Volume 2013 (2013), Article ID 589362, 3 pages
Note on a Class of Subsets of AG(3, q) with Intersection Numbers 1, q and n with respect to the Planes
Dipartimento di Ingegneria Civile, Seconda Università degli Studi di Napoli, Real Casa dell’Annunziata, Via Roma 29, 81031 Aversa, Italy
Received 28 June 2013; Accepted 8 October 2013
Academic Editor: Changzheng Qu
Copyright © 2013 Vito Napolitano. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We give a new and correct proof of a result of O. Ferri and S. Ferri (1995) on -caps of in this paper; moreover we prove that sets of of type with respect to the planes of have size at most with equality if and only if is a cap.
Let denote either a finite projective space or a finite affine space, and let be a set of nonnegative integers with . A subset of is of class with respect to the subspaces of dimension of if any subspaces intersect either in or points, and is of type with respect to the subspaces of dimension if for every integer , there exists a -subspace meeting in exactly points. The numbers are called intersection numbers of . As usual, by a -set we mean a set of size .
In the literature one can find many papers devoted to the study of -sets of given type, not only in affine and projective geometries (cf., e.g., [1–15]), and most of these results are characterizations of classical geometric objects. Recently, characterizations of Hermitian varieties and quadrics of as -sets with given intersection numbers with respect to more than one family of subspaces (e.g., with respect to planes and solids) have been considered [16, 17].
A cap of an affine or projective space of dimension ≥3 is a subset of points no three of which are collinear.
In 1995, O. Ferri and S. Ferri  gave a characterization of -caps of in terms of sets with three given intersection numbers with respect to the planes. Their result reads as follows.
Theorem 1 (see ). Let be a subset of with points and of type . Then, and is a cap of .
Unfortunately, a step of the proof of that theorem is not correct; in fact it contains a counting argument which does not give the contradiction they want (see  page 71 line +7). However, the statement of the result is true as we are going to prove in Lemma 4.
In this paper we will prove the following slight extension of the O. Ferri and S. Ferri result.
Theorem 2. Let be a set of of size and with three intersection numbers 1, , and . Then , and if and only if . Moreover, if the set is a cap.
Thus, it follows that the sets of type of have size at most and that equality holds if and only if they are caps.
2. Proof of Theorem 1
In this section, first, we briefly recall the basic equations for a -set of with three intersection numbers, and then we will assume that and we will give the proof of Theorem 1.
2.1. The Basic Equations for -Sets with Intersection Numbers 1, , and
Let , , denote the number of planes intersecting in exactly -points (such numbers are called characters of ).
Double counting gives
From (1) it follows that
2.2. -Sets of with Three Intersection Numbers
From now on, is a -set of with and with intersection numbers 1, , and with respect to the planes.
Since , there are at least two distinct points in . Let and be two points of , and let be the line connecting them. Put . Namely, .
Let denote the number of planes on intersecting in points. Counting via the planes passing through , we have
Being and it follows that ; that is, .
Lemma 3. If , then , , and is a -cap.
Proof. Let denote the number of points of a line meeting in at least two points. For (5) gives . Thus, and . If , then , and so there would be no plane intersecting in exactly points, against the assumptions. Thus , , , and is a cap.
Lemma 4 (see S. Ferri and O. Ferri ). If , then .
Proof. From it follows that (2) becomes and (3) . Assume that . Let , with a prime number, and an integer. We have that is a divisor of and , but this is impossible since the difference of these numbers is . Thus, .
Let us end with the following easy consequence of Theorem 2.
Corollary 5. A subset of points of of type has size , and if and only if is a cap of .
This research was partially supported by G.N.S.A.G.A. of INdAM.
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