On non-hyperelliptic flocks
Let \(K\) be the quadratic cone of \(PG (3,q)\) with vertex \(v = (0,0,0,1)\) and equation \(X_0 X_1 = X_2^2\). A flock \(F\) of \(K\) is a partition of \(K \backslash \{v\}\) into \(q\) disjoint irreducible conics. Associated to each flock \(F\) is a translation plane \(T(F)\) of order \(q^2\) and dimension at most two over its kernel. When \(T(F)\) is a semifield (resp. likeable) plane, \(F\) is called a semifield (resp. likeable) flock. For \(q\) even, the semifield and likeable flocks have been classified, so the authors restrict their attention to the case that \(q = p^r\), where \(p\) is an odd prime. If \(f\) and \(g\) are two functions of \(GF (q)\) into itself such that \(f(0) = g(0) = 0\), for each \(t \in GF (q)\) let \(\pi_t\) be the plane with equation \(tX_0 - