- Email: [email protected]
The Bruck–Bose map and Beukenhout–Metz unitals
Journal of Statistical Planning and Inference 72 (1998) 89–98
The Bruck–Bose map and Beukenhout–Metz unitals Rey Casse ∗ , Catherine T. Quinn Department of Pure Mathematics, The University of Adelaide, G.P.O. Box 498, Adelaide, SA 5001, Australia
To R.C. Bose, whose work in geometry has in uenced a generation of geometers world-wide
Abstract In this paper we survey some old and new results concerning Buekenhout–Metz unitals and the Bruck and Bose representation of translation planes, in particular the Desarguesian plane of c 1998 Published by Elsevier Science B.V. All rights reserved. order q2 . AMS classi cation: 51E20 Keywords: Buekenhout–Metz unital; Baer subplane; Spread; Translation plane; Ruled cubic surface
1. The Bruck–Bose map 1.1. Regular spreads Embed PG(3; q) in PG(3; q 2 ) and let t be a line in PG(3; q 2 ). The line t intersects PG(3; q) in (q + 1) points, one point or is disjoint from PG(3; q); choose t so that t is disjoint from PG(3; q). The Frobenius map is de ned as follows: : PG(3; q 2 ) → PG(3; q 2 ); P = (a; b; c) 7→ P q = (aq ; bq ; cq ) for P a point in PG(3; q 2 ) with homogeneous coordinates (a; b; c). We call P = P q the conjugate of the point P and the conjugate L of a line L in PG(3; q 2 ) is the line containing all the conjugates of points on L. ∗
Corresponding author. E-mail: [email protected].
c 1998 Published by Elsevier Science B.V. All rights reserved. 0378-3758/98/$ – see front matter PII: S 0 3 7 8 - 3 7 5 8 ( 9 8 ) 0 0 0 2 4 - X
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is the unique involution of PG(3; q 2 ) xing PG(3; q) pointwise. Let t be the conjugate of t. Join every point P on t to its conjugate P. The q 2 + 1 lines li so obtained have the property li = li , i = 1 : : : q 2 + 1, and the li ∩ PG(3; q) are therefore lines of PG(3; q). As the q 2 + 1 lines of this set are pairwise disjoint, they partition the points of PG(3; q) and therefore constitute a spread S of PG(3; q). Recall that in PG(3; q), given three pairwise skew lines m1 ; m2 ; m3 then a line incident with all three lines is called a transversal of m1 ; m2 ; m3 and that there exist q + 1 (pairwise skew) transversals of m1 ; m2 ; m3 . Such a set of q + 1 lines is called a regulus of PG(3; q). A spread of PG(3; q) is a set of q 2 +1 skew lines of PG(3; q), one through each point of PG(3; q). A spread is called regular if for every line m of PG(3; q) which is not a line of the spread, the q + 1 lines of the spread which meet m form a regulus of PG(3; q). Theorem 1.1 (Bruck, 1969). The spread S of PG(3; q); constructed above; is a regular spread of PG(3; q). Theorem 1.2 (Bruck, 1969). Every regular spread of PG(3; q) can be represented in this manner for a unique pair of lines t, t in PG(3; q 2 ). 1.2. The Bruck–Bose construction Let ∞ be a 3-space of PG(4; q). Let S ={t1 ; : : : ; tq 2 +1 } be a spread of ∞ . De ne an incidence structure (S) as follows: Points (i) Points of PG(4; q)\∞ ; (ii) lines ti of S. Lines (a) Planes of PG(4; q)\∞ through the lines of S, (b) {t1 ; t2 ; : : : ; tq 2 +1 } = l∞ . Incidence Natural incidence. Result 1.3 (Andre, 1954; Bruck and Bose, 1964; 1966). (S) is a translation plane of order q 2 with translation line l∞ . Every translation plane of order q 2 whose kernel contains GF(q) is obtainable from such a construction. (S) is isomorphic to the Desarguesian plane PG(2; q 2 ) of order q 2 if and only if S is a regular spread. The construction above is a particular case of the Bruck–Bose construction of translation planes. We denote by the map associating a translation plane (S) of order q 2 with its corresponding structure in PG(4; q) de ned with the spread S. We call the Bruck–Bose Map, and call (S) the Bruck–Bose representation of (S).
