Characterizations of Veronese and Segre varieties. (English) Zbl 1248.51005
The authors give a synopsis of the methods of characterizing Veronese and Segre varieties which have been developed during the last thirty years. Many published sources are given, they also refer to results from hitherto unpublished papers and preprints. Most of the results involve finite objects, on several occasions results are carried over to the infinite case.
Characterizations of Veroneseans using tangent spaces are mentioned in the papers [F. Mazocca and N. Melone, Discrete Math. 48, 243–252 (1984; Zbl 0537.51014); J. W. P. Hirschfeld and the first author, General Galois geometries. Oxford: Clarendon Press. (1991; Zbl 0789.51001); the authors, Eur. J. Comb. 25, No. 2, 275–285 (2004; Zbl 1040.51013)]. The latter result has been generalized to skewfields by Schillewaert and the second author, while Segre varieties are treated in [the authors, “Characterization of Segre varieties”, Preprint]. They point out characterizations using subvarieties presented in [G. Tallini, Atti Accad. Naz. Lincei, VIII. Ser., Rend., Cl. Sci. Fis. Mat. Nat. 24, 19–23, 135–138 (1958; Zbl 0080.14003)], and, in a general setting, in [the authors, Quart. J. Math. 55, 99–113 (2004; Zbl 1079.51004)]. Intersection numbers are used in [E. Ferrara Dentice and G. Marino, Discrete Math. 308, No. 2–3, 299–302 (2008; Zbl 1151.51011); Hirschfeld and the first author, loc. cit.; the authors, J. Comb. Theory, Ser. A 110, No. 2, 217–221 (2005; Zbl 1072.51011)] to characterize Veronese varieties \({\mathcal V}^4_2\) in \(PG(5,q)\), and for higher order Veroneseans they refer to [J. Schillewaert and the authors, “A characterization of the finite Veronesean by intersection properties”, Ann. Comb. 16, No. 2, 331–348 (2012; doi:10.1007/s00026-012-0136-7)]. Point sets of quadric Veroneseans, finite or infinite, may be also be characterized as incidence structures (cf. [the authors, Q. J. Math. 55, No. 1, 99–113 (2004; Zbl 1079.51004); Combinatorica 31, No. 5, 615–629 (2011; Zbl 1299.51003)]). By [C. Zanella, Bull. Belg. Math. Soc. Simon Stevin 3, No. 1, 65–79 (1996; Zbl 0859.51007); the authors, “Characterization of Segre varieties”, Preprint], the same is true for Segre varieties.
Characterizations of Veroneseans using tangent spaces are mentioned in the papers [F. Mazocca and N. Melone, Discrete Math. 48, 243–252 (1984; Zbl 0537.51014); J. W. P. Hirschfeld and the first author, General Galois geometries. Oxford: Clarendon Press. (1991; Zbl 0789.51001); the authors, Eur. J. Comb. 25, No. 2, 275–285 (2004; Zbl 1040.51013)]. The latter result has been generalized to skewfields by Schillewaert and the second author, while Segre varieties are treated in [the authors, “Characterization of Segre varieties”, Preprint]. They point out characterizations using subvarieties presented in [G. Tallini, Atti Accad. Naz. Lincei, VIII. Ser., Rend., Cl. Sci. Fis. Mat. Nat. 24, 19–23, 135–138 (1958; Zbl 0080.14003)], and, in a general setting, in [the authors, Quart. J. Math. 55, 99–113 (2004; Zbl 1079.51004)]. Intersection numbers are used in [E. Ferrara Dentice and G. Marino, Discrete Math. 308, No. 2–3, 299–302 (2008; Zbl 1151.51011); Hirschfeld and the first author, loc. cit.; the authors, J. Comb. Theory, Ser. A 110, No. 2, 217–221 (2005; Zbl 1072.51011)] to characterize Veronese varieties \({\mathcal V}^4_2\) in \(PG(5,q)\), and for higher order Veroneseans they refer to [J. Schillewaert and the authors, “A characterization of the finite Veronesean by intersection properties”, Ann. Comb. 16, No. 2, 331–348 (2012; doi:10.1007/s00026-012-0136-7)]. Point sets of quadric Veroneseans, finite or infinite, may be also be characterized as incidence structures (cf. [the authors, Q. J. Math. 55, No. 1, 99–113 (2004; Zbl 1079.51004); Combinatorica 31, No. 5, 615–629 (2011; Zbl 1299.51003)]). By [C. Zanella, Bull. Belg. Math. Soc. Simon Stevin 3, No. 1, 65–79 (1996; Zbl 0859.51007); the authors, “Characterization of Segre varieties”, Preprint], the same is true for Segre varieties.
