On Quasi-Hermitian Varieties A. Aguglia,1 A. Cossidente,2 G. Korchmáros 2 1 Dipartimento di Matematica,Politecnico, di Bari,Via Amendola 126/B, Bari, I-70126, Italy, E-mail: aguglia@poliba.it 2 Dipartimento di Matematica, e Informatica, Università degli Studi della Basilicata, Campus Macchia Romana, Viale dell’Ateneo Lucano, 10, Potenza, I-85100, Italy, E-mail: cossidente@unibas.it; korchmaros@unibas.it Received April 19, 2011; revised April 18, 2012 Published online in Wiley Online Library (wileyonlinelibrary.com). DOI 10.1002/jcd.21317 Abstract: Quasi-Hermitian varieties V in PG(r, q 2 ) are combinatorial generalizations of the (nondegenerate) Hermitian variety H(r, q 2 ) so that V and H(r, q 2 ) have the same size and the same intersection numbers with hyperplanes. In this paper, we construct a new family of quasi-Hermitian varieties. The isomorphism problem for the associated strongly regular graphs C 2012 Wiley Periodicals, Inc. J. Combin. Designs : 1–15, 2012 is discussed for r = 2.  Keywords: Hermitian variety; quadric; two-character sets 1. INTRODUCTION A two-character set in the projective space PG(r, q) is a set S of n points with the property that the intersection number with any hyperplane only takes two values, n − w1 and n − w2 . Then the positive constants w1 and w2 are called the weights of the two-character set. Embed now PG(r, q) as a hyperplane  in PG(r + 1, q). The linear representation graph Ŵd∗ (S) is the graph having as vertices the points of PG(r + 1, q) \  and where two vertices are adjacent whenever the line defined by them meets S. It follows that Ŵr∗ (S) has v = q r+1 vertices and valency k = (q − 1)n. In 1972, Delsarte [8] proved that this graph is strongly regular if S is a two-character set [3, 8]. If the two-character set S is also transitive, i.e., it has a transitive automorphism group, then the strongly regular graph Ŵr∗ (S) is also symmetric. The other parameters of the graph Ŵr∗ (S) are λ=k 2 +3k−q(w1 +w2 )−kq(w1 +w2 ) + q 2 w1 w2 ,μ = k 2 + k − kq(w1 +w2 )+q 2 w1 w2 . Regarding the coordinates of the elements of S as columns of the generator matrix of a code L of length n and dimension r + 1, then the two-character set property of S Journal of Combinatorial Designs  C 2012 Wiley Periodicals, Inc. 1 2 AGUGLIA, COSSIDENTE, AND KORCHM translates into the fact that the code L has two (nonzero) weights (w1 and w2 ). Such a code is said to be a projective two-weight code. The weights of the code are exactly the weights of the two-character set. An updated overview of this subject is found in [4, 5]. In this paper, we are interested in certain two-character sets in PG(r, q 2 ) which have the same size as a Hermitian variety. If a two-character set V and a Hermitian variety in PG(r, q 2 ) have the same intersection numbers with hyperplanes then V is a quasiHermitian variety. Note that a quasi-Hermitian variety is not necessarily an algebraic variety. The intersection numbers of a quasi-Hermitian variety with hyperplanes are (q r + (−1)r−1 )(q r−1 − (−1)r−1 ) , q2 − 1 and (q r + (−1)r−1 )(q r−1 − (−1)r−1 ) + (−1)r−1 q r−1 . q2 − 1 Obviously, a Hermitian variety can be viewed as a classical quasi-Hermitian variety. Recently, De Winter and Schillewaert [9] constructed two infinite families of nonclassical quasi-Hermitian varieties. Both consist of quasi-Hermitian varieties arising from a Hermitian variety H(r, q 2 ) by modifying some point-hyperplane incidences at the points in just one tangent space to H(r, q 2 ). They used the pivoting method originally developed for the construction of quasi-quadrics in [6], and also a Hermitian analogue of a theorem of Delanote; see [7]. As far as we know, these are the only known quasi-Hermitian varieties. In this paper, we construct a new infinite family of nonclassical quasi-Hermitian varieties. Our procedure is different from that used in [9], since it does not consist in some local modification of H(r, q 2 ). In fact, we modify many point-hyperplane incidences. To do this, we keep the points of PG(r, q 2 ) but replace the hyperplanes with their images under a quadratic transformation. In other words, we