. Math., 41, 1, 45-89 (1977) · Zbl 0353.14004 · doi:10.1007/BF01390164 |
| [18] | Campbell, AD, Nets of conics in the real domain, Amer. J. Math., 50, 3, 459-466 (1928) · JFM 54.0680.04 · doi:10.2307/2370813 |
| [19] | Campbell, AD, Nets of conics in the Galois fields of order \(2^n\), Bull. Amer. Math. Soc., 34, 4, 481-489 (1928) · JFM 54.0164.09 · doi:10.1090/S0002-9904-1928-04572-X |
| [20] | Cartwright, DA; Erman, D.; Velasco, M.; Viray, B., Hilbert schemes of 8 points, Algebra Number Theory, 3, 7, 763-795 (2009) · Zbl 1187.14005 · doi:10.2140/ant.2009.3.763 |
| [21] | Casnati, G.; Notari, R., On some Gorenstein loci in \(\rm Hilb_6({\mathbb{P} }^4_k)\), J. Algebra, 308, 2, 493-523 (2007) · Zbl 1120.14003 · doi:10.1016/j.jalgebra.2006.09.023 |
| [22] | Casnati, G.; Notari, R., On the Gorenstein locus of the punctual Hilbert scheme of degree 11, J. Pure Appl. Algebra, 218, 9, 1635-1651 (2014) · Zbl 1287.13013 · doi:10.1016/j.jpaa.2014.01.004 |
| [23] | Casnati, G.; Elias, J.; Notari, R.; Rossi, ME, Poincaré series and deformations of Gorenstein local algebras, Comm. Algebra, 41, 3, 1049-1059 (2013) · Zbl 1274.13032 · doi:10.1080/00927872.2011.636643 |
| [24] | Casnati, G.; Jelisiejew, J.; Notari, R., Irreducibility of the Gorenstein loci of Hilbert schemes via ray families, Algebra Number Theory, 9, 7, 1525-1570 (2015) · Zbl 1349.14011 · doi:10.2140/ant.2015.9.1525 |
| [25] | Cheltsov, I.: Calabi problem for smooth Fano threefolds (2021). Slides for a talk https://www.maths.ed.ac.uk/cheltsov/talk2021x.pdf |
| [26] | Ciamberlini, C., Sulla condizione necessaria e sufficiente affinchè un triangolo sia acutangolo, rettangolo o ottusangolo, Boll. Unione Mat. Ital., 5, 37-41 (1943) · Zbl 0060.33306 |
| [27] | Ciliberto, C.; Ottaviani, G., The Hessian map, Int. Math. Res. Not. IMRN, 2022, 8, 5781-5817 (2022) · Zbl 1492.14093 · doi:10.1093/imrn/rnaa288 |
| [28] | Conca, A., Gröbner bases for spaces of quadrics of codimension 3, J. Pure Appl. Algebra, 213, 8, 1564-1568 (2009) · Zbl 1177.13062 · doi:10.1016/j.jpaa.2008.11.017 |
| [29] | Cook, RJ; Thomas, AD, Line bundles and homogeneous matrices, Q. J. Math. Oxford Ser., 30, 120, 423-429 (1979) · Zbl 0437.14004 · doi:10.1093/qmath/30.4.423 |
| [30] | Cossec, F., Dolgachev, I., Liedtke, C.: Enriques Surfaces, I (2021). http://www.math.lsa.umich.edu/ idolga/EnriquesOne.pdf |
| [31] | D’Alì, A., The Koszul property for spaces of quadrics of codimension three, J. Algebra, 490, 256-282 (2017) · Zbl 1406.13014 · doi:10.1016/j.jalgebra.2017.06.032 |
| [32] | Damon, J.: Thom-Mather theory at 50 years of age (2022) (preprint) |
| [33] | Dimca, A., Geometric approach to the classification of pencils, Geom. Dedicata, 14, 2, 105-111 (1983) · Zbl 0518.14018 · doi:10.1007/BF00181618 |
| [34] | Dimca, A., Topics on Real and Complex Singularities: An Introduction, Advanced Lectures in Mathematics (1987), Braunschweig: Friedr. Vieweg & Sohn, Braunschweig · Zbl 0628.14001 · doi:10.1007/978-3-663-13903-4 |
| [35] | Dimca, A.; Gibson, CG, Classification of equidimensional contact unimodular map germs, Math. Scand., 56, 1, 15-28 (1985) · Zbl 0579.32014 · doi:10.7146/math.scand.a-12085 |
| [36] | Dixon, A., Note on the reduction of a ternary quantic to a symmetrical determinant, Proc. Cambridge Philos. Soc., 11, 350-351 (1902) · JFM 33.0140.04 |
| [37] | Dolgachev, I.: Lectures on Invariant Theory. London Mathematical Society Lecture Note Series, vol. 296. Cambridge University Press, Cambridge (2003) · Zbl 1023.13006 |
| [38] | Dolgachev, IV, Classical Algebraic Geometry: A Modern View (2012), Cambridge: Cambridge University Press, Cambridge · Zbl 1252.14001 · doi:10.1017/CBO9781139084437 |
| [39] | Dolgachev, I.; Kanev, V., Polar covariants of plane cubics and quartics, Adv. Math., 98, 2, 216-301 (1993) · Zbl 0791.14013 · doi:10.1006/aima.1993.1016 |
| [40] | Dolgachev, I., Kondo, S.: Enriques Surfaces, II (2021). http://www.math.lsa.umich.edu/ idolga/EnriquesTwo.pdf |
| [41] | Domokos, M.; Fehér, LM; Rimányi, E., Equivariant and invariant theory of nets of conics with an application to Thom polynomials, J. Singul., 7, 1-20 (2013) · Zbl 1295.58012 |
| [42] | Dye, S.; Kohn, K.; Rydell, F.; Sinn, R., Maximum likelihood estimation for nets of conics, Matematiche (Catania), 76, 2, 399-414 (2021) · Zbl 1481.14015 |
| [43] | Dyment, Z.M.: Maximal commutative nilpotent subalgebras of a matrix algebra of the sixth degree. Vesci Akad. Navuk BSSR Ser. Fiz.-Mat. Navuk 1966(3), 53-68 (1966). (in Russian) · Zbl 0168.28601 |
| [44] | Edwards, SA; Wall, CTC, Nets of quadrics and deformations of \(^{3 \langle 3\rangle }\) singularities, Math. Proc. Cambridge Philos. Soc., 105, 1, 109-115 (1989) · Zbl 0698.57008 · doi:10.1017/S0305004100001407 |
| [45] | 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 |
| [46] | Elias, J.; Rossi, ME, Isomorphism classes of short Gorenstein local rings via Macaulay’s inverse system, Trans. Amer. Math. Soc., 364, 9, 4589-4604 (2012) · Zbl 1281.13015 · doi:10.1090/S0002-9947-2012-05430-4 |
| [47] | Ellingsrud, G.; Peine, R.; Strømme, SA; Ghione, F., On the variety of nets of quadrics defining twisted cubics, Space Curves, 84-96 (1987), Berlin: Springer, Berlin · Zbl 0659.14027 · doi:10.1007/BFb0078179 |
| [48] | Emsalem, J., Géométrie des points épais, Bull. Soc. Math. France, 106, 4, 399-416 (1978) · Zbl 0396.13017 · doi:10.24033/bsmf.1879 |
| [49] | Emsalem, J., Iarrobino, A.: Réseaux de Coniques et algèbres de longueur sept associées. Université de Paris VII, Paris (1977) (preprint) |
| [50] | Emsalem, J.; Iarrobino, A., Some zero-dimensional generic singularities: finite algebras having small tangent spaces, Compositio Math., 36, 2, 145-188 (1978) · Zbl 0393.14002 |
| [51] | Erman, D.; Velasco, M., A syzygetic approach to the smoothability of zero-dimensional schemes, Adv. Math., 224, 3, 1143-1166 (2010) · Zbl 1194.14012 · doi:10.1016/j.aim.2010.01.009 |
| [52] | Feng, R.; Shen, L-Y, Computing the intersections of three conics according to their Jacobian curve, J. Symbolic Comput., 73, 175-191 (2016) · Zbl 1323.51009 · doi:10.1016/j.jsc.2015.06.004 |
| [53] | Fevola, C., Mandelshtam, Y., Sturmfels, B.: Pencils of quadrics: old and new. Matematiche (Catania) 76(2), 319-335 (2021). arXiv:2009.04334 · Zbl 1480.14037 |
| [54] | Frium, HR; Mullen, GL, The group law on elliptic curves on Hesse form, Finite Fields with Applications to Coding Theory, Cryptography and Related Areas, 123-151 (2002), Berlin: Springer, Berlin · Zbl 1057.14038 · doi:10.1007/978-3-642-59435-9_10 |
| [55] | Gordan, P., Ueber Büschel von Kegelschnitten, Math. Ann., 19, 4, 529-552 (1882) · JFM 14.0080.01 · doi:10.1007/BF01446669 |
| [56] | Grace, J.H., Young, A.: The Algebra of Invariants. Cambridge Library Collection. Cambridge University Press, Cambridge (2010). Reprint of the 1903 original |
| [57] | Happel, D., Klassifikationstheorie endlich-dimensionaler algebren in der Zeit von 1880 bis 1920, Enseign. Math., 26, 1-2, 91-102 (1980) · Zbl 0439.01008 |
| [58] | Hartshorne, R.: Deformation Theory. Graduate Texts in Mathematics, vol. 257. Springer, New York (2010) · Zbl 1186.14004 |
| [59] | Hassett, B.; Tschinkel, Yu, Varieties of planes on intersections of three quadrics, Eur. J. Math., 7, 2, 613-632 (2021) · Zbl 1467.14038 · doi:10.1007/s40879-020-00424-x |
| [60] | Hodge, WVD; Pedoe, D., Methods of Algebraic Geometry, Quadrics and Grassmann Varieties (1952), Cambridge: Cambridge University Press, Cambridge · Zbl 0048.14502 |
| [61] | Huibregtse, ME, Some elementary components of the Hilbert scheme of points, Rocky Mountain J. Math., 47, 4, 1169-1225 (2017) · Zbl 1386.14023 · doi:10.1216/RMJ-2017-47-4-1169 |
| [62] | Iarrobino, A., Reducibility of the families of 0-dimensional schemes on a variety, Invent. Math., 15, 72-77 (1972) · Zbl 0227.14006 · doi:10.1007/BF01418644 |
| [63] | Iarrobino, A., The number of generic singularities, Rice Univ. Stud., 59, 1, 49-51 (1973) · Zbl 0272.14002 |
| [64] | Iarrobino, A.: Deforming complete intersection Artin algebras. Appendix: Hilbert function of \({\mathbb{C}}[x,y]/I\). In: Orlik, P. (ed.) Singularities, Part 1. Proceedings of Symposia in Pure Mathematics, vol. 40, pp. 593-608. American Mathematical Society, Providence (1983) · Zbl 0563.13010 |
| [65] | Iarrobino, A.: Hilbert scheme of points: Overview of last ten years. In: Bloch, S.J. (ed.) Algebraic Geometry: Bowdoin 1985, Part 2. Proceedings of Symposia in Pure Mathematics, vol. 46.2, pp. 297-320. American Mathematical Society, Providence (1987) · Zbl 0646.14002 |
| [66] | Ishitsuka, Y.; Ito, T., The local-global principle for symmetric determinantal representations of smooth plane curves, Ramanujan J., 43, 1, 141-162 (2017) · Zbl 1390.14086 · doi:10.1007/s11139-016-9775-3 |
| [67] | Jelisiejew, J., Classifying local Artinian Gorenstein algebras, Collect. Math., 68, 1, 101-127 (2017) · Zbl 1369.13033 · doi:10.1007/s13348-016-0183-1 |
| [68] | Jelisiejew, J., Elementary components of Hilbert schemes of points, J. London Math. Soc., 100, 1, 249-272 (2019) · Zbl 1441.14023 · doi:10.1112/jlms.12212 |
| [69] | Jordan, C., Réduction d’un réseau de formes quadratiques ou bilinéaire, J. Math. Pures Appl., 6, 403-438 (1906) · JFM 37.0136.01 |
| [70] | Kaygorodov, I., Rakhimov, I., Said Husain, Sh.K.: The algebraic classification of nilpotent associative commutative algebras. J. Algebra Appl. 19(11), 2050220 (2020) · Zbl 1460.16007 |
| [71] | Kline, M., Mathematical Thought from Ancient to Modern Times (1972), New York: Oxford University Press, New York · Zbl 0277.01001 |
| [72] | Lavrauw, M.; Popiel, T.; Sheekey, J., Nets of conics of rank one in \({\rm PG}(2, q), q\) odd, J. Geom., 111, 3, 36 (2020) · Zbl 1452.51002 · doi:10.1007/s00022-020-00548-1 |
| [73] | Lee, M., Patel, A., Tseng, D.: Equivariant degenerations of plane curve orbits (2019). arXiv:1903.10069 |
| [74] | Mather, JN, Stability of \(C^\infty\) mappings: IV: Classification of stable germs by \(R\)-algebras, Inst. Hautes Études Sci. Publ. Math., 37, 223-248 (1969) · Zbl 0202.55102 · doi:10.1007/BF02684889 |
| [75] | Mazzola, G., Generic finite schemes and Hochschild cocycles, Comment. Math. Helv., 55, 2, 267-293 (1980) · Zbl 0463.14004 · doi:10.1007/BF02566686 |
| [76] | Michałek, M.; Moon, H.; Sturmfels, B.; Ventura, E., Real rank geometry of ternary forms, Ann. Mat. Pura Appl., 196, 3, 1025-1054 (2017) · Zbl 1390.14179 · doi:10.1007/s10231-016-0606-3 |
| [77] | Molk, J.: Encyclopédie des sciences mathématiques. III 3. Géométrie algébrique plane. Edition française, Gauthier-Villars, Paris, 1898-1904; Edition Allemande: F. Meyers, ed., Teubner, Leipzig (1914); Reprint (2000) Editions J. Gabay (Paris) |
| [78] | Okonek, Ch; Van de Ven, A., Cubic forms and complex 3-folds, Enseign. Math., 41, 3-4, 297-333 (1995) · Zbl 0869.14018 |
| [79] | Onda, N.: Isomorphism types of commutative algebras of rank 7 over an arbitrary algebraically closed field of characteristic not 2 or 3 (2021). arXiv:2107.04959 |
| [80] | Ottaviani, G., An invariant regarding Waring’s problem for cubic polynomials, Nagoya Math. J., 193, 95-110 (2009) · Zbl 1205.14064 · doi:10.1017/S0027763000026040 |
| [81] | Ottaviani, G., Five lectures on projective invariants, Rend. Semin. Mat. Univ. Politec. Torino, 71, 1, 119-194 (2013) · Zbl 1312.13012 |
| [82] | Pavlov, I.A.: Commutative nilpotent matrix algebras of class \(n-3\): I. Vesci Akad. Navuk BSSR Ser. Fiz.-Mat. Navuk 1968(4), 39-46 (1968). (in Russian) · Zbl 0167.03402 |
| [83] | Pavlov, I.A.: Commutative nilpotent matrix algebras of class \(n-3\): II. Vesci Akad. Navuk BSSR Ser. Fiz.-Mat. Navuk 1968(5), 10-16 (1968). (in Russian) · Zbl 0227.16013 |
| [84] | Poonen, B.: Isomorphism types of commutative algebras of finite rank over an algebraically closed field. In: Lauter, K.E., Ribet, K.A. (eds.) Computational Arithmetic Geometry. Contemporary Mathematics, vol. 463, pp. 111-120. American Mathematical Society, Providence (2008). (Revised): http://www-math.mit.edu/ poonen/papers/dimension6.pdf · Zbl 1155.13015 |
| [85] | Poonen, B., The moduli space of commutative algebras of finite rank, J. Eur. Math. Soc. (JEMS), 10, 3, 817-836 (2008) · Zbl 1151.14011 · doi:10.4171/JEMS/131 |
| [86] | Reid, M.: The Complete Intersection of Two or More Quadrics. Ph.D. Thesis, Trinity College, Cambridge (1972). https://homepages.warwick.ac.uk/ masda/3folds/qu.pdf |
| [87] | Ronga, F., Applications polynômiales de degré deux du plan projectif complexe dans lui-même, Enseign. Math., 22, 1-2, 41-54 (1976) · Zbl 0325.58003 |
| [88] | Rosanes, MJ, Erweiterung eines bekannten Satzes auf Formen von beliebig vielen Veränderlichen, Math. Ann., 23, 3, 412-415 (1884) · JFM 16.0731.01 · doi:10.1007/BF01446396 |
| [89] | Rydh, D.; Skjelnes, R., An intrinsic construction of the principal component of the Hilbert scheme, J. London Math. Soc., 82, 2, 459-481 (2010) · Zbl 1198.14005 · doi:10.1112/jlms/jdq042 |
| [90] | Segre, C.; Rosanes, MJ, Sur les invariants simultanés de deux formes quadratiques, Math. Ann., 24, 1, 152-156 (1884) · JFM 16.0096.01 · doi:10.1007/BF01446445 |
| [91] | Salmon, G., A Treatise on the Higher Plane Curves: Intended as a Sequel to “A Treatise on Conic Sections” (1960), New York: Chelsea Publishing, New York |
| [92] | Satriano, M., Staal, A.P.: Small elementary components of Hilbert schemes of points (2021) arXiv:2112.01481 |
| [93] | Segre, B., Intorno alla geometria sopra un campo di caratteristica due, Rev. Fac. Sci. Univ. Istanbul. Sér. A, 21, 97-123 (1956) · Zbl 0073.36802 |
| [94] | Segre, C., Studio sulle quadriche in uno spazio lineare ad un numero qualunque di dimensioni, Mem. della R. Acc. delle Scienze di Torino, 36, 2, 3-86 (1883) |