Reviewer: Horst Szambien (Garbsen)
MSC:
| 51M35 | Synthetic treatment of fundamental manifolds in projective geometries (Grassmannians, Veronesians and their generalizations) |
| 51E20 | Combinatorial structures in finite projective spaces |
| 51E22 | Linear codes and caps in Galois spaces |
| 51E25 | Other finite nonlinear geometries |
| 51E30 | Other finite incidence structures (geometric aspects) |
| 05B25 | Combinatorial aspects of finite geometries |
Citations:
Zbl 0537.51014; Zbl 0789.51001; Zbl 1040.51013; Zbl 0080.14003; Zbl 1079.51004; Zbl 1151.51011; Zbl 1072.51011; Zbl 0859.51007; Zbl 1299.51003References:
| [1] | Ferrara Dentice E., Marino G.: Classication of Veronesean caps. Discret. Math. 308, 299–302 (2008) · Zbl 1151.51011 · doi:10.1016/j.disc.2006.11.042 |
| [2] | Ferri O.: Su di una caratterizzazione grafica della superficie di Veronese di un $${\(\backslash\)mathcal{S}_{5,q}}$$ . Atti Accad. Naz. Lincei Rend. 61, 603–610 (1976) |
| [3] | Hirschfeld J.W.P., Thas J.A.: General Galois Geometries. Oxford Mathematical Monographs, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York (1991) · Zbl 0789.51001 |
| [4] | Mazzocca F., Melone N.: Caps and Veronese varieties in projective Galois spaces. Discret. Math. 48, 243–252 (1984) · Zbl 0537.51014 · doi:10.1016/0012-365X(84)90186-9 |
| [5] | Melone N., Olanda D.: Spazi pseudoprodotto e varietà di C Segre. Rend. Mat. Appl. 1(7), 381–397 (1981) · Zbl 0492.51006 |
| [6] | Schillewaert, J., Thas, J.A., Van Maldeghem, H.: A characterization of the finite Veronesean by intersection properties. Ann. Combin. (to appear) · Zbl 1261.51008 |
| [7] | Tallini, G.: Una proprietà grafica caratteristica delle superficie di Veronese negli spazi finiti (Note I; II). Atti Accad. Naz. Lincei Rend. 24, 19–23, 135–138 (1976) · Zbl 0080.14003 |
| [8] | Thas J.A., Van Maldeghem H.: Classification of finite Veronesean caps. Eur. J. Combin. 25, 275–285 (2004) · Zbl 1040.51013 · doi:10.1016/S0195-6698(03)00113-6 |
| [9] | Thas J.A., Van Maldeghem H.: Characterizations of the finite quadric Veroneseans $${\(\backslash\)mathcal{V}_n\^{2\^n}}$$ . Quart. J. Math. 55, 99–113 (2004) · Zbl 1079.51004 · doi:10.1093/qmath/hag035 |
| [10] | Thas J.A., Van Maldeghem H.: On Ferri’s characterization of the finite quadric Veronesean $${\(\backslash\)mathcal{V}_2\^4}$$ . J. Combin. Theory Ser. A 110, 217–221 (2005) · Zbl 1072.51011 · doi:10.1016/j.jcta.2004.10.009 |
| [11] | Thas, J.A., Van Maldeghem, H.: André embeddings of affine planes. In: Bruen, A.A., Wehlau, D.L. (eds.) Error-Correcting Codes, Finite Geometries and Cryptography. Contemporary Mathematics 523, 123–131 (2010) · Zbl 1237.51009 |
| [12] | Thas, J.A., Van Maldeghem, H.: Generalized Veronesean embeddings of projective spaces. Combinatorica (to appear) · Zbl 1299.51003 |
| [13] | Akça, Z., Bayar, A., Ekmekçi, S., Kaya, R., Thas, J.A., Van Maldeghem, H.: Generalized lax Veronesean embeddings of projective spaces Ars Combin. (to appear) · Zbl 1265.51005 |
| [14] | Thas, J.A., Van Maldeghem, H.: Characterizations of Segre varieties (Preprint) · Zbl 1270.51015 |
| [15] | Zanella C.: Universal properties of the Corrado Segre embedding. Bull. Belg. Math. Soc. Simon Stevin 3, 65–79 (1996) · Zbl 0859.51007 |
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