A new combinatorial characterization of (quasi)-Hermitian surfaces

In this paper, we present a combinatorial characterization of a quasi-Hermitian surface as a set H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {H}}$$\end{document} of points of PG(3,q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{PG}(3,q)$$\end{document}, q=p2h\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q=p^{2h}$$\end{document}h≥1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h\ge 1$$\end{document}, p a prime number and q≠4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q\ne 4$$\end{document}, having the same size as the Hermitian surface and containing no plane, such that either a line is contained in H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {H}}$$\end{document} or intersects H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {H}}$$\end{document} in at most q+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sqrt{q}+1$$\end{document} points and every plane intersects H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {H}}$$\end{document} in at least qq+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q\sqrt{q}+1$$\end{document} points. Moreover, if there is no external line, the set H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {H}}$$\end{document} is a Hermitian surface.


Introduction
Hermitian varieties over finite fields have particular properties which make them nice and interesting geometric objects connected to other combinatorial structures such as e.g. (linear) codes, strongly regular graphs, (two-class) association schemes and block designs (cf e.g. [2,3,11]). Thus, there is a wide literature devoted to Hermitian varieties, in particular results of characterization (cf e.g. [4,[7][8][9] and the references cited therein).
These results either use only purely geometric-combinatorial considerations (an approach which is called the Beniamino Segre point of view in (finite) geometry [1]) or they use algebraic geometry over finite fields. An example of the latter approach is the paper [5] in which the authors give a characterization of Hermitian surfaces of PG (3, q), q = p 2 h , p prime, as surfaces of degree √ q + 1 without F q -plane components and of maximal size (i.e. meeting the Homma-Kim elementary upper bound for surfaces of degree d with no F qplane components in PG (3, q), cf [6]). In the proof of this result the plane sections of the surface play a special role, and since most of recent geometriccombinatorial characterizations of Hermitian surfaces are based on the sizes (or the possible number of sizes) of the plane sections, this result suggest in a geometric-combinatorial approach to study Hermitian surfaces via their plane sections without assuming that the possible sizes of the sections are exactly two or are precisely those of a Hermitian surface.
In this paper, we give a new combinatorial characterization of the Hermitian surface in PG(3, q) following this idea.
Before to state our result, let's quickly recall the definitions of Hermitian variety and quasi-Hermitian surface. A (nonsingular) Hermitian variety of PG(n, q), q a square of a prime power, over F q is a variety projectively equivalent to the variety For n = 3, we have that a (nonsingular) Hermitian surface is a set of (q √ q + 1)(q + 1) points of PG(3, q), q = p 2 h , h ≥ 1, p prime, intersected by any line in 1, √ q + 1 or q + 1 points and by every plane either in q √ q + 1 or q √ q + q + 1 points.
A subset of points of PG(3, q) is a quasi-Hermitian surface if the sizes of its intersections with the planes of PG(3, q) are exactly two and they are precisely those of a Hermitian surface of PG (3, q), that is q √ q + 1 and q √ q + q + 1 (cf [4,9]).
Let H be a subset of points of PG(r, q), an external line is a line such that ∩ H = ∅.
Our result reads as follows.
Theorem I Let H be a subset of (q √ q + 1)(q + 1) points of PG(3, q), with q = p 2 h , h ≥ 1, p a prime number and q = 4, such that (ii) H contains no plane and |π ∩ H| ≥ q √ q + 1 for every plane π such that π ∩ H = ∅. Then H is a quasi-Hermitian surface of PG (3, q). If there is no external line, then H is a Hermitian surface.
Note that the set of points on q √ q +1 pairwise skew lines has size (q √ q +1)(q + 1) and it is intersected by any plane either in q √ q+1 points or q √ q+q+1 points and has external lines, that is it is a quasi-Hermitian surface with external lines. Moreover Property (i), which is fulfilled by a surface of order √ q + 1, together with the assumption that H contains no plane and has the same number of points as a Hermitian surface will allow us to prove, with only combinatorial considerations, that the plane sections of H have size at most q √ q + q + 1, a step which is similar to the one of the proof of the above mentioned Homma and Kim characterization of Hermitian surfaces, in which they use a result of B. Segre [ [10], Teorema I page 29 and Osservazione IV page 33] on the number of F q -rational points of (possibly reducible) plane curves of a given order. Furthermore, as in the Homma-Kim result, the case q = 4 occurs like a special one and whereas they solved it with an extra argument, we leave it as an open case.

