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A combinatorial characterization of the Baer and the unital cone in $$\mathrm{PG}(3,q^2)$$. (English) Zbl 1452.51001
The characterisation of point sets by their intersection numbers is a classical problem in finite geometry.
A set of points $$S$$ in $$\mathrm{PG}(n,q)$$ is said to be of type $$(m_1,m_2,\dots,m_s)_d$$ if the size of the intersection of any $$d$$-space with $$S$$ is contained in $$\{m_1,m_2,\dots,m_s\}$$ and all possibilities $$m_i$$ occur as an intersection for some $$d$$-space.
It is a well-known result of A. Bruen [Bull. Am. Math. Soc. 76, 342–344 (1970; Zbl 0207.02601)] and A. A. Bruen and J. A. Thas [Geom. Dedicata 6, 193–203 (1977; Zbl 0367.05009)] that a set of type $$(1,q+1)_1$$ in $$\mathrm{PG}(2,q^2)$$ is either a Baer subplane or a unital.
In this paper, the authors deal with two particular examples of point sets with three intersection numbers with respect to planes. The first example is a Baer cone (a cone with vertex a point and base a Baer subplane), the second is a unital cone (a cone with vertex a point and base a unital). Both examples form $$2$$-blocking sets in $$\mathrm{PG}(3,q^2)$$. A 2-blocking set in $$\mathrm{PG}(3,q^2)$$ is a blocking set with respect to lines.
The two main results of this paper show that a 2-blocking set of type $$(q^2 +1,q^2 +q+1,q^3 +q^2 +1)_2$$ is a Baer cone and a 2-blocking set of type $$(q^2 +1,q^3 +1,q^3 +q^2 +1)_2$$ is a unital cone.
The proofs are combinatorial. The hypothesis that the point sets form 2-blocking sets enables the authors to make use of the known results about planar blocking sets in $$\mathrm{PG}(2,q^2)$$.
##### MSC:
 5.1e+21 Combinatorial structures in finite projective spaces 5.1e+22 Blocking sets, ovals, $$k$$-arcs
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##### References:
 [1] Aguglia, A., Characterizing Hermitian varieties in three- and four-dimensional projective spaces, J. Aust. Math. Soc., 107, 1, 1-8 (2019) · Zbl 1420.51007 [2] Batten, LM; Dover, JM, Some sets of type (m, n) in cubic order planes, Des. Codes Cryptogr., 16, 3, 211-213 (1999) · Zbl 0937.51007 [3] Batten, LM; Dover, JM, Blocking semiovals of type (1, m + 1, n + 1), SIAM J. Discrete Math., 14, 4, 446-457 (2001) · Zbl 1004.51013 [4] Beelen, P.; Datta, M., Maximum number of points on intersection of a cubic surface and a non-degenerate Hermitian surface, Moscow Math. J., 20, 3, 453-474 (2020) [5] Bruen, AA, Baer subplane and blocking set, Bull. Am. Math. Soc., 76, 342-344 (1970) · Zbl 0207.02601 [6] Bruen, AA, Blocking sets in finite projective planes, SIAM J. Appl. Math., 21, 380-392 (1971) · Zbl 0252.05014 [7] Bruen, AA; Thas, JA, Blocking sets, Geom. Dedicata, 6, 193-203 (1977) · Zbl 0367.05009 [8] Coykendall, J.; Dover, J., Sets with few intersection numbers from Singer subgroup orbits, Eur. J. Combin., 22, 455-464 (2001) · Zbl 0993.51001 [9] Hamilton, N., Penttila, T.: Sets of type (a, b) from subgroups of $$\Gamma L(1, p^R)$$. J. Algebraic Combin. 13, 67-76 (2001) · Zbl 0979.51005 [10] Hirschfeld, JWP, Finite Projective Spaces of Three Dimensions (1985), New York: Oxford University Press, New York [11] Hirschfeld, JWP, Projective Geometries over Finite Fields (1998), New York: Oxford University Press, New York [12] Hirschfeld, JWP; Thas, JA, Sets of type (1, n, q + 1) in PG(d, q), Proc. London Math. Soc., 41, 3, 254-278 (1980) · Zbl 0388.51005 [13] Lane-Harvard, L.; Penttila, T., Some strongly regular graphs with the parameters of Paley graphs, Australas. J. Combin., 61, 138-141 (2015) · Zbl 1309.05186 [14] Innamorati, S., Zannetti, M., Zuanni, F.: A note on sets of type $$(0, mq, 2mq)_2$$ in $$PG(3, q)$$. J. Geom. 111, 26 (2020). 10.1007/s00022-020-00539-2 · Zbl 1441.51003 [15] Napolitano, V.: On quasi-Hermitian varieties in $$PG(3, q^2)$$. Discrete Math. 339(2), 511-514 (2016) · Zbl 1331.51009 [16] Tallini, G.: Graphic characterization of algebraic varieties in a Galois space. Teorie Combinatorie, Vol. II, Atti dei Convegni Lincei, 17, Roma, 3-15 settembre 1973, 153-165 (1976) [17] Scafati, M.; Baker, CA; Batten, LM, Tallini: The k-set of PG(r, q) from the character point of view, Finite Geometries, 321-326 (1985), New York: Marcel Dekker Inc., New York [18] Thas, JA, A combinatorial problem, Geom. Dedicata, 1, 236-240 (1973) · Zbl 0252.50022 [19] Ueberberg, J., On regular v, n-arcs in finite projective spaces, J. Combin. Des., 1, 395-409 (1993) · Zbl 0885.05043 [20] Yassen, N., Yahya, K.: A geometric construction of complete $$(k, r)$$-arcs in $$PG(2,7)$$ and the related projective $$[n,3, d]_7$$ Codes. Raf. J. Comput. Math’s. 12(1), 24-40 (2018)
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