use a nonstandard model  of PG(r, q 2 ), where (i) points of  are those of PG(r, q 2 ); (ii) hyperplanes of  are certain hyperplanes and quadrics of PG(r, q 2 ); so that H viewed as a pointset of , be a quasi-Hermitian variety. The choice of such a nonstandard model requires some preliminary results of independent interest which generalize analogous results stated in [1] for r = 2. We also prove that no collineation of PG(r, q 2 ) maps our quasi-Hermitian variety to one constructed in [9]. In the lowest dimension case, r = 2, our construction provides classical unitals and Bukenehot–Metz unitals. In Section 6, we discuss some properties of the associated strongly regular graphs. In particular, the automorphism group G of the strongly regular graph arising from the classical unital U is the inherited group, see Remark 6. Theorem 6.5 states that two nonisomorphic strongly regular graphs are obtained from two unitals whenever one is classical while the other is not. It may be that this theorem extends to any dimension r although the higher dimensional generalization of the Journal of Combinatorial Designs DOI 10.1002/jcd 3 ON QUASI-HERMITIAN VARIETIES key Proposition 6.1 might be much more complicated for quasi-Hermitian varieties and, consequently, dealing with such a phenomena might require new technical tools. Our notation and terminology are standard. For generalities on Hermitian varieties in projective spaces, the reader is referred to [14, 19]. Basic facts on rational transformations of projective spaces are found in [15], Section 3.3]. 2. A NONSTANDARD MODEL OF PG(r, q 2 ) Fix a projective frame in PG(r, q 2 ) with homogeneous coordinates (X0 , X1 , . . . , Xr ), and consider the affine plane AG(r, q 2 ) whose infinite hyperplane ∞ has equation X0 = 0. Then, AG(r, q 2 ) has affine coordinates (x1 , x2 , . . . , xr ), where xi = Xi /X0 for i ∈ {1, . . . , r}. Fix a nonzero element a ∈ GF(q 2 ). For m = (m1 , . . . , mr−1 ) ∈ GF(q 2 )r−1 , d ∈ GF(q 2 ) and a ∈ GF(q 2 )∗ , let Qa (m, d) denote the quadric of equation   2 + m1 x1 + · · · + mr−1 xr−1 + d. xr = a x12 + · · · + xr−1 (1) Consider the incidence structure a = (P, ) whose points are the points of AG(r, q 2 ) and whose hyperplanes are the hyperplanes through the infinite point P∞ (0, 0, . . . , 0, 1) together with the quadrics Qa (m, d), where m and d range over GF(q 2 )r−1 and GF(q 2 ), respectively. Lemma 2.1. For every nonzero a ∈ GF(q 2 ), the incidence structure a = (P, ) is an affine space isomorphic to AG(r, q 2 ). Proof. The birational transformation ϕ given by    2 ϕ : (x1 , . . . , xr−1 , xr ) → x1 , . . . , xr−1 , xr − a x12 + · · · + xr−1 , (2) transforms the hyperplanes through P∞ (0, 0, . . . , 0, 1) into themselves, whereas the hyperplane of equation xr = m1 x1 + · · · + mr−1 xr−1 + d is mapped into the quadric Qa (m, d). Therefore, ϕ determines an isomorphism a ≃ AG(r, q 2 ), and the assertion is proven.  Completing a with its points at infinity in the usual way gives a projective space isomorphic to PG(r, q 2 ). 3. THE CONSTRUCTION The Hermitian variety H is assumed in an affine canonical form  q+1 q+1  xrq − xr = (bq − b) x1 + · · · + xr−1 , Journal of Combinatorial Designs DOI 10.1002/jcd (3) 4 AGUGLIA, COSSIDENTE, AND KORCHM where b ∈ GF(q 2 ) \ GF(q). The set of the infinity points of H is  q+1   q+1 F = (0, x1 , . . . , xr )x1 + · · · + xr−1 = 0 (4) and it can be viewed as a Hermitian cone of PG(r − 1, q 2 ) projecting a Hermitian variety of PG(r − 2, q 2 ). Theorem 3.1. Let a ∈ GF (q 2 )∗ and b ∈ GF (q 2 ) \ GF (q). The affine algebraic variety B of equation  q+1  2q   2q  q+1  2 xrq − xr + a q x1 + · · · + xr−1 − a x12 + · · · + xr−1 = (bq − b) x1 + · · · + xr−1 , (5) together with the infinity points (4) of H is quasi-Hermitian variety V of PG(r, q 2 ) provided that the following conditions are satisfied. For odd q, (1) r is odd and 4a q+1 + (bq − b)2 = 0, or (2) r is even and 4a q+1 + (bq − b)2 is a nonsquare in GF(q). For even q, (i) r is odd, or (ii) r is even and Tr (a q+1 /(bq + b)2 ) = 0. Proof. Any point P = (ξ1 , . . . , ξr ) in the model a can be regarded as a point of 2 AG(r, q 