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De nition 1.4 (Hirschfeld, 1979). A Baer subplane of PG(2; q 2 ) is a subplane of order q. A line l of PG(2; q 2 ) intersects a Baer subplane B either in 1 point or in q + 1 points. In the latter case the set b = l ∩ B of size q + 1 is called a Baer subline of l and l is called a line of B. If the line at in nity l∞ (the line which corresponds to S in the Bruck–Bose representation) is a line of B we say B is secant to the line at in nity, otherwise B is tangent to the line at in nity. In Bruck and Bose (1964), it is shown that a plane of PG(4; q)\∞ which contains no line of S (call such a plane a transversal plane) represents a Baer subplane of PG(2; q 2 ) secant to the line of in nity of the plane. In fact, for PG(2; q 2 ) every Baer subplane secant to the line of in nity is represented in Bruck–Bose by a transversal plane of PG(4; q)\∞ . As a consequence Baer sublines in PG(2; q 2 ) which contain a unique point on l∞ are represented in Bruck–Bose by a line of PG(4; q)\∞ and conversely each line of PG(4; q)\∞ is the Bruck–Bose representation of a Baer subline in PG(2; q 2 ). Note that there exist examples of non-Desarguesian translation planes (S) (with S therefore not a regular spread) with a Baer subplane secant to the line at in nity whose Bruck–Bose representation is not a transversal plane (see, for example, Freeman, 1980). In PG(2; q 2 ) each Baer subplane B tangent to the line at in nity l∞ is represented in Bruck–Bose by a ruled cubic surface with line directrix a line of the spread S (see Bose et al., 1980; Vincenti, 1980; see Section 2 for de nitions and further details). Bernasconi and Vincenti (1981) showed that if a ruled cubic surface V2 3 in PG(4; q) is the Bruck–Bose representation of a Baer subplane of a translation plane (S) then (S) is isomorphic to PG(2; q 2 ) (i.e. the structure V2 3 in PG(4; q) induces a regular spread in ∞ ). (See Section 2 for an introduction to the ruled cubic surface in PG(4; q) and an alternative proof of the Bernasconi and Vincenti (1981) result.) The Bruck–Bose representation of Baer subplanes of PG(2; q 2 ) is therefore determined. For the remainder of the paper we shall be working with PG(2; q 2 ) and the Bruck– Bose representation of PG(2; q 2 ), and therefore in Bruck–Bose the spread S of ∞ is a given regular spread. We now discuss in greater detail the representations in PG(4; q) of Baer subplanes of PG(2; q 2 ) which intersect l∞ in a unique point and Baer sublines which are disjoint from l∞ . Lemma 1.5 (Lefevre-Percsy, 1982). A Baer subline b; containing no point on l∞ ; of a line l in PG(2; q 2 ); is represented in PG(4; q) by a non-degenerate conic C in the plane l representing l. The conics in PG(4; q)\∞ which represent Baer sublines of PG(2; q 2 ) shall be called Baer conics. Note that a Baer conic is necessarily disjoint from ∞ .
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The representation of Baer subplanes which intersect l∞ in a unique point is determined in Bose et al. (1980) and Vincenti (1980); an alternative proof can be found in Quinn and Casse (1995). The variety we call a ruled cubic V2 3 is called a twisted ladder in Bose et al. (1980). Lemma 1.6. Let B be a Baer subplane in PG(2; q 2 ) such that B intersects l∞ in the unique point P. Then B corresponds to a ruled cubic surface B in PG(4; q)\∞ with B ∩ ∞ = p; the line directrix of B; and p is the line of the spread S of ∞ representing the point P of B on the line at in nity. Every non-degenerate conic on such a ruled cubic surface B is a Baer conic. Since there exist conics in PG(4; q)\∞ which do not represent Baer sublines (Metz, 1979), it follows that there exist ruled cubic surfaces, with directrix a line of S, which do not represent Baer subplanes B of PG(2; q2 ) intersecting l∞ in a unique point; those which do represent such Baer subplanes will be called Baer ruled cubics. Some obvious properties of a Baer ruled cubic surface are (with the notation of Lemma 1.6): (1) Through each line ti of S (ti 6= p) there passes a unique plane which intersects V23 in a Baer conic. (2) The q(q + 1) points of V23 , not on p, lie on q2 Baer conics, and any two of these conics intersect in a unique point. 2. The ruled cubic surface in PG(4; q) Consider a conic C 2 and a line l skew to the plane of C 2 . Set up a projectivity between l and C 2 , and join corresponding points by lines. The ruled surface so obtained is of order 3, and is denoted V23 . By conveniently choosing the coordinate system in PG(4; q), let the point (0; 0; 0; x; y) lying on l correspond to the point (x2 ; xy; y2 ; 0; 0) on C 2 . Thus V23 can be expressed as {(x2 ; xy; y2 ; zx; zy); x; y ∈ GF(q); z ∈ GF(q) ∪ {∞}} and is a set of q + 1 pairwise disjoint lines called generators of the resulting surface V23 . The line l contained in V23 is the unique line incident with each generator of V23 and l is called the line directrix of the surface. The conic C 2 has one point on each generator and is called a conic directrix of the surface. There exist q2 conic directrices on V23 and the conic directrices pairwise intersect in a point. De ne : V23 → PG(2; q) by (x2 ; xy; y2 ; zx; zy) 7→ (x; y; z):
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Thus contracts l into the point (0; 0; 1), and : V23 \l → PG(2; q)\{(0; 0; 1)} is one-to-one and onto. We can de ne −1 : PG(2; q) → V23 which dilates (0; 0; 1) into the line l, and −1 : PG(2; q)\{(0; 0; 1)} → V23 \l is one-to-one and onto. Now if B is a Baer subplane of PG(2; q2 ) intersecting l∞ in one point L only, we have:
