ARS MATHEMATICA CONTEMPORANEA
ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 17 (2019) 515-524 https://doi.org/10.26493/1855-3974.1763.6cb
(Also available at http://amc-journal.eu)
Symplectic semifield spreads of PG(5, qf), q even
Valentina Pepe *
Sapienza University of Rome, Italy
*
Received 23 July 2018, accepted 14 July 2019, published online 24 November 2019
Abstract
Let q > 2-34t be even. We prove that the only symplectic semifield spread of PG(5, qf'), whose associate semifield has center containing Fq, is the Desarguesian spread. Equiva-lently, a commutative semifield of order q3t, with middle nucleus containing Fqt and center containing Fq, is a field. We do that by proving that the only possible Fq -linear set of rank 3t in PG(5, qt) disjoint from the secant variety of the Veronese surface is a plane of PG(5,qt).
Keywords: Semifields, spreads, symplectic polarity, linear sets, Veronese variety. Math. Subj. Class.: 05B25, 51E15, 51E23, 14M12
1 Introduction
Let PG(r - 1, q) be the projective space of dimension r - 1 over the finite field Fq of order q. An (n — 1)-spread S of PG(2n — 1, q), which we will call simply spread from now on, is a partition of the point-set in (n — 1)-dimensional subspaces. With any spread S it is associated a translation plane A(S) of order qn via the Andre-Bruck-Bose construction (see e.g. [7, Section 5.1]). Translation planes associated with different spreads of PG(2n — 1, q) are isomorphic if and only if there is a collineation of PG(2n — 1, q) mapping one spread to the other (see [1] or [16, Chapter 1]). A spread S is said to be Desarguesian if A(S) is isomorphic to AG(2, qn) and hence a plane coordinatized by the field of order qn. The spread S is said to be a semifield spread if A(S) is a plane of Lenz-Barlotti class V and this is equivalent to saying that A(S) is coordinatized by a semifield.
*The author acknowledges the support of the project "Polinomi ortogonali, strutture algebriche e geometriche inerenti a grafi e campi finiti" of the SBAI Department of Sapienza University of Rome. E-mail address: valepepe@sbai.uniroma1.it (Valentina Pepe)
©® This work is licensed under https://creativecommons.org/licenses/by/4.0/
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A finite semifield S = (S, +, *) is a finite algebra satisfying all the axioms for a skew-field except (possibly) associativity of multiplication. The subsets
N; = {a e S : (a* b) *c = a* (b*c), Vb,c e S}, Nm = {b e S : (a* b) *c = a* (b*c), Va,c e S}, Nr = {c e S : (a* b) *c = a* (b*c), Va,b e S} and K = {a e N; n Nm n Nr : a*b = b*a, Vb e S}
are fields and are known, respectively, as the left nucleus, the middle nucleus, the right nucleus and the center of the semifield. A finite semifield is a vector space over its nuclei and its center.
If A(S) is coordinatized by the semifield S, then S has order qn and its left nucleus contains Fq.
Semifields are studied up to an equivalence relation called isotopy, which corresponds to the study of semifield planes up to isomorphisms (for more details on semifields see, e.g., [7]).
The spread S is said to be symplectic if the elements of S are totally isotropic with respect to a symplectic polarity of PG(2n -1, q). If A(S) is coordinatized by the semifield S, then S is called symplectic semifield and if its center contains Fs < Fq, then from S we get by the cubical array (see [13]) a semifield isotopic to a commutative semifield with middle nucleus containing Fq and center containing Fs ([11]).
Let q be even. For n = 2, there is the following remarkable theorem due to Cohen and Ganley.
Theorem 1.1 ([6]). A commutative semifield of order q2 with middle nucleus containing Fq is afield.
For n > 2, the only known commutative semifields, that are not a field, are the KantorWilliams symplectic pre-semifields of order qn and n > 1 odd ([12]) and their commutative Knuth derivatives ([11]). Symplectic semifield spreads in characteristic 2 with odd dimension over F2 give arise to Z4-linear codes and extremal line sets in Euclidean spaces ([4]).