2 ) with coordinates xi = ξi , for 1 ≤ i ≤ r − 1 and xr = ξr + a(ξ12 + · · · + ξr−1 ). Therefore H and V coincide in . Bearing in mind the definition of , we only need the following lemma where Qa (m, d) denotes the quadric with equation (1). The quadric Qa (m, d) and H have either Lemma 3.2. (q r + (−1)r−1 )(q r−1 − (−1)r−1 ) − |H(r − 2, q 2 )| q2 − 1 or (q r + (−1)r−1 )(q r−1 − (−1)r−1 ) + (−1)r−1 q r−1 − |H(r − 2, q 2 )| q2 − 1 common points in AG(r, q 2 ). Proof. The number of solutions (x1 , . . . , xr ) ∈ GF(q 2 )r of the system   q+1 q q+1  xr − xr = (bq − b) x1 + · · · + xr−1   2 a x12 + · · · + xr−1 + m1 x1 + · · · + mr−1 xr−1 − xr + d = 0 (6) gives the number of common points of H with Qa (m, d). To solve this system, recover the value of xr from the second equation and substitute it in the first. The result is Journal of Combinatorial Designs DOI 10.1002/jcd 5 ON QUASI-HERMITIAN VARIETIES  q+1  2q 2q  q+1  q q a q x1 + · · · + xr−1 + (b − bq ) x1 + · · · + xr−1 + m1 x1   q q 2 + · · · + mr−1 xr−1 + d q − a x12 + · · · + xr−1 − m1 x1 + · · · + (7) − mr−1 xr−1 − d = 0. Consider now GF(q 2 ) as a vector space over GF(q), fix a basis {1, ε} with ε ∈ GF(q 2 ) \ GF(q), and write the elements in GF(q 2 ) as linear combinations with respect to this basis, that is, xi = xi0 + xi1 ε, with xi0 , xi1 ∈ GF(q). Thus, (7) becomes an equation over GF(q). We investigate separately the even q and odd q cases. For odd q choose a primitive element β of GF(q 2 ) and let ε = β (q+1)/2 . Then, ε q = −ε and ε 2 is a primitive element of GF(q). With this choice of ε, (7) becomes  2  2 (b1 + a 1 )ε 2 x11 + 2a 0 x10 x11 + (a 1 − b1 ) x10 + · · · + (b1 + a 1 )ε 2  1 2  0 2 0 1 × xr−1 + 2a 0 xr−1 xr−1 + (a 1 − b1 ) xr−1 + m01 x11 + m11 x10 (8) 0 1 + · · · + m0r−1 xr−1 + m1r−1 xr−1 + d 1 = 0. 0 1 It is convenient to represent the solutions (x10 , x11 , . . . , xr−1 , xr−1 ) of (8) as points of the affine space AG(2(r − 1), q) over GF(q). In fact, (8) turns out to be the equation of a (possibly degenerate) affine quadric of AG(2(r − 1), q) whose number of points equals the number of points in AG(r, q 2 ) which lie in H ∩ Qm,d . Now compute this number. For this purpose, count first the number of infinite points of . These points are those of the quadric ∞ in PG(2(r − 1) − 1, q) with quadratic form associated to the symmetric 2(r − 1) × 2(r − 1) matrix ⎛ (a 1 − b1 ) a0 ... 0 0 a0 .. . (b1 + a 1 )ε 2 ... 0 0 .. . 0 0 ... (a 1 − b1 ) a0 0 0 ... a0 (b1 + a 1 )ε 2 ⎜ ⎜ ⎜ A=⎜ ⎜ ⎜ ⎝ ⎞ ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎠ Since a q = a 0 − εa 1 and bq − b = 2εb1 , det(A) = {(a 0 )2 − ε 2 [(a 1 )2 − (b1 )2 ]}r−1 = [a q+1 + (bq − b)2 /4]r−1 . By assumption (i) and (ii), ∞ is a nonsingular quadric and it is either hyperbolic or elliptic according as r is odd or even; see [16], Theorem 22.2.1]. First suppose that r is odd. If is nonsingular then the number N of its affine points is N= If (q r−1 + 1)(q r−1 − 1) (q r−2 + 1)(q r−1 − 1) − = q r−2 (q r−1 − 1). q −1 q −1 is singular then N= is a cone, and q(q r−2 + 1)(q r−1 − 1) (q r−2 + 1)(q r−1 − 1) − + 1 = q r−2 (q r−1 − 1) + q r−1 . q −1 q −1 Journal of Combinatorial Designs DOI 10.1002/jcd 6 AGUGLIA, COSSIDENTE, AND KORCHM Now assume that r is even. When N= If is nonsingular, (q r−1 + 1)(q r−1 − 1) (q r−1 + 1)(q r−2 − 1) − = q r−2 (q r−1 + 1). q −1 q −1 is singular then N= q(q r−1 + 1)(q r−2 − 1) (q r−1 + 1)(q r−2 − 1) − + 1 = q r−2 (q r−1 + 1) − q r−1 , q −1 q −1 and our lemma follows for odd q. For even q, choose ε ∈ GF(q 2 ) \ GF(q) such that ε 2 + ε + v = 0, for some v ∈ GF(q) \ {1} with Tr (v) = 1. Then ε2q + ε q + v = 0. Therefore, (εq + ε)2 + (ε q + ε) = 0, whence ε q + ε + 1 = 0. With this choice of ε, (7) reads  2  2   (a 1 + b1 ) x10 + [(a 0 + a 1 ) + ν(a 1 + b1 )] x11 + b1 x10 x11 + m11 x10 + m01 + m11 x11  0 2  1 2 0 1 + · · · + (a 1 + b1 ) xr−1 + [(a 0 + a 1 ) + ν(a 1 + b1 )] xr−1 + b1 xr−1 xr−1   0 1 + m1r−1 xr−1 + m0r−1 + m1r−1 xr−1 + d 1 = 0. (9) The discussion of the (possibly degenerate) quadric may similarly be carried out. However, according to the remark before [16], Theorem 22.2.1], precaution is needed when quadrics and their classifications are studied in even characteristic. For this reason we provide some technical details. Consider the algebraic number field Q(θ), where θ q + θ + 1 = 0, and set ν = θ 2 + θ. Further, let α 0 , α 1 , β 0 , β 1 , denote indeterminates, and define the matrix A(α 0 , α 1 , β 0 , β 1 ) of order 2(r − 1) with enters in the polynomial ring Q(θ)[α 0 , α 1 , β 0 , β 1 ]: ⎛ 1 ⎞ β1 ... 