(L) = (0; 0; 1): Note that the map : B → PG(2; q) is one-to-one, onto and it can be shown that preserves collinearity. Hence is a collineation. Consider a V23 in PG(4; q) generated by a line l, where l is a line in ∞ , and a conic C 2 in a plane , where C 2 is disjoint from ∞ . Let t be the line ∞ ∩ which therefore contains no real points of C 2 . Let C 2 ∩ t = {P; P}. Denote by g; g the generators of V23 through the points P; P respectively. Note that every conic Ci on V23 necessarily intersects both g and g. Let C1 = C 2 , where 3 C1 ; : : : ; Cq2 are the conics on V2 . If Ci ∩ g = {Xi }, then Ci ∩ g = {X i }; i = 1; : : : ; q2 , and hence {Xi X i ∩ ht; li | i = 1; : : : ; q2 } ∪ {l} = {Xi X i ∩ ∞ | i = 1; : : : ; q2 } ∪ {l} is a spread S0 of ∞ . By Section 1.1 the spread S0 in ∞ is a regular spread by construction. Hence the translation plane whose Bruck–Bose representation is de ned with the spread S0 is isomorphic to PG(2; q2 ); we shall refer to this plane as PG(2; q2 ). Let B be the subset of this plane such that B = V23 ; we now show B is a Baer subplane of PG(2; q2 ). By the above construction of the spread S0 in ∞ , there exists a plane about each spread element which contains a conic of V23 . Since the ruled cubic surface is properly contained in PG(4; q), all remaining planes about a spread element intersect the ruled cubic surface in at most one point and therefore in exactly one point (the number of points of V23 o one of its conics and the number of remaining planes about a spread
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element being equal). Thus in PG(2; q2 ) the set of q2 + q + 1 points of B constitutes a blocking set and since a blocking set in PG(2; q2 ) with this cardinality is a subgeometry isomorphic to PG(2; q), the set B is a Baer subplane of PG(2; q2 ). We have shown: Theorem 2.1. Given a ruled cubic surface V23 with line directrix l in PG(4; q) and a hyperplane ∞ such that V23 ∩ ∞ = l; then there exists a regular spread S0 of ∞ , containing the line l, such that V23 is the Bruck–Bose representation of a Baer subplane of (S0 ) ∼ = PG(2; q2 ). This result was originally proved by Bernasconi and Vincenti (1981). See also Vincenti (1983). 3. Buekenhout–Metz unitals • A unital is a 2 − (q3 + 1; q + 1; 1) design. • A unital in a translation plane q2 of order q2 is a set U of q3 + 1 points of q2 such that each line contains exactly 1 or q + 1 points of U. A line l is a tangent or secant line of U if l ∩ U contains 1 or q + 1 points, respectively. • A unital in a translation plane q2 of order q2 is called parabolic or hyperbolic if l∞ , the translation line of q2 , is respectively a tangent or secant line of the unital. One class of unitals in the Desarguesian plane PG(2; q2 ) consists of the Classical Unitals. The classical unitals are projectively equivalent and the canonical form of a classical unital is H : xq+1 + yq+1 + z q+1 = 0: Some properties of the classical unital H to note: (1) Every secant line of H intersects H in a Baer subline (Lefevre-Percsy, 1982). (2) If B is a Baer subplane of PG(2; q2 ) then |B ∩ H | = 1; q + 1 or 2q + 1 and the corresponding intersection sets are a point, a line or a conic, and a line pair of B, respectively (Bruen and Hirschfeld, 1986). A Buekenhout–Metz unital (see Buekenhout, 1976) is a unital U de ned in a translation plane q2 of order q2 , where q 2 is de ned with a spread S, and to within collineation has the following construction in Bruck–Bose: Let O be an ovoid in PG(4; q)\∞ intersecting ∞ in a unique point A, where the tangent plane of O at A does not contain the spread line t incident with A. Let V be a point on t, V distinct from A. Let U be the structure containing the spread line t and all points of PG(4; q)\∞ on the ovoidal cone with vertex V and base O. −1 In q2 , the set of points U = (U ) is a unital, called a Buekenhout–Metz unital. We sometimes say U is Buekenhout–Metz re(T; l∞ ), if T is the unique point of U on the line at in nity.
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For the remainder of the paper we shall abbreviate Buekenhout–Metz to B–M. The class of B–M unitals in PG(2; q2 ) contains the classical unitals and some (nonclassical) unitals, when q¿2. Some known properties and results concerning Buekenhout–Metz unitals in PG(2; q2 ): (1) A classical unital U is B–M re(T; lT ), where T is any point of U and lT is the (unique) tangent line of U on T (Lefevre-Percsy, 1981; 1982). (2) If q is even and q = 22s+1 , s¿0, taking the ovoid O as a Suzuki-Tits ovoid gives a non-classical B–M unital in PG(2; q2 ) (Buekenhout, 1976). (3) The ovoidal cone in PG(4; q) associated with a classical unital has an elliptic quadric as the base ovoid (Buekenhout, 1976). (4) There exist (for all q¿2) non-classical B–M unitals whose associated ovoidal cone has base ovoid an elliptic quadric (Metz, 1979). (5) A B–M unital U with elliptic quadric as base is classical if and only if there exists a secant line of U, not through T (the unique point of U on the line at in nity), which intersects U in a Baer subline (Lefevre-Percsy, 1982). In PG(2; q 2 ), let U be a non-classical B–M unital re(T; l∞ ) with base ovoid an elliptic quadric. Since U is non-classical, there exists a secant line l, not on T , such that C = l ∩ U is not a Baer subline. In Bruck–Bose (with respect to l∞ ), the set of points C is therefore a non-Baer conic. From Theorem 2.1 there exists a regular spread S0 of ∞ containing the spread line t, with respect to which C is a Baer conic. Hence, by Lefevre-Percsy (1982) (Result 5 above) we have: Theorem 3.1 (Casse and Quinn, preprint). A non-classical B–M unital U in PG(2; q 2 ) with elliptic quadric as base can be made classical by a change of spreads (by the procedure stated above).