Most of the above mentioned results are obtained with an algebraic approach, whereas ours is mainly geometric. For small n, the study of semifield spreads has shown to be a good way to classify semifields.
Let M (n, Fq) be the set of all n x n matrices over Fq. Without loss of generality, we may always assume that S(to) := {(0, y) : y e Fj;} and S(0) := {(x, 0) : x e FJ^} belong to S, hence we may write S = {S(A) : A e C} U S(to), with S(A) := {(x, xA) : x e F;}, with C c M(n, Fq) such that |C| = qn and C contains the zero matrix. The set C is called the spread set associated with S. In order to have a semi-field spread, the non-zero elements of C must be invertible and C must be a subgroup of the additive group of M(n, Fq) ([7, Section 5.1]), hence C is a vector space over some subfield of Fq. If we choose the symplectic polarity induced by the alternating bilinear form ^((xi, yi), (x2y2)) = xiyif - yixj, x4, yi e F;, then the subspace S(A) e S is totally isotropic if and only if A is symmetric. The symmetric matrices form an n(n2+1) -dimensional subspace of M (n, Fq) that then induces a PG ^ n(n2+1) - 1, qj. The rank-1 symmetric matrices form the Veronese variety V of degree 2 of PG ^n(n2+1) - 1, qj (this
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is the so called determinant representation of the Veronese variety of degree 2, see [8, Example 2.6]). Hence the singular symmetric matrices form the (n - 2)-th secant variety, say Vn_2, of the Veronese variety. If C is an Fs-vector space, q = s4, then dimFs C = nt and
it defines a subset L of PG(""+11 - 1 , q called Fs-linear set of rank nt (for a complete overview on linear sets see [18]). So to a symplectic semifield spread of PG(2n - 1, q) there corresponds an Fs-linear set L, q = s4, of PG ^"("+1) - 1, qj of rank tn such that L n V"_2 = 0 (see also [15]). We recall the associated semifield has left nucleus containing Fq and if Fs is the maximum subfield with respect to L is linear, then the center of the semifield is isomorphic to Fs. So the isotopic commutative semifield we get has middle nucleus containing Fq and center isomorphic to Fs.
In this article, we are focused on the case n = 3, i.e., on symplectic semifield spreads of PG(5, q), when q is even. In such a case, only two non-sporadic examples are known: the Desarguesian spread and one of its cousin (see [10]), so they are both obtained by slicing the so called Desarguesian spread of Q+(7, q). In the former case, the associated translation plane is the Desarguesian plane, hence it is coordinatized by the finite field of order q3 and the relevant linear set is actually linear on Fq. In the latter case, the semifield spread is associated to a spread set C that is an F2-linear set L of PG(5, q), where F2 is the maximum subfield of Fq for which L is linear, and the associate semifield has order q3 and center F2 .
In [5], it is proven that the only symplectic semifield spread of PG(5, q2), q > 214, whose associate semifield has center containing Fq, is the Desarguesian spread, meaning that a commutative semifield of order q6, with middle nucleus containing Fq2 and center containing Fq is a field, provided q is not too small. That was done by studying the intersection of the five non-equivalent Fq-linear sets of PG(5, q2) with the secant variety V1 of the Veronese variety and the only one that can have empty intersection with V1 is a plane. A classification of the Fq-linear sets of PG(5, q4) of rank 3t is not feasible, as the number of non-equivalent ones quickly grows with t. In fact, the present paper, we had a slightly different approach which allowed us to generalize the result of [5] in PG(5, q4) for any t: by field reduction, a PG(5, q4) can be seen as PG(6t - 1, q), a linear set of rank 3t as a subspace = PG(3t - 1, q) and V1 an algebraic variety, say Vf, of codimension t in PG(6t - 1, q). Hence, we have studied when a subspace of dimension 3t - 1 can have empty intersection with V (over Fq), regardless the geometric feature of the linear set in PG(5, q4).