0 0 2(α + β 1 ) ⎜ ⎟ ⎜ ⎟ β1 2(α 0 + α 1 + ν(α 1 + β 1 )) . . . 0 0 ⎜ ⎟ ⎜ ⎟ .. .. ⎜ ⎟ . . ⎜ ⎟ ⎜ ⎟ 1 1 1 0 0 . . . 2(α + β ) β ⎝ ⎠ 0 0 ... β1 2(α 0 + α 1 + ν(α 1 + β 1 )) Formally, A(α 0 , α 1 , β 0 , β 1 ) is twice the usual matrix associated to the quadric equation (9). Its determinant is equal to ∞ of det A(α 0 , α 1 , β 0 , β 1 ) = [4(α 1 + β 1 )(α 0 + α 1 + ν(α 1 + β 1 )) + (β 1 )2 ]r−1 . This function can be regarded as being in the polynomial ring GF(q 2 )[α 0 , α 1 , β 0 , β 1 ]. Evaluating it in (a 0 , a 1 , b0 , b1 ) gives b12 . Here b1 = 0, by our assumption bq = b. From [16], Theorem 22.2.1 (i)], the quadric must be nondegenerate. Further, [16], Theorem 22.2.1 (ii)] and the successive Lemma 22.2.2 explain how to decide weather is hyperbolic or elliptic. Let B(α 0 , α 1 , β 0 , β 1 ) the matrix obtained from A(α 0 , α 1 , β 0 , β 1 ) by omitting all entries from its diagonal, and define α= det B(α 0 , α 1 , β 0 , β 1 ) − (−1)r−1 det A(α 0 , α 1 , β 0 , β 1 ) . 4 det B(α 0 , α 1 , β 0 , β 1 ) Journal of Combinatorial Designs DOI 10.1002/jcd 7 ON QUASI-HERMITIAN VARIETIES A straightforward computation shows that (β 1 )2(r−1) ± (4(α 1 + β 1 )(α 0 + α 1 + ν(α 1 + β 1 )) + (β 1 )2 )r−1 4 (β 1 )2(r−1) (r − 1)(α 1 + β 1 )(α 0 + α 1 + ν(α 1 + β 1 )(β 1 )2(r−2) = + 2u, (β 1 )2(r−1) α= where ± stands for + or − according as r is even or odd, and u = w/(β 1 )2(r−1) with w ∈ Q(θ)[α 0 , α 1 , β 0 , β 1 ]. If r odd, then r − 1 is even and hence α = 2v/w with v ∈ Q(θ)[α 0 , α 1 , β 0 , β 1 ]. Now, evaluating α in (a 0 , a 1 , b0 , b1 ) gives zero. Therefore, ∞ is hyperbolic. If r even, then r − 1 is odd, and hence evaluating α in (a 0 , a 1 , b0 , b1 ) gives (a 1 + b1 )(a 0 + a 1 + ν(a 1 + b1 )) . (b1 )2 From [16], Theorem 22.2.1(ii)], the trace Tr = TrGF(q 2 )|GF(2) of this is either 0 or 1 according as ∞ is hyperbolic or elliptic. Since  02    (a ) + a 0 a 1 + ν(a 1 )2 a q+1 Tr = Tr , (b1 )2 (bq + b)2 our assumption yields Tr  (a 0 )2 + a 0 a 1 + ν(a 1 )2 (b1 )2  = 0. Furthermore, Tr  a0 + a1 b1 2 +  a0 + a1 b1  = 0. From these equations, Tr  (a 1 + b1 )(a 0 + a 1 + ν(a 1 + b1 )) (b1 )2 This completes the proof of Theorem 3.1.  = 1.   Theorem 3.3. The quasi-Hermitian variety V defined in Theorem (3.1) is not projectively equivalent to the Hermitian variety of PG(r, q 2 ). Proof. First assume r = 2. In this case, V coincides with a (nonclassical) Buekenhout– Metz unital and hence V is not projectively equivalent to the Hermitian curve of PG(2, q 2 ); see [1, 12]. In the case where r > 2 let π be the plane of affine equations x2 = · · · = xr−1 = 0, Journal of Combinatorial Designs DOI 10.1002/jcd 8 AGUGLIA, COSSIDENTE, AND KORCHM and let U denote the intersection of V and π. We can choose homogeneous projective coordinates in π in such a way that U is the set of points U = {(1, t, −at 2 + bt q+1 + r)|t, r ∈ GF(q)} ∪ {(0, 0, 1)}. Since such a set is a (nonclassical) Buekenhout–Metz unital of π, the theorem follows.  4. A LINEAR COLLINEATION GROUP PRESERVING V In this section, we determine a subgroup of the stabilizer of V in P GL(r + 1, q 2 ). For each γ = (γ1 , . . . , γr−1 ) ∈ GF(q 2 )r−1 let ψγ be the collineation of PG(r, q 2 ) associated with the nonsingular matrix ⎛ 1 γ1 ⎜ ⎜0 1 ⎜ ⎜0 0 ⎜ ⎜. ⎜ .. ⎜ ⎜ ⎝0 0 0 0 γ2 . . . γr−1 0 ... 0 1 ... 0 0 ... 1 0 ... 0    q+1 q+1 ⎞ 2 −a γ12 + · · · + γr−1 + b γ1 + · · · + γr−1 ⎟ q ⎟ (b − bq )γ1 − 2aγ1 ⎟ q ⎟ q (b − b )γ2 − 2aγ2 ⎟ ⎟. .. ⎟ . ⎟ ⎟ q q (b − b )γr−1 − 2aγr−1 ⎠ 1 for odd q and ⎛ 1 γ1 γ2 ... γr−1 ⎜ ⎜0 ⎜ ⎜0 ⎜ ⎜. ⎜ .. ⎜ ⎜ ⎝0 1 0 ... 0 0 1 ... 0 0 0 ... 1 0 0 ... 0 0    q+1 q+1 ⎞ 2 a γ12 + · · · + γr−1 + b γ1 + · · · + γr−1 ⎟ q ⎟ (b − bq )γ1 ⎟ q ⎟ q (b − b )γ2 ⎟ ⎟. .. ⎟ . ⎟ ⎟ q (b − bq )γr−1 ⎠ 1 for even q. Furthermore, for each s ∈ GF(q), let φs denote the collineation associated with the nonsingular matrix ⎛ 1 0 ... 