| [95] | Sernesi, E.: Deformations of Algebraic Schemes. Grundlehren der Mathematischen Wissenschaften, vol. 334. Springer, Berlin (2006) · Zbl 1102.14001 |
| [96] | Serre, J.P.: Cours d’arithmétique. Collection SUP: “Le Mathématicien”. 2 Presses Universitaires de France, Paris (1970) · Zbl 0225.12002 |
| [97] | Shafarevich, IR, Deformations of commutative algebras of class 2, Leningrad Math. J., 2, 6, 1335-1351 (1991) · Zbl 0743.13009 |
| [98] | Shafarevich, IR, Degeneration of semisimple algebras, Comm. Algebra, 29, 9, 3943-3960 (2001) · Zbl 1086.14508 · doi:10.1081/AGB-100105983 |
| [99] | Sivatski, AS, On zeros of a system of quadratic forms in 3 variables, J. Algebra, 449, 237-245 (2016) · Zbl 1385.11015 · doi:10.1016/j.jalgebra.2015.10.014 |
| [100] | Sturmfels, B., Algorithms in Invariant Theory: Texts and Monographs in Symbolic Computation (2008), Vienna: Springer, Vienna · Zbl 1154.13003 |
| [101] | Suprunenko, D.A.: On maximal commutative subalgebras of the full linear algebra. Uspehi Mat. Nauk (N.S) 11(69), 181-184 (1956). (in Russian) · Zbl 0070.03102 |
| [102] | Suprunenko, D.A., Tyškevič, R.I.: Commutative Matrices. Academic Press (1968) |
| [103] | Szachniewicz, M.: Non-reducedness of the Hilbert schemes of few points (2021) arXiv:2109.11805 |
| [104] | Szafarczyk, R.: New elementary components of the Gorenstein locus of the Hilbert scheme of points (2022) arXiv:2206.03732 |
| [105] | Tjøtta, E.: Rational curves on the space of determinantal nets of conics (1998) arXiv:math/9802037 |
| [106] | Tjurin, A.N.: An invariant of a net of quadrics Izv. Akad. Nauk SSSR Ser. Mat. 39, 23-27 (1975). (in Russian) |
| [107] | Tjurin, AN, The intersection of quadrics, Russian Math. Surveys, 30, 6, 51-105 (1975) · Zbl 0339.14020 · doi:10.1070/RM1975v030n06ABEH001530 |
| [108] | Verra, A., Superfici di Enriques et reti di quadriche, Ann. Mat. Pura Appl., 130, 307-320 (1982) · Zbl 0508.14030 · doi:10.1007/BF01761500 |
| [109] | Verra, A., On Enriques surface as a fourfold cover of \({\mathbb{P}}^2\), Math. Ann., 266, 2, 241-250 (1983) · Zbl 0506.14032 · doi:10.1007/BF01458446 |
| [110] | Verra, A., Edge and Fano on nets of quadrics, Eur. J. Math., 4, 3, 1264-1277 (2018) · Zbl 1425.14024 · doi:10.1007/s40879-018-0252-y |
| [111] | Wall, CTC, Nets of conics, Math. Proc. Cambridge Philos. Soc., 81, 3, 351-364 (1977) · Zbl 0351.14032 · doi:10.1017/S0305004100053421 |
| [112] | Wall, CTC, Nets of quadrics, and theta-characteristics of singular curves, Philos. Trans. R. Soc. London Ser. A, 289, 1357, 229-269 (1978) · Zbl 0382.14011 · doi:10.1098/rsta.1978.0060 |
| [113] | Wall, C.T.C.: Singularities of nets of quadrics. Compositio Math. 42(2), 187-212 (1980/81) · Zbl 0423.14002 |
| [114] | Wang, X., Maximal linear spaces contained in the based loci of pencils of quadrics, Algebr. Geom., 5, 3, 359-397 (2018) · Zbl 1419.14042 · doi:10.14231/AG-2018-011 |
| [115] | Weierstrass, K.: Zur Theorie der bilinearen und quadratischen Formen. Berliner Monatsberichte, 310-338 (1868) · JFM 01.0054.04 |
| [116] | Wilson, AH, The canonical types of nets of modular conics, Am. J. Math., 36, 2, 187-210 (1914) · JFM 45.0228.01 · doi:10.2307/2370239 |
| [117] | Zanella, C., On finite Steiner surfaces, Discrete Math., 312, 3, 652-656 (2012) · Zbl 1235.51008 · doi:10.1016/j.disc.2011.05.035 |
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.