2 Preliminary results
2.1 Fq-linear sets and the Fq-linear representation of PG(r — 1 ,qt)
The set L c PG(V, Fqt) = PG(r - 1,q4), with V an r-dimensional vector space over Fqt, is said to be an Fq-linear set of rank m if it is defined by the non-zero vectors of an Fq-vector subspace U of V of dimension m, i.e.
L = Lu = j(u)v : u e U \{0}}.
If r = m and (Ly) = PG(r - 1, q4), then Ly = PG(r - 1, q). In this case, Ly is said to be a subgeometry (of order q) of PG(r - 1, q4). Throughout this paper, we shall extensively use the following result: a subset E of PG(r - 1, q4) is a subgeometry of order q if and only if there exists an Fq-linear collineation a of PG(r - 1, q4) of order t such
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that E = Fix a, where Fix a is the set of points fixed by a. This is a straightforward consequence of the fact that there is just one conjugacy class of Fq-linear collineations of order t in PrL(r, q4), namely that of
^ . (xi^ X]^ . . . , xr-1) 1 ^ (^C^ . . . , 1).
In particular, all subgeometries = PG(r -1, q) of PG(r -1, q4) are projectively equivalent to the subgeometry induced by {(x0, x1,..., xr-1) : x, G Fq}. A subspace n of PG(r-1, q4) defines a subspace of Fix a = PG(r - 1,q) of the same dimension if and only if n = n (see [14, Lemma 1]). It will be more convenient for us to explicitly state the following equivalent result.
Notation. Let F be any field containing Fq. Throughout the paper we will denote by n(F) the unique subspace of PG(r - 1, F) containing n.
Lemma 2.1. If we consider PG(r - 1, q) embedded as a subgeometry of PG(r - 1, q4) and n is a subspace of PG(r - 1, q) of dimension s - 1, then the subspace n(Fqt) of PG(r - 1, q4) containing n has dimension s - 1 as well.
Analogously, if W is an algebraic variety of PG(r - 1, q4), then W n Fix a c W n Wn • • • n Wand hence W n Fix a has the same dimension and degree of W if and only if W = W
Remark 2.2. An algebraic variety W is said to be a variety of PG(r - 1, q) if it consists of the set of zeros of polynomials fi, /2,..., fk G Fq [xo , X* 1, . . . , 'X r — 1 1, and we will write W = V(f1, f2,..., fk). By dimension and degree of W we will mean the dimension and degree of the variety when considered as variety of PG(r - 1, Fq), with Fq the algebraic closure of Fq.
In the remaining part of this section, we will describe the setting we adopt to study the Fq -linear sets of PG(V, Fqt) = PG(r - 1,q4).
When we regard V as an Fq-vector space, dimFq V = rt and hence PG(V, q) = PG(rt - 1, q). Furthermore, a point (v)F t G PG(r - 1, q4) corresponds to the (t - 1)-dimensional subspace of PG(rt - 1,q) given by {Av : A G Fqt}. This is the so-called Fq-linear representation of (v)F t and the set S, consisting of the (t - 1)-subspaces of PG(rt - 1, q) that are the linear representation of the points of PG(r - 1, q4), is a partition of the point set of PG(rt - 1, q). Such a partition S is called Desarguesian spread of PG(rt - 1, q). In this setting, a linear set is the subset of the Desarguesian spread S with non-empty intersection with the projective subspace ny of PG(rt - 1, q) induced by U.
We shall adopt the following cyclic representation of PG(rt - 1, q) in PG(rt - 1, q4). Let PG(rt - 1, q4) = PG(V', q4), with V' the standard rt-dimensional vector space over Fqt and let e, the ¿-th element of the canonical base of V'. Consider the semi-linear collineation a with accompanying automorphism x ^ xq and such that e» ^ ei+r, where the subscript are taken mod rt. Then a is an Fq-linear collineation of order t and Fix a = {(x, xq,..., xq ) : x = (x0, x1,..., xr-1), xj G Fqt, x = 0} is isomorphic to PG(rt - 1, q). The elements of S are the subspaces nP := (P, ,..., Pfft 1) n Fix a, with P G n0 = PG(r - 1, q4) and n0 defined by x, =0 Vi > r - 1 (see [14]). Let n, be njf. In the following, we shall identify a point P of n0 = PG(r - 1, q4) with the spread
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element nP. We observe that P is just the projection of nP from (ni, n2,..., nt_i} on n0. If Lu is a linear set of rank m, then it is induced by an (m - 1)-dimensional subspace ny c PG(rt - 1, q) = Fix a and it can be viewed both as the subset of n0 that is the projection of nu from (n1, n2,..., nt_1} on n0 as well as the subset of S consisting of the elements nP such that nP n nu = 0. We stress out that we have defined the subspaces nu and nP as subspaces of Fix a = PG(rt - 1, q). Let F be any field containing Fqt, then the projection of nu(F) on n0 from (n1, n2,..., nt_1} is (Lu}F.
Let H be a hypersurface of PG(r - 1, q4) and let f G Fqt [x0, x1,..., xr-1] a polynomial defining H, i.e., H = V(f). In the linear representation of PG(r - 1, q4) = n0, the points of H correspond to the spread elements nP such that P G H, hence it is the intersection of the variety V (f, f,..., f) of PG(rt - 1, q4) with Fix a, where, by abuse of notation, we extend the action of a also to polynomials. We observe that the variety V(f, fa,..., fa ) is the join of the varieties H, Ha,..., Ha (see [8, Chapter 8]) and hence it has dimension t(dim H +1) - 1 = t(r - 1) - 1 = tr - t - 1 and degree deg(H)4. We observe that V(f, fa,..., fa ) it is defined by t polynomials and dim V (f,f,...,f) = tr - t - 1 = dimPG(rt - 1, q4) - t, hence V (f,f,..., f) is a complete intersection (see [8, Example 11.8]). We will denote the join of the varieties W1, W2,..., Wk by Join(W1, W2,..., Wk).
Let TP (W) be the tangent space to the algebraic variety W at the point P G W.
Proposition 2.3 (Terracini's Lemma [20]). Let W = Join(Y1, Y2) and let P = (P1, P2} G W with Pi G Yi. Then (Tp (Y1), Tp2 (Y2)} C Tp(W).
The variety V(f, fa,..., f1) is the join of the varieties , i = 0,1,..., t - 1. We recall that is a hypersurface of ni, hence TPi ) is a hypersurface of ni for a
non-singular point Pi G . By ni n (n¿, j = i} = 0, we get
dim(Tp0 (H), Tpi (H),..., Tpt_i (Hff'-1)} = rt - 1 -1
for non-singular points P0, P1,..., Pt_ 1. Since for a non-singular point P G V(f, fa,...,
ffft-1), dim TP (V (f,f, ...,ffft-1)) = rt - 1 - t, we have
(Tpo(H),TP1 (H),... ,Tpt-1 (Hfft-1)} = Tp(V(f,f,..., f"t-1))
for a non- singularP G V(f,f,...,f).
Let Sing(W) be the set of the singular points of a variety W; we recall that Sing(W) is a subvariety of W. From the discussion above, it is clear that
t_1
Sing(V(f,f,...,f))= (J Si,
i=0
with Si = Join(Sing(Hffi), H^, j = i).
2.2 The Veronese surface and its secant variety
In this section we denote by Pn-1 the (n - 1)-dimensional projective space over a generic field F.
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The Veronese map of degree 2
v2 : (x0,xbx2) € P2 i—> (. .., x1,. ..) € P5
is such that x1 ranges over all monomials of degree 2 in x0, x1, x2. The image V := v2(P2) is the quadric Veronese surface, a variety of dimension 2 and degree 4. A section H n V, where H is a hyperplane of P5, consists of the points of v2 (C), where C is a conic of P2.
If we use the so-called determinantal representation of V (see [8, Example 2.6]), then we can take P5 as induced by the subspace of M(3, F) consisting of symmetric matrices and v2(x0,x1,x2) = A such that aij = XiXj, i.e., V consists of the rank 1 matrices of M (3, F).
Hence, the secant variety of V, say V1, consists of the symmetric matrices of rank at most 2, i.e., V1 consists of the singular symmetric 3 x 3 matrices. So V1 is a hypersurface of P5 of degree 3. It is well known that the singular points of V1 are the points of V.
The automorphism group G of V is the lifting of G = PGL(3, F) acting in the obvious way: v2(p)^ = v2(pg) V# € PGL(3, F). The group G obviously fixes V1.
The maximal subspaces contained in V1 are planes and they are of three types: the span of v2(^), with I a line of P2, the tangent planes TP(V) for P € V, and, when the characteristic of F is even, the nucleus plane nN.
Let the characteristic of F be even. The plane nN of P5 consists of the symmetric matrices with zero diagonal, hence nN is contained in V1. By the Jacobi's formula, det A = tr(adj(A)), where tr(M) is the trace of a matrix M and adj(M) is the adjoint matrix of M. Let Eij be the 3 x 3 matrix with 1 in the ij-position and 0 elsewhere, so we have det A = tr(adj(A) ) = tr(adj(A)(Eij + Eji)) = 0 Vi = j. It follows that a hyperplane is tangent to V1 if and only if it contains nN. Also, each point of nN is the nucleus of a point of a unique conic v2 (¿).
If P € V1, then the tangent hyperplane H to V1 at P is such that H n V = v2(^2), where I = (p1,p2) if P € nN and hence P € (v2(p1), v2(p2)), or I is such that P is the nucleus of v2(^) if P € nN. The tangent plane at v2 (p) to V is the intersection of three hyperplanes K, K2, K3 such that Ki n V = v2(^i U 4), where £j, (!i are lines through p.
If F is an algebraically closed field, then any subspace of P5 of dimension at least 1 has non-empty intersection with V1. If F = Fq, then there are subspaces of larger dimension disjoint from V1 and, by the Chevalley-Warning Theorem, we know that they can have dimension at most 2. For q even we have the following result.
Theorem 2.4 ([5]). Let q > 4 be even, then there exists one orbit of planes under the action of G disjoint from V1.
3 Proof of the main result
Through this section, we assume q to be even. Let Fq be the algebraic closure of Fq.
We adopt the Fq-linear representation of PG(5, q4), i.e., we regard the points of PG(5, q4) as elements of a Desarguesian spread of PG(6t - 1, q) and Lv as the subset of the spread with non-empty intersection with a (3t - 1)-dimensional subspace n^ of PG(6t - 1, q); also, we consider PG(6t - 1, q) as subgeometry of PG(6t - 1, q4) (cf. Section 2). Let f be the polynomial with coefficients in F2 such that V1 = V(f), hence the Fq-linear representation of V1 is V(f, f,..., f) n Fix a. Let Vf be V(f, f,..., f).
We have that V1 n Lu = 0 ^ Vf n = 0 ^ Vf n Fix an ny (Fqt) = 0. Let W be nU (Fq) n V f. We observe that W = Wa, hence dim W = dim W n Fix a. We stress
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out that W is defined by polynomials in Fqt [x0,x 1,..., x6t-1] but it might not contain any Fqt -rational point. The linear representation of nN is the (3t - 1)-dimensional subspace nN of Fix a that is partitioned by the spread elements {np : P G nN}. As Ly n nN = 0, we must have ny n nN = 0 and hence, by Lemma 2.1, ny (Fqt) n nN (Fqt) = 0 and
nu (Fq) n nN (Fq) = 0.
Theorem 3.1. Let P G W, then dim Tp (Vt) n (Fq) = dim Tp (V t) - 3t. Proof. The subspace ny (Fq) has codimension 3t, hence
dimTP(Vt) n ny(Fq) > dimTP(Vt) - 3t. Let P G (Po, Pi,..., Pt-i) with Pi g n¿(Fq). We have that
Tp (Vt) = (Tp0 (Vi ),Tp (Vf),...,Tp- (Vf'-))
and nN cTft (Vf) Vi, hence nN (Fq) c Tp (Vt )._Since ny (Fq) n nN (Fq) = 0 and dimnN(Fq) = 31 - 1, we have dimTp(Vt) n n v(Fq) < dimTp(Vt) - 3t, hence the statement follows.	□
Corollary 3.2. We have dim W = 2t - 1, hence W is a complete intersection.
Proof. If P is non-singular for Vt, then dim Tp(Vt) = dim(Vt) = 5t - 1, whereas
dim Tp (Vt) > 5t - 1 for P G Sing(Vt). As W = Vt n (Fqi , Tp (W) = Tp (Vt) n ny (Fqt). By Theorem 3.1,
dimTp(Vt) n ny(Fq) = dimTp(Vt) - 3t > 2t - 1,
and
dim Tp (Vt) n ny (Fq) > 2t - 1
only if P G Sing(Vt). Hence dim W = 2t - 1. We observe that 2t -1 = dimny(Fq) -t, hence W is a complete intersection.	□
Corollary 3.3. Sing(W) = Sing(Vt) n (Fq).
Proof. By Theorem 3.1, dimTp(W) = dimTp(Vt) - 3t, hence dim Tp(W) > dim W = 2t - 1 if and only if dim Tp (Vt) > 5t - 1 = dim( Vt), i.e., P G Sing(Vt).	□
If a variety Y is a complete intersection and dim Y - dimSing(Y) > 2, then Y is normal (see [19, Chapter 2, Section 5.1] for the general definition of normal varieties). An important tool for our proof is the following reformulation of the Hartshorne connectedness theorem ([9]).
Theorem 3.4 ([3, Theorem 2.1]). If Y is a normal complete intersection, then Y is absolutely irreducible.
Theorem 3.5. If W is reducible and Ly n V 1 = 0, then Ly is a plane which is isomorphic to PG(2, q4) disjoint from V1.
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Proof. If W is reducible, then W is not normal and hence dim Sing(W) = dim W — 1 = 2t — 2. A point P G W is singular if and only if P G Sing(V4 ) n ny (Fq). We have
Sing(Vt) = Ut-o1 Si, with
Si = Join(Sing(Vf ), Vf*, j = i) = Join(V, Vf* , j = i)
(see Section 2), so Sf * = Si and hence
dim Sing(Vt) n ny (Fq) = dimS0 n ny(Fq) = 2t — 2.
Let P G So n nu (Fq) with P = (P0, ... ,Pt-o ), Po G V, P G Vf, i = 1, 2,..., t — 1, then the tangent space TP(S0 n ny(Fq)) is
(Tp0 (V), Tp (Vf ),..., Tp- (Vft-1 )) n nu(Fq)
= Kl n K n K3 n Hi n • • • n h4*_o n ny(Fq),
with Ki, Hj hyperplanes of PG(6t — 1, q4) such that Ki projects on the hyperplane Ki of n0 for i = 1,2,3, H1 projects on the hyperplane Hj of n Vj = 1, 2,..., t — 1,
Ko n K2 n K3 = Tp0(V) and Hj = TPj (Vfj). We can take Ko,K2,Ks such that Ko n V = v2(^), K2 n V = v2(^2) and K3 n V = v2(^ o U ¿2). Hence, K\ n K2l n H n • • • n Htl_ o contains nN and so dim K2l n K| n Hi n • • • n H4- o n ny (Fq) is the smallest possible, i.e., 2t — 2. Hence,
Ki d K2l n K3 n Hi n • • • n H_o n ny(Fq)
and the projection of ny (Fq) on n0 is a subspace n0 such that the tangent space of P0 at
V	n n0 has codimension 2 in n0. So either the codimension of n0 n V in n0 is 2 or n0 n V has codimension 3 in n0 but it has singular points. Suppose we are in the latter case. The Veronese variety V is smooth, hence n0 can by a 3 or 4-dimensional subspace of n0. If n0 is a hyperplane of n0 and n0 n V (Fq ) has singular points, then n0 n V is is either v2(^2) or v2(^ o U ¿2). In the first case, n0 contains . A plane = PG(2, q4) is a Fq-linear set of rank 3t, so n0(Fqt ) = PG(4, q4) contains two linear sets of rank 3t that must intersect by Grassmann, i.e., n Vo = 0. If n0 n V = v2(^ o U ¿2), then n0 contains the tangent space at V of the point P = v2 (^ o n ) and it is the unique tangent space at V contained in n0. Let t be the collineation induced by the field automorphism x ^ xq , then both n0 and V(Fq) are fixed by t, hence ï> (V)T = Tp(V) and, by Lemma 2.1, Tp (V) contains a PG(2, q4). Again, by Grassmann, n Vo = 0. Suppose that n0 is a 3-dimensional space, so it contains 4 points counted with their multiplicity and at least one of them is multiple. If P is a multiple point and it is Fqt -rational, i.e., P = PT, then n0 contains a line tangent to
V	at P that it is fixed by t and hence contains a PG(1,q4), so, by Grassmann, nVo = 0. So a multiple point P must be Fqst -rational, but also PT G n0 n V would be, hence s = 2 and we have n0 n V = {P, PT }, with P g n0(q24). The line joining P and PT is set-wise fixed by t and so it contains a PG(1, q4), yielding again n Vo = 0. So suppose that the codimension of n0 n V(Fq) in n0 is 2. Hence n0 is either a 3-dimensional space or a plane. Suppose that n0 is a 3-dimensional space and so dimn0 n V (Fq ) = 3 — 2 = 1. Since n0 n V(Fq) is the Veronese embedding of the intersection of two distinct conics, n0 contains the Veronese embedding of a line I and it cannot contain the embedding of any other line. Hence v2(^)T c n0 implies v2(^)T = v2(^) and so (v2(^)) contains a
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plane = PG(2, qf). By Grassmann, LU n Vi = 0. Hence n0(qf) is a plane and so
Lu = n0(q4).	□
Theorem 3.6. If W is absolutely irreducible and q > 2 • 34t, then W n Fix a has at least one point.
Proof. By [2, Corollary 7.4], an absolutely irreducible algebraic variety of PG(n - 1, q) with dimension r and degree S for q > max{2(r + 1)S2,2S4} has at least one Fq-rational point. By r = 2t - 1 and S < 3f = deg Vf, we have the statement.	□
We conclude the section with our main result.
Theorem 3.7. Let q > 2 • 34t be even. The only symplectic semifield spread of PG(5, qf) whose associate semifield has center containing Fq, is the Desarguesian spread.
Proof. By Theorems 3.6 and 3.5, we have that the only Fq-linear set of rank 3t disjoint from V1 is a plane. The planes disjoint from V1 form a unique orbit under the action of G (see Theorem 2.4). In this case, the linear set is Fqt -linear as well, hence the semifield associated to the spread is 3-dimensional over its center. By [17], in even characteristic this implies that the semifield is a field, hence the spread is Desarguesian.	□
Corollary 3.8. Let q > 2 • 34t be even. Then a commutative semifield of order q3t, with middle nucleus containing Fqt and center containing Fq, is afield.
Remark 3.9. We emphasize that the hypothesis of even characteristic is crucial for all our arguments: only for even q the variety V1 contains the plane nN, and using L n nN = 0 we can prove that W is a complete intersection, i.e. W has codimension t, and the singular points of W are just the ones coming from V.
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