0 s ⎞ ⎜0 1 . . . 0⎟ ⎜ ⎟ ⎜ ⎟ ⎜ .. .. ⎟. ⎜. .⎟ ⎜ ⎟ ⎜ ⎟ 0 0 . . . 1 0 ⎝ ⎠ 0 0 ... Journal of Combinatorial Designs DOI 10.1002/jcd 0 1 9 ON QUASI-HERMITIAN VARIETIES The collineations ψγ and φs fix the point P∞ , preserve both V and the infinite hyperplane. A straightforward computation shows that φs ψγ = ψγ φ s , φs φt = φs+t = φt φs , q ′q ′q ψγ ψγ ′ = ψγ +γ ′ φ−(b(γ1q γ1′ +···+γr−1 ′ γr−1 )+bq (γ1 γ1 +···+γr−1 γr−1 )) , q ′q ′q ψγ φs ψγ ′ φt = ψγ +γ ′ φs+t−(b(γ1q γ1′ +···+γr−1 ′ γr−1 )+bq (γ1 γ1 +···+γr−1 γr−1 )) . It follows that the set S = {φs ψγ |s ∈ GF(q), γ ∈ GF(q 2 )r−1 } (10) is a nonabelian collineation group of order q 2r−1 that fixes P∞ and stabilizes V. Furthermore, K = {φs |s ∈ GF(q)} is an elementary abelian p-group of order q that fixes P∞ , each point on the hyperplane at infinity, and stabilizes V. Finally, since the stabilizer of any affine point of V in S consists only of the identity collineation then S is transitive on the set of affine points of V. Lemma 4.1. Let δ ∈ GF (q 2 )∗ and let μδ denote the collineation induced by the nonsingular matrix ⎛ 1 0 ... 0 ⎜0 δ ... ⎜ ⎜ ⎜ .. ⎜. ⎜ ⎜ ⎝0 0 . . . δ 0 0 ... 0 0 ⎞ 0⎟ ⎟ ⎟ .. ⎟. .⎟ ⎟ ⎟ 0⎠ δ2 μδ leaves V invariant if and only if δ ∈ GF (q). Proof. We first observe that μδ fixes the point P∞ and in general leaves invariant the set of points at infinity in V. q+1 q+1 2 ) + b(x1 + · · · + xr−1 ) + m) deNow, let M(1, x1 , . . . , xr−1 , −a(x12 + · · · + xr−1 note any affine point of V where m ∈ GF(q) is arbitrarily chosen. The image μδ (M) is on V if and only if  q+1  q+1 q+1  q+1  bδ 2 x1 + · · · + xr−1 + δ 2 m − bδ q+1 x1 + · · · + xr−1 ∈ GF(q), q+1 q+1 that is b(δ 2 − δ q+1 )(x1 + · · · + xr−1 ) + δ 2 m ∈ GF(q) for all (x1 , . . . , xr−1 ) ∈ GF (q 2 )r−1 and for all m ∈ GF(q). Taking (x1 , . . . , xr−1 ) = (0, . . . , 0) and m = 1, we have that δ 2 ∈ GF(q). Thus, since q+1 q+1 x1 + · · · + xr−1 ∈ GF(q) for all (x1 , . . . , xr−1 ) ∈ GF(q 2 )r−1 we have that μδ leaves V invariant if and only if b(δ 2 − δ q+1 ) ∈ GF(q) and δ 2 ∈ GF(q). Since b ∈ / GF(q), the necessary and sufficient condition becomes δ 2 = δ q+1 , that is q−1 δ = 1 and hence δ ∈ GF(q).  Journal of Combinatorial Designs DOI 10.1002/jcd 10 AGUGLIA, COSSIDENTE, AND KORCHM Theorem 4.2. Let J be the cyclic subgroup generated by μδ , where δ is some primitive element of GF (q). Then J is a collineation group of order q − 1, fixes P∞ and preserves V. Furthermore, J normalizes the group S, as defined in (10), and intersects S trivially. Proof. This follows from the above lemma and the following relation μ−1 δ ψγ φs μδ = ψδγ φsδ 2 .  Corollary 4.3. The semidirect product H of S by the cyclic group J is a collineation group of order (q − 1)q 2r−1 that stabilizes V and acts transitively on the set of affine points of V. Remark 4.4. 5. For r = 2, H coincides with the stabilizer in P GL(3, q 2 ) of V ; see [12]. THE ISOMORPHISM PROBLEM FOR QUASI-HERMITIAN VARIETIES Two quasi-Hermitian varieties in PG(r, q 2 ) are said to be isomorphic if there is a collineation of PG(r, q 2 ) mapping one onto the other. In this section, we show that our quasi-Hermitian varieties are not isomorphic to those constructed in [9]. One of the families in [9] consists of quasi-Hermitian varieties obtained from H(r, q 2 ) by using the Hermitian analog of the pivoting method. Let P be a point on H(r, q 2 ). The tangent space T of H(r, q 2 ) at P meets the Hermitian variety in a cone with vertex P and base a nonsingular Hermitian variety H (r − 2, q 2 ). Replacing H(r − 2, q 2 ) by a quasi-Hermitian variety H ′ in PG(r − 2, q 2 ) gives a set which is defined to be the pivoted set P of H(r, q 2 ) with respect to P . In [9], the authors proved that every pivoted set P of H(r, q 2 ) with respect to a point P od H(r, q 2 ) is a quasi-Hermitian variety in PG(r, q 2 ). The second family in [9] can be viewed as a generalization of the first one. In this case, r ≥ 5 must be odd and the pivoting applies in the tangent space of an (r − 3)/2dimensional isotropic subspace of H(r, q 2 ). In both cases, the difference between H(r, q 2 ) and a quasi-Hermitian variety P arising by pivoting is only in the tangent space of a point. Thinking of this tangent space as the infinite hyperplane of AG(r, q 2 ), we may say that H(r, q 2 ) and P coincide in their affine parts. Theorem 5.1. The quasi-Hermitian variety V defined in Theorem (3.1) is not isomorphic to a quasi-Hermitian variety P constructed by the pivoting method. Proof. By a way of contradiction assume that there is a projectivity φ from V onto P. From the proof of Theorem 3.3, it follows that some affine planar section of V is a (nonclassical) Buekenhout–Metz unital. By Corollary 4.3, the automorphism group of V is transitive on the set of affine points of V thus, trough every affine point A of V there pass a plane πA which intersects V in a (nonclassical) Buekenhout–Metz unital UA . Take an affine point A of V such that φ(A) ∈ / T . In this case, the intersection of φ(πA ) with the hyperplane T is a line ℓ and hence the nonclassical unital φ(UA ) and the Hermitian curve C = φ(πA ) ∩ H(r, q 2 ) would differ only by one or q + 1 collinear points accordingly to ℓ is either tangent or secant to C, and this is impossible.  Journal of Combinatorial Designs DOI 10.1002/jcd ON QUASI-HERMITIAN VARIETIES 11 Remark 5.2. From the proof of Theorem 5.1, the isomorphism problem between our quasi-Hermitian variety and those obtained by multiply pivoting, say s pivoting, is related to the problem of classifying the possible intersections between a Hermitian curve and a Buekenhout–Metz unital in PG(2, q 2 ). This classification is well known to be an open and difficult problem. Knowing the maximum size of the intersection between a Hermitian curve and a Buekenhout–Metz unital in PG(2, q 2 ) would give some information about our isomorphism problem. Let μ denote the maximum size of the intersection between a Hermitian curve and a Buekenhout–Metz unital. With the argument used in the proof of Theorem 5.1, one can show that if s < (q 3 + 1 − μ)/(q + 1) then P cannot be isomorphic to V. 6. STRONGLY REGULAR GRAPHS ARISING FROM UNITALS IN PG(2, q 2 ) In the lowest dimensional case, r = 2, our construction provides either a classical unital or a Bukenehot–Metz unital. We deal with the associated strongly regular graphs, especially with the relative isomorphism problem. Let U be a unital in PG(2, q 2 ). Embed PG(2, q 2 ) in PG(3, q 2 ) as a plane , and regard PG(3, q 2 ) \  as a three-dimensional affine space AG(3, q 2 ) with infinite plane . The linear representation graph Ŵ2∗ (U) associated to U is the strongly regular graph whose vertices are the points of AG(3, q 2 ) and whose edges are the pairs {P , Q} where P and Q are two distinct vertices collinear with a point of U. We establish some property of the strongly regular graph Ŵ2∗ (U) arising from a classical unital U. A key issue is to determine all cliques of Ŵ2∗ (U). We will use the fact that no O’Nan configuration lies on a classical unital; see [18] and also [2], Section 4.2]. Recall that an O’Nan configuration consists of six points, namely the four vertices of a nondegenerate quadrangle together with two of their three diagonal points. Proposition 6.1. Let U be a classical unital. Then every maximal clique of Ŵ2∗ (U) has size q 2 . More precisely, maximal cliques of Ŵ2∗ (U) are of two types: (i) the set of all points of a line of AG(3, q 2 ) whose infinite point lies on U. (ii) the set of all affine Baer subplanes of AG(3, q 2 ) whose infinite line is a chord of U. Proof. We begin by showing that every clique of Ŵ2∗ (U) is contained in a plane. For this purpose, suppose on the contrary that the vertices of a nondegenerate tetrahedron A1 A2 A3 A4 of AG(3, q 2 ) belong to a clique. For 1 ≤ i < k ≤ 4, let Aij be the infinite point of the edge Ai Aj . First observe that {A12 , A13 , A23 } is a collinear triple. In fact, they lie on the infinite line of the face A1 A2 A3 . Similarly, {A13 , A14 , A34 } and {A12 , A14 , A24 } are collinear triples. Hence, the above six points Aij determine an O’Nan configuration on U, a contradiction. Obviously, lines with infinite point in U are maximal cliques