Some preliminary definitions and results
Let k be a positive integer and let K be a non-empty set of k points of PG(3, q) different from PG(3, q), S 2 be the family of all the planes of PG(3, q) and let c i , i = 0, . . . , q 2 + q + 1 denote the number of planes intersecting K in exactly i points. Even if one may find the following countings in some papers devoted to the study k-sets of PG(r, q) via their intersections with the family of all the d-dimensional subspaces d ≥ 2, to make the paper self-contained we prefer recall them briefly. Let Double counting arguments give and put When one needs to be more precise on the size of the intersections of a set of points of a projective space with the planes and/or lines or the number of these sizes is small, one may use the following notions.
Let P = PG(n, q) be the n-dimensional (desarguesian) projective space of order q, and m 1 , . . . , m s be s integers such that 0 ≤ m 1 < · · · < m s and for any integer h, Hence an Hermitian surface of PG(3, q), with q a square, is a (q √ q + 1)(q + 1)set of line-type (1, √ q + 1, q + 1) 1 and plane-type (q √ q + 1, q √ q + q + 1) 2 , and a quasi-Hermitian surface is a set of points of PG(3, q), q square, of plane-type (q √ q + 1, q √ q + q + 1) 2 Let K be a subset of points of P, a line (plane) intersecting K in exactly i points is called i-line (plane). If i = 1 a 1-line (plane) is called tangent.
We end this section by recalling a characterization result of Hermitian surfaces in terms of their line-class and plane-type.

The proof
Let q = p 2h , with p a prime number and h ≥ 1. Let H be a subset of points of PG(3, q) satisfying the assumptions of Theorem I.
Note that by the assumption that H contains no plane it follows that if π is a plane, then through any point of π there pass at most √ q + 1 (q + 1)-lines.
Proof. Let π be a plane of PG (3, q). If π contains no (q + 1)-line let p be a point in π ∩ H. Counting the number of points of π through the lines on p gives v π ≤ 1 + (q + 1) √ q < q √ q + q + 1. So we may assume that π contains a (q + 1)-line, say . If contains a point p such that is the only (q + 1)-line through , then v π ≤ q + 1 + q √ q as required. Thus, we may assume that in π all the points of are on at least two (q + 1)-lines. Then in π there is no line with at most √ q − 1 points. Otherwise, since any two lines of π meet each other, counting the number of (q + 1)-lines of π meeting a line with at most √ q − 1 points gives that this number is at most ( On the other hand, since any point of is on at least two (q + 1)-lines of π, one has that there are at least q + 2 (q + 1)-lines, a contradiction. So, any line of π has at least √ q points in common with H. If contains a point on at least three On the other hand, counting the number of points of π ∩ H through the lines on a point of π not in H gives v π ≤ (q + 1)( which is not possible, since q = 4. 1 So every point of is on exactly two (q + 1)-lines. Hence the number of (q + 1)lines of π is q + 2. Every point of each of these (q + 1)-lines is on at least two (q + 1)-lines, otherwise we are done, by the previous argument. Since there are exactly q +2 (q +1)-line it follows that on each point of one of these (q +1)-lines there pass exactly two (q + 1)-lines.
If there is a point of π ∩ H on no (q + 1)-line, then the assertion is true. So on each point of π ∩ H there is at least one (q + 1)-line and so exactly two (q + 1)-lines. Counting in double way the incident point-line pairs (p, m) where m is a (q + 1)-line gives 2v π = (q + 2)(q + 1).
Let p be a point of π \ H, and let y be the number of √ q-lines through p. Then, and so q ≤ 2 √ q. Hence q = 4 (and y = 0) against the assumption.

Remark 3.2.
Actually, in the proof of the above Lemma we have used neither the fact that the plane is embedded in PG (3, q) nor that the plane is desarguesian. Indeed, we have proved the following result: Let π n be a projective plane of square order n and let X be a subset of points of π n such that • there is at least a point of π n outside X, • every line of π n with | ∩ X| ≥ √ n + 2 is contained in X, then |X| ≤ n √ n + n + 1 n = 4 (and so π n is desarguesian) |X| = 15 and X is the complement of a set of line-type (0, 2) 1 in PG (2,4), that is of a 6-arc. Proof. If there is a point p ∈ H such that every line on it shares at most √ q +1 points with H then which is not possible.
It follows that there is no external plane.

Proposition 3.4.
If is a (q + 1)-line, then every plane through intersects H into exactly q √ q + q + 1 points.
Thus, q √ q +q +1 is the maximal size for the intersections of H with the planes. Now, we are ready to complete the proof of Theorem I.
Let m the minimal size of the intersections of H with the planes of PG (3, q).
Since there is no external plane then m ≥ q √ q + 1 and g(k, m, q √ q + q + 1) ≤ 0.
And if there is no external line, by Theorem 2.2 it follows that Theorem I is completely proved.

Author contributions
The author wrote and reviewed the paper.
Funding Information Open access funding provided by Università degli Studi della Campania Luigi Vanvitelli within the CRUI-CARE Agreement.
Data availability statement No datasets were generated or analyzed during the current study.

Declarations
Conflict of interest I declare that that there is no conflict of interest.
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