4. Characterisations of unitals in PG(2; q 2 ) A number of authors have made signi cant contributions to the study of characterisations of unitals in PG(2; q 2 ). The following ve are typical. Theorem 4.1 (Faina and Korchmaros, 1983; Lefevre-Percsy, 1982). Let U be a unital in PG(2; q 2 ), where q¿2. If each line of the plane which is secant to U intersects it in a Baer subline, then U is a classical unital. Theorem 4.2 (Thas, 1992). In PG(2; q 2 ) a unital U is a classical unital if and only if the tangents of U at collinear points of U are concurrent. Theorem 4.3 (Lefevre-Percsy, 1981). Let U be a unital in PG(2; q 2 ) where q¿2 and let l be some tangent line to U. If all Baer sublines having a point on l intersect U in 0; 1; 2 or q + 1 points, then U is a B–M unital.
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Theorem 4.4 (Casse et al., to appear; Quinn and Casse, 1995). Let U be a unital in PG(2; q 2 ); q¿2. Then U is a B–M unital if and only if there exists a point T of U such that the points of U on each of the q 2 secant lines to U through T form a Baer subline. Buekenhout (1976) de ned a class of unitals called Buekenhout unitals. A Buekenhout unital is, to within collineation, a hyperbolic unital and the construction is given in Section 3, Theorem 4, of the paper Buekenhout (1976). Barwick (1994) gives a characterisation of Buekenhout unitals in PG(2; q 2 ). Theorem 4.5 (Barwick, 1994). If U is a Buekenhout unital in PG(2; q 2 ) then U is classical. Finally we note that the known unitals in PG(2; q 2 ) are Buekenhout–Metz unitals (see Casse et al., to appear, for more detail). 5. Buekenhout–Metz Unitals and Baer subplanes Bruen and Hirschfeld (1986) gave many combinatorial results for the intersection of a set of type (m; n) and a set of type (m0 ; n0 ) in a projective plane of order q, including speci c results when the two sets concerned are a Baer subplane and a unital. Gruning (1987) gave similar results including the following result which has proved to be very useful. Theorem 5.1 (Bruen and Hirschfeld, 1986; Gruning, 1987). Let B be a Baer subplane and let U be a unital in a projective plane q 2 of order q 2 . Denote by b1 the number of lines of B which when extended are tangent lines of U and let |B ∩ U| denote the number of points in the intersection of B and U. Then, |B ∩ U| + b1 = 2(q + 1): This result tells us for a unital U and Baer subplane B in a projective plane q 2 of order q 2 , 06|B ∩ U|62(q + 1): The values |B ∩ U| in fact takes has previously only been determined for a classical unital U and a Baer subplane B in PG(2; q 2 ). Bruen and Hirschfeld (1986) used an algebraic proof to obtain the following result (which we mentioned earlier): Theorem 5.2 (Bruen and Hirschfeld, 1986). In PG(2; q 2 ), for U a classical unital and B a Baer subplane we have |B ∩ U| = 1; q + 1
or
2q + 1
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where the intersection sets are a unique point, q + 1 points of a line of B or q + 1 points of a conic in B and a line pair in B respectively. Using the various results described above, the following theorem can be proved. Theorem 5.3 (Casse and Quinn, preprint). Let U be a non-classical Buekenhout–Metz unital re(T; l∞ ) in PG(2; q 2 ) and let B be a Baer subplane in PG(2; q 2 ). Then; (i) If |B ∩ l∞ | = q + 1 then |B ∩ U| = 1; q + 1 or 2q + 1. (ii) If B ∩ l∞ = {T } then |B ∩ U| = q + 1 or 2q + 1. (iii) If B ∩ l∞ = {P} with P 6= T , and if U has an elliptic quadric as base, then for q¿13, 16|B ∩ U|62q + 1. Thus, in PG(2; q 2 ), when it concerns the intersection with a Baer subplane, a nonclassical B–M unital re(T; l∞ ) with elliptic quadric as base “looks” like a classical unital from the point T .
References Andre, J., 1954. Uber nicht-Desarguessche Ebenen mit transitiver Translationsgruppe. Math. Z. 60, 156 –186. Barwick, S.G., 1994. A Characterization of the Classical Unital. Geom. Dedicata 52, 175 –180. Bernasconi, C., Vincenti, R., 1981. Spreads induced by varieties V23 of PG(4; q) and Baer subplanes. Bollettino UMI 18-B, 821– 830. Bose, R.C., Freeman, J.W., Glynn, D.G., 1980. On the intersection of two Baer subplanes in a nite projective plane. Utilitas Math. 17, 65 –77. Bruck, R.H., 1969. Construction problems of nite projective planes. In: Combinatorial Mathematics and its Applications. University of North Carolina Press, Chapel Hill. Bruck, R.H., Bose, R.C., 1964. The construction of translation planes from projective spaces. J. Algebra 1, 85 –102. Bruck, R.H., Bose, R.C., 1966. Linear representations of projective planes in projective spaces. J. Algebra 4, 117–172. Bruen, A.A., Hirschfeld, J.W.P., 1986. Intersections in Projective space I: Combinatorics. Math. Z. 193, 215 –225. Buekenhout, F., 1976. Existence of unitals in nite translation planes of order q 2 with a kernel of order q. Geom. Dedicata 5, 189 –194. Casse, L.R.A., O’Keefe, C.M., Penttila, T., Characterisations of Buekenhout–Metz unitals. Geom. Dedicata, to appear. Casse, L.R.A., Quinn, C.T., Buekenhout–Metz unitals and Baer subplanes in PG(2; q 2 ). Preprint. Faina, G., Korchmaros, G., 1983. A graphic characterization of Hermitian curves. Ann. Discrete Math. 18, 335 –342. Freeman, J.W., 1980. Reguli and Pseudoreguli in PG(3; s 2 ). Geom. Dedicata 9, 267–280. Gruning, K., 1987. A class of unitals of order q which can be embedded in two di erent planes of order q 2 . J. Geom. 29, 61–77. Hirschfeld, J.W.P., 1979. Projective Geometries over Finite Fields. Clarendon Press, Oxford. Lefevre-Percsy, C., 1981. Characterization of Buekenhout–Metz unitals. Arch. Math. 36, 565 –568. Lefevre-Percsy, C., 1982. Characterization of hermitian curves. Arch. Math. 39, 476 – 480. Metz, R., 1979. On a class of unitals. Geom. Dedicata 8, 125 –126.