of Ŵ2∗ (U). This gives case (i). In particular, neither a single point, nor a pair of points are maximal cliques. Now, take a further maximal clique C of Ŵ2∗ (U), and fix two points, say P , Q ∈ C. Furthermore, let R denote another point from C. Let α be the unique plane of AG(3, q 2 ) containing C. The infinite line ℓ of α is not a tangent of U. Therefore, ℓ ∩ U is a Baer subline of . Let β be the unique affine Baer subplane through P , Q whose infinite line is ℓ ∩ U. Since both lines P R and QR meet ℓ ∩ U, their common point lies in β, as well. Journal of Combinatorial Designs DOI 10.1002/jcd 12 AGUGLIA, COSSIDENTE, AND KORCHM This shows that R ∈ β. Therefore, C is contained in β. Obviously, any two points lying on β are adjacent. This gives case (ii).  Proposition 6.2. Let U be a classical unital. Then the total number of maximal cliques of Ŵ2∗ (U) is equal to q 4 (q 2 + 1)(q 3 + 1). Proof. The number of chords of U is q 4 − q 3 + q 2 . Each of these chords is the infinite line of q 2 planes of AG(3, q 2 ). Each such plane contains exactly q 2 (q + 1) affine Baer subplanes whose infinite line is the set of the common points of a given chord of U. Therefore, the number of maximal cliques of type (ii) is equal to (q 4 − q 3 + q 2 )q 4 (q + 1). Furthermore, each of the q 3 + 1 points of U is the infinite point of q 4 lines of AG(3, q 2 ). Therefore, the number of maximal cliques of type (i) is (q 3 + 1)q 4 . By Proposition 6.1 all maximal cliques of Ŵ2∗ (U) are obtained in this way. This completes the proof.  Remark 6.3. Let q = 3. From Proposition 6.2, the number of maximal cliques of Ŵ2∗ (U) for a classical unital U is 22, 680. Now, fix a primitive element of GF(9) and let U ′ be the Buekenhout–Metz unital consisting of the point P∞ = (0, 0, 1) together with the points Pt = (1, t, at 2 + bt 4 + r) with t ranging over GF(9) and r ranging over GF(3), such that 4a 4 + (b3 − b)2 is a nonsquare in GF(3). A computer aided exhaustive search shows that Ŵ2∗ (U ′ ) has maximal cliques of sizes 4 and 9. More precisely, the number N4 of its maximal cliques of size 4 is 3, 149, 280 and the number of its maximal cliques of size 9 is N9 = 5, 184. Proposition 6.1 has the following consequence. Proposition 6.4. Let U be a classical unital. Then the number of vertices which are adjacent to three distinct vertices lying on a Baer subline of AG(3, q 2 ) with infinite point in U is equal to (q 2 + 1)(q 2 − q) + q − 3. Proof. Consider an affine Baer subline ℓ together with three of its points, say P , Q, and R. From Proposition 6.1, each maximal clique containing P , Q, R is either an affine Baer subplane or the line ℓ̄ of AG(3, q 2 ) containing ℓ. Observe that the infinite point L of ℓ lies on U. Choose an affine Baer subplane π containing ℓ. Since every chord of U is a Baer subline, the infinite line of π is a chord of U through L. The (affine) points of π other than those lying on ℓ count q 2 − q. Since there are q 2 such chords, we have the same number of affine Baer subplanes through ℓ. Doing so, q 2 (q 2 − q) points adjacent to P , Q, R are obtained in this way. Therefore, the required number is q 2 (q 2 − q) plus q 2 − 3, the number of points of ℓ̄ distinct from P , Q, R.  Our main result in this section is the following theorem. Theorem 6.5. In PG(2, q 2 ), let U be the classical unital, and let U ′ be any unital. ∗ ∗ ∼ Then Ŵ2 (U) = Ŵ2 (U ′ ) if and only if U ′ is classical. Proof. Take a unital U ′ in PG(2, q 2 ) other than U and assume that Ŵ2∗ (U) ∼ = Ŵ2∗ ((U)′ ). 2 Let γ be a bijection on the pointset of AG(3, q ) with the following property: for any two distinct points P , Q ∈ AG(3, q 2 ), the infinite point of the line ℓ through P and Q lies on U if and only if the infinite point of the line ℓ′ through P ′ = P γ and Q′ = Qγ lies on U ′ . Journal of Combinatorial Designs DOI 10.1002/jcd 13 ON QUASI-HERMITIAN VARIETIES Assume that there exist three points P , Q, R lying on an affine Baer subline with infinite point on U such that the corresponding points P ′ , Q′ , R ′ by γ are not collinear. For a line r through P ′ , let m(P ′ , r) denote the number of points on r that are adjacent to each of the points P ′ , Q′ , and R ′ . Let m(P ′ ) the maximum value of m(P ′ , r) as r ranging over the lines through P ′ . Since |U| = q 3 + 1, Proposition 6.4 yields that m(P ′ )(q 3 + 1) > (q 2 + 1)(q 2 − q) + q − 3. Therefore, m(P ′ ) > q4 − q3 + q2 − 2 > q − 1, q3 + 1 (11) whence m(P ′ ) ≥ q. Let t be a line through P ′ such that m(P ′ , t) ≥ q. Since P ′ Q′ R ′ is a nondegenerate triangle, r does not contain both Q′ and R ′ , and Q′ may be assumed not to lie on r. Project r from Q′ to , and let s be the resulting line in . The intersection U ∩ s contains the projection of each of the points of t adjacent to P ′ together with the projection of P ′ . Since |U ∩ s| = q + 1, this is only possible when m(P ′ ) = q and U ∩ s consists of all these points. On the other hand, as the infinite point of r has not been counted so far, this would imply that |U ∩ s| > q + 1, a contradiction. Therefore, γ sends any affine Baer subline with infinite point on U to an affine Baer subline with infinite point on U ′ . We show next that γ takes any affine Baer subplane π whose infinite line is a chord of U to either an affine Baer subplane π ′ whose infinite line is a chord of U ′ , or a line of AG(3, q 2 ) with infinite point on U ′ . In π, let (T , t) be a nonincident point–line pair. It may be that T ′ = T γ is on the line r of AG(3, q 2 ) containing t ′ = t γ . If this happens for every T ∈ π then γ takes π to a line of AG(3, q 2 ) with infinite point on U ′ . Now, suppose T ′ ∈ r to be off r and look at the Baer subplane β determined by the T ′ = T γ and t ′ = t γ . Every point V ∈ π other than those on the parallel line to t through T is on a (unique) line v of π through T and incident t. Since the infinite point of v is on U, v ′ = v γ is a Baer subline and this implies that V ′ = V γ lies in β. Now replace t with another Baer subline of π not parallel to t and repeat the above argument. It turns out that V ∈ π implies that V ′ ∈ β. Therefore, β = π ′ with p′ = π γ . We show now that the above claim holds true when π is replaced by a line ℓ of AG(3, q 2 ) with infinite point on U. Take a Baer subline t on ℓ together with a point T ∈ ℓ disjoint from t. Arguing as before, either ℓ′ = ℓγ is a line with infinite point on U ′ , or T ′ is disjoint from t ′ . In the latter case, look at the affine Baer subplane π ′ through T ′ and t ′ . Since ℓ together with its affine Baer sublines is isomorphic to an affine Baer subplane, the previous proof involving V and v can be used to prove that π ′ = ℓ′ . We are in a position to show that every chord of U ′ is a (projective) Baer subline in . Let ℓ be a chord of U ′ , and take a triangle in AG(3, q 2 ) such that the infinite points of its sides are on ℓ. From Proposition 6.1, there is an affine Baer subplane α containing the vertices of this triangle. The infinite line of α contains the infinite points of the sides of the triangle. Therefore, ℓ is the infinite line of an affine Baer subplane. The last assertion implies that U ′ itself is a classical unital, see [13, 17] and [2, Chapter 7.1].  Remark 6.6. The automorphism group Aut(Ŵ2∗ (U)) of the strongly regular graph Ŵ2∗ (U) arising from the classical unital U contains the inherited group G, that is, the subgroup of the collineation group of AG(3, q 2 ) which preserves U where G/N ∼ = P ŴU (3, q 2 ) 2 6 where N is the (normal) subgroup of order (q − 1)q consisting of all translations and Journal of Combinatorial Designs DOI 10.1002/jcd 14 AGUGLIA, COSSIDENTE, AND KORCHM dilations of AG(3, q 2 ). The group Aut(Ŵ2∗ (U)) may be much larger than the inherited group, and this actually occurs for q = 3. In fact, a computer aided computation shows for q = 3 that Aut(Ŵ2∗ (U))/N = P SU (4, 9). A corollary of Theorem 6.5 is the following result. Corollary 6.7. Let U ′ be any two-character set in P G(r, q ′ ) for a prime-power ′ ∗ ′ q . If Ŵr (U ) is isomorphic to Ŵ2∗ (U) where U is a classical unital in P G(2, q 2 ), then q ′ = q 2 , r = 2 and U ′ is also a classical unital. Proof. Obviously, Ŵr∗ (U ′ ) and Ŵ2∗ (U) have the same parameters. 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