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Quinn, C.T., Casse, L.R.A., 1995. Concerning a characterisation of Buekenhout–Metz unitals. J. Geom. 52, 159 –167. Thas, J.A., 1992. A combinatorial characterization of hermitian curves. J. Algebra, Combin. 1, 97–102. Vincenti, R., 1980. Alcuni tipi di varieta V23 di S4; q e sottopiani di Baer. BUMI 2 (Suppl.), 31– 44. Vincenti, R., 1983. A Survey on varieties of PG(4; q) and Baer subplanes of translation planes. Annals Discrete Math. 18, 775 –780.
The Bruck–Bose map and Beukenhout–Metz unitals Rey Casse ∗ , Catherine T. Quinn Department of Pure Mathematics, The University of Adelaide, G.P.O. Box 498, Adelaide, SA 5001, Australia
To R.C. Bose, whose work in geometry has in uenced a generation of geometers world-wide
Abstract In this paper we survey some old and new results concerning Buekenhout–Metz unitals and the Bruck and Bose representation of translation planes, in particular the Desarguesian plane of c 1998 Published by Elsevier Science B.V. All rights reserved. order q2 . AMS classi cation: 51E20 Keywords: Buekenhout–Metz unital; Baer subplane; Spread; Translation plane; Ruled cubic surface
1. The Bruck–Bose map 1.1. Regular spreads Embed PG(3; q) in PG(3; q 2 ) and let t be a line in PG(3; q 2 ). The line t intersects PG(3; q) in (q + 1) points, one point or is disjoint from PG(3; q); choose t so that t is disjoint from PG(3; q). The Frobenius map is de ned as follows: : PG(3; q 2 ) → PG(3; q 2 ); P = (a; b; c) 7→ P q = (aq ; bq ; cq ) for P a point in PG(3; q 2 ) with homogeneous coordinates (a; b; c). We call P = P q the conjugate of the point P and the conjugate L of a line L in PG(3; q 2 ) is the line containing all the conjugates of points on L. ∗
Corresponding author. E-mail: [email protected].
c 1998 Published by Elsevier Science B.V. All rights reserved. 0378-3758/98/$ – see front matter PII: S 0 3 7 8 - 3 7 5 8 ( 9 8 ) 0 0 0 2 4 - X
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is the unique involution of PG(3; q 2 ) xing PG(3; q) pointwise. Let t be the conjugate of t. Join every point P on t to its conjugate P. The q 2 + 1 lines li so obtained have the property li = li , i = 1 : : : q 2 + 1, and the li ∩ PG(3; q) are therefore lines of PG(3; q). As the q 2 + 1 lines of this set are pairwise disjoint, they partition the points of PG(3; q) and therefore constitute a spread S of PG(3; q). Recall that in PG(3; q), given three pairwise skew lines m1 ; m2 ; m3 then a line incident with all three lines is called a transversal of m1 ; m2 ; m3 and that there exist q + 1 (pairwise skew) transversals of m1 ; m2 ; m3 . Such a set of q + 1 lines is called a regulus of PG(3; q). A spread of PG(3; q) is a set of q 2 +1 skew lines of PG(3; q), one through each point of PG(3; q). A spread is called regular if for every line m of PG(3; q) which is not a line of the spread, the q + 1 lines of the spread which meet m form a regulus of PG(3; q). Theorem 1.1 (Bruck, 1969). The spread S of PG(3; q); constructed above; is a regular spread of PG(3; q). Theorem 1.2 (Bruck, 1969). Every regular spread of PG(3; q) can be represented in this manner for a unique pair of lines t, t in PG(3; q 2 ). 1.2. The Bruck–Bose construction Let ∞ be a 3-space of PG(4; q). Let S ={t1 ; : : : ; tq 2 +1 } be a spread of ∞ . De ne an incidence structure (S) as follows: Points (i) Points of PG(4; q)\∞ ; (ii) lines ti of S. Lines (a) Planes of PG(4; q)\∞ through the lines of S, (b) {t1 ; t2 ; : : : ; tq 2 +1 } = l∞ . Incidence Natural incidence. Result 1.3 (Andre, 1954; Bruck and Bose, 1964; 1966). (S) is a translation plane of order q 2 with translation line l∞ . Every translation plane of order q 2 whose kernel contains GF(q) is obtainable from such a construction. (S) is isomorphic to the Desarguesian plane PG(2; q 2 ) of order q 2 if and only if S is a regular spread. The construction above is a particular case of the Bruck–Bose construction of translation planes. We denote by the map associating a translation plane (S) of order q 2 with its corresponding structure in PG(4; q) de ned with the spread S. We call the Bruck–Bose Map, and call (S) the Bruck–Bose representation of (S).
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De nition 1.4 (Hirschfeld, 1979). A Baer subplane of PG(2; q 2 ) is a subplane of order q. A line l of PG(2; q 2 ) intersects a Baer subplane B either in 1 point or in q + 1 points. In the latter case the set b = l ∩ B of size q + 1 is called a Baer subline of l and l is called a line of B. If the line at in nity l∞ (the line which corresponds to S in the Bruck–Bose representation) is a line of B we say B is secant to the line at in nity, otherwise B is tangent to the line at in nity. In Bruck and Bose (1964), it is shown that a plane of PG(4; q)\∞ which contains no line of S (call such a plane a transversal plane) represents a Baer subplane of PG(2; q 2 ) secant to the line of in nity of the plane. In fact, for PG(2; q 2 ) every Baer subplane secant to the line of in nity is represented in Bruck–Bose by a transversal plane of PG(4; q)\∞ . As a consequence Baer sublines in PG(2; q 2 ) which contain a unique point on l∞ are represented in Bruck–Bose by a line of PG(4; q)\∞ and conversely each line of PG(4; q)\∞ is the Bruck–Bose representation of a Baer subline in PG(2; q 2 ). Note that there exist examples of non-Desarguesian translation planes (S) (with S therefore not a regular spread) with a Baer subplane secant to the line at in nity whose Bruck–Bose representation is not a transversal plane (see, for example, Freeman, 1980). In PG(2; q 2 ) each Baer subplane B tangent to the line at in nity l∞ is represented in Bruck–Bose by a ruled cubic surface with line directrix a line of the spread S (see Bose et al., 1980; Vincenti, 1980; see Section 2 for de nitions and further details). Bernasconi and Vincenti (1981) showed that if a ruled cubic surface V2 3 in PG(4; q) is the Bruck–Bose representation of a Baer subplane of a translation plane (S) then (S) is isomorphic to PG(2; q 2 ) (i.e. the structure V2 3 in PG(4; q) induces a regular spread in ∞ ). (See Section 2 for an introduction to the ruled cubic surface in PG(4; q) and an alternative proof of the Bernasconi and Vincenti (1981) result.) The Bruck–Bose representation of Baer subplanes of PG(2; q 2 ) is therefore determined. For the remainder of the paper we shall be working with PG(2; q 2 ) and the Bruck– Bose representation of PG(2; q 2 ), and therefore in Bruck–Bose the spread S of ∞ is a given regular spread. We now discuss in greater detail the representations in PG(4; q) of Baer subplanes of PG(2; q 2 ) which intersect l∞ in a unique point and Baer sublines which are disjoint from l∞ . Lemma 1.5 (Lefevre-Percsy, 1982). A Baer subline b; containing no point on l∞ ; of a line l in PG(2; q 2 ); is represented in PG(4; q) by a non-degenerate conic C in the plane l representing l. The conics in PG(4; q)\∞ which represent Baer sublines of PG(2; q 2 ) shall be called Baer conics. Note that a Baer conic is necessarily disjoint from ∞ .
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The representation of Baer subplanes which intersect l∞ in a unique point is determined in Bose et al. (1980) and Vincenti (1980); an alternative proof can be found in Quinn and Casse (1995). The variety we call a ruled cubic V2 3 is called a twisted ladder in Bose et al. (1980). Lemma 1.6. Let B be a Baer subplane in PG(2; q 2 ) such that B intersects l∞ in the unique point P. Then B corresponds to a ruled cubic surface B in PG(4; q)\∞ with B ∩ ∞ = p; the line directrix of B; and p is the line of the spread S of ∞ representing the point P of B on the line at in nity. Every non-degenerate conic on such a ruled cubic surface B is a Baer conic. Since there exist conics in PG(4; q)\∞ which do not represent Baer sublines (Metz, 1979), it follows that there exist ruled cubic surfaces, with directrix a line of S, which do not represent Baer subplanes B of PG(2; q2 ) intersecting l∞ in a unique point; those which do represent such Baer subplanes will be called Baer ruled cubics. Some obvious properties of a Baer ruled cubic surface are (with the notation of Lemma 1.6): (1) Through each line ti of S (ti 6= p) there passes a unique plane which intersects V23 in a Baer conic. (2) The q(q + 1) points of V23 , not on p, lie on q2 Baer conics, and any two of these conics intersect in a unique point. 2. The ruled cubic surface in PG(4; q) Consider a conic C 2 and a line l skew to the plane of C 2 . Set up a projectivity between l and C 2 , and join corresponding points by lines. The ruled surface so obtained is of order 3, and is denoted V23 . By conveniently choosing the coordinate system in PG(4; q), let the point (0; 0; 0; x; y) lying on l correspond to the point (x2 ; xy; y2 ; 0; 0) on C 2 . Thus V23 can be expressed as {(x2 ; xy; y2 ; zx; zy); x; y ∈ GF(q); z ∈ GF(q) ∪ {∞}} and is a set of q + 1 pairwise disjoint lines called generators of the resulting surface V23 . The line l contained in V23 is the unique line incident with each generator of V23 and l is called the line directrix of the surface. The conic C 2 has one point on each generator and is called a conic directrix of the surface. There exist q2 conic directrices on V23 and the conic directrices pairwise intersect in a point. De ne : V23 → PG(2; q) by (x2 ; xy; y2 ; zx; zy) 7→ (x; y; z):
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Thus contracts l into the point (0; 0; 1), and : V23 \l → PG(2; q)\{(0; 0; 1)} is one-to-one and onto. We can de ne −1 : PG(2; q) → V23 which dilates (0; 0; 1) into the line l, and −1 : PG(2; q)\{(0; 0; 1)} → V23 \l is one-to-one and onto. Now if B is a Baer subplane of PG(2; q2 ) intersecting l∞ in one point L only, we have:
(L) = (0; 0; 1): Note that the map : B → PG(2; q) is one-to-one, onto and it can be shown that preserves collinearity. Hence is a collineation. Consider a V23 in PG(4; q) generated by a line l, where l is a line in ∞ , and a conic C 2 in a plane , where C 2 is disjoint from ∞ . Let t be the line ∞ ∩ which therefore contains no real points of C 2 . Let C 2 ∩ t = {P; P}. Denote by g; g the generators of V23 through the points P; P respectively. Note that every conic Ci on V23 necessarily intersects both g and g. Let C1 = C 2 , where 3 C1 ; : : : ; Cq2 are the conics on V2 . If Ci ∩ g = {Xi }, then Ci ∩ g = {X i }; i = 1; : : : ; q2 , and hence {Xi X i ∩ ht; li | i = 1; : : : ; q2 } ∪ {l} = {Xi X i ∩ ∞ | i = 1; : : : ; q2 } ∪ {l} is a spread S0 of ∞ . By Section 1.1 the spread S0 in ∞ is a regular spread by construction. Hence the translation plane whose Bruck–Bose representation is de ned with the spread S0 is isomorphic to PG(2; q2 ); we shall refer to this plane as PG(2; q2 ). Let B be the subset of this plane such that B = V23 ; we now show B is a Baer subplane of PG(2; q2 ). By the above construction of the spread S0 in ∞ , there exists a plane about each spread element which contains a conic of V23 . Since the ruled cubic surface is properly contained in PG(4; q), all remaining planes about a spread element intersect the ruled cubic surface in at most one point and therefore in exactly one point (the number of points of V23 o one of its conics and the number of remaining planes about a spread
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element being equal). Thus in PG(2; q2 ) the set of q2 + q + 1 points of B constitutes a blocking set and since a blocking set in PG(2; q2 ) with this cardinality is a subgeometry isomorphic to PG(2; q), the set B is a Baer subplane of PG(2; q2 ). We have shown: Theorem 2.1. Given a ruled cubic surface V23 with line directrix l in PG(4; q) and a hyperplane ∞ such that V23 ∩ ∞ = l; then there exists a regular spread S0 of ∞ , containing the line l, such that V23 is the Bruck–Bose representation of a Baer subplane of (S0 ) ∼ = PG(2; q2 ). This result was originally proved by Bernasconi and Vincenti (1981). See also Vincenti (1983). 3. Buekenhout–Metz unitals • A unital is a 2 − (q3 + 1; q + 1; 1) design. • A unital in a translation plane q2 of order q2 is a set U of q3 + 1 points of q2 such that each line contains exactly 1 or q + 1 points of U. A line l is a tangent or secant line of U if l ∩ U contains 1 or q + 1 points, respectively. • A unital in a translation plane q2 of order q2 is called parabolic or hyperbolic if l∞ , the translation line of q2 , is respectively a tangent or secant line of the unital. One class of unitals in the Desarguesian plane PG(2; q2 ) consists of the Classical Unitals. The classical unitals are projectively equivalent and the canonical form of a classical unital is H : xq+1 + yq+1 + z q+1 = 0: Some properties of the classical unital H to note: (1) Every secant line of H intersects H in a Baer subline (Lefevre-Percsy, 1982). (2) If B is a Baer subplane of PG(2; q2 ) then |B ∩ H | = 1; q + 1 or 2q + 1 and the corresponding intersection sets are a point, a line or a conic, and a line pair of B, respectively (Bruen and Hirschfeld, 1986). A Buekenhout–Metz unital (see Buekenhout, 1976) is a unital U de ned in a translation plane q2 of order q2 , where q 2 is de ned with a spread S, and to within collineation has the following construction in Bruck–Bose: Let O be an ovoid in PG(4; q)\∞ intersecting ∞ in a unique point A, where the tangent plane of O at A does not contain the spread line t incident with A. Let V be a point on t, V distinct from A. Let U be the structure containing the spread line t and all points of PG(4; q)\∞ on the ovoidal cone with vertex V and base O. −1 In q2 , the set of points U = (U ) is a unital, called a Buekenhout–Metz unital. We sometimes say U is Buekenhout–Metz re(T; l∞ ), if T is the unique point of U on the line at in nity.
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For the remainder of the paper we shall abbreviate Buekenhout–Metz to B–M. The class of B–M unitals in PG(2; q2 ) contains the classical unitals and some (nonclassical) unitals, when q¿2. Some known properties and results concerning Buekenhout–Metz unitals in PG(2; q2 ): (1) A classical unital U is B–M re(T; lT ), where T is any point of U and lT is the (unique) tangent line of U on T (Lefevre-Percsy, 1981; 1982). (2) If q is even and q = 22s+1 , s¿0, taking the ovoid O as a Suzuki-Tits ovoid gives a non-classical B–M unital in PG(2; q2 ) (Buekenhout, 1976). (3) The ovoidal cone in PG(4; q) associated with a classical unital has an elliptic quadric as the base ovoid (Buekenhout, 1976). (4) There exist (for all q¿2) non-classical B–M unitals whose associated ovoidal cone has base ovoid an elliptic quadric (Metz, 1979). (5) A B–M unital U with elliptic quadric as base is classical if and only if there exists a secant line of U, not through T (the unique point of U on the line at in nity), which intersects U in a Baer subline (Lefevre-Percsy, 1982). In PG(2; q 2 ), let U be a non-classical B–M unital re(T; l∞ ) with base ovoid an elliptic quadric. Since U is non-classical, there exists a secant line l, not on T , such that C = l ∩ U is not a Baer subline. In Bruck–Bose (with respect to l∞ ), the set of points C is therefore a non-Baer conic. From Theorem 2.1 there exists a regular spread S0 of ∞ containing the spread line t, with respect to which C is a Baer conic. Hence, by Lefevre-Percsy (1982) (Result 5 above) we have: Theorem 3.1 (Casse and Quinn, preprint). A non-classical B–M unital U in PG(2; q 2 ) with elliptic quadric as base can be made classical by a change of spreads (by the procedure stated above).
4. Characterisations of unitals in PG(2; q 2 ) A number of authors have made signi cant contributions to the study of characterisations of unitals in PG(2; q 2 ). The following ve are typical. Theorem 4.1 (Faina and Korchmaros, 1983; Lefevre-Percsy, 1982). Let U be a unital in PG(2; q 2 ), where q¿2. If each line of the plane which is secant to U intersects it in a Baer subline, then U is a classical unital. Theorem 4.2 (Thas, 1992). In PG(2; q 2 ) a unital U is a classical unital if and only if the tangents of U at collinear points of U are concurrent. Theorem 4.3 (Lefevre-Percsy, 1981). Let U be a unital in PG(2; q 2 ) where q¿2 and let l be some tangent line to U. If all Baer sublines having a point on l intersect U in 0; 1; 2 or q + 1 points, then U is a B–M unital.
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Theorem 4.4 (Casse et al., to appear; Quinn and Casse, 1995). Let U be a unital in PG(2; q 2 ); q¿2. Then U is a B–M unital if and only if there exists a point T of U such that the points of U on each of the q 2 secant lines to U through T form a Baer subline. Buekenhout (1976) de ned a class of unitals called Buekenhout unitals. A Buekenhout unital is, to within collineation, a hyperbolic unital and the construction is given in Section 3, Theorem 4, of the paper Buekenhout (1976). Barwick (1994) gives a characterisation of Buekenhout unitals in PG(2; q 2 ). Theorem 4.5 (Barwick, 1994). If U is a Buekenhout unital in PG(2; q 2 ) then U is classical. Finally we note that the known unitals in PG(2; q 2 ) are Buekenhout–Metz unitals (see Casse et al., to appear, for more detail). 5. Buekenhout–Metz Unitals and Baer subplanes Bruen and Hirschfeld (1986) gave many combinatorial results for the intersection of a set of type (m; n) and a set of type (m0 ; n0 ) in a projective plane of order q, including speci c results when the two sets concerned are a Baer subplane and a unital. Gruning (1987) gave similar results including the following result which has proved to be very useful. Theorem 5.1 (Bruen and Hirschfeld, 1986; Gruning, 1987). Let B be a Baer subplane and let U be a unital in a projective plane q 2 of order q 2 . Denote by b1 the number of lines of B which when extended are tangent lines of U and let |B ∩ U| denote the number of points in the intersection of B and U. Then, |B ∩ U| + b1 = 2(q + 1): This result tells us for a unital U and Baer subplane B in a projective plane q 2 of order q 2 , 06|B ∩ U|62(q + 1): The values |B ∩ U| in fact takes has previously only been determined for a classical unital U and a Baer subplane B in PG(2; q 2 ). Bruen and Hirschfeld (1986) used an algebraic proof to obtain the following result (which we mentioned earlier): Theorem 5.2 (Bruen and Hirschfeld, 1986). In PG(2; q 2 ), for U a classical unital and B a Baer subplane we have |B ∩ U| = 1; q + 1
or
2q + 1
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where the intersection sets are a unique point, q + 1 points of a line of B or q + 1 points of a conic in B and a line pair in B respectively. Using the various results described above, the following theorem can be proved. Theorem 5.3 (Casse and Quinn, preprint). Let U be a non-classical Buekenhout–Metz unital re(T; l∞ ) in PG(2; q 2 ) and let B be a Baer subplane in PG(2; q 2 ). Then; (i) If |B ∩ l∞ | = q + 1 then |B ∩ U| = 1; q + 1 or 2q + 1. (ii) If B ∩ l∞ = {T } then |B ∩ U| = q + 1 or 2q + 1. (iii) If B ∩ l∞ = {P} with P 6= T , and if U has an elliptic quadric as base, then for q¿13, 16|B ∩ U|62q + 1. Thus, in PG(2; q 2 ), when it concerns the intersection with a Baer subplane, a nonclassical B–M unital re(T; l∞ ) with elliptic quadric as base “looks” like a classical unital from the point T .
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Quinn, C.T., Casse, L.R.A., 1995. Concerning a characterisation of Buekenhout–Metz unitals. J. Geom. 52, 159 –167. Thas, J.A., 1992. A combinatorial characterization of hermitian curves. J. Algebra, Combin. 1, 97–102. Vincenti, R., 1980. Alcuni tipi di varieta V23 di S4; q e sottopiani di Baer. BUMI 2 (Suppl.), 31– 44. Vincenti, R., 1983. A Survey on varieties of PG(4; q) and Baer subplanes of translation planes. Annals Discrete Math. 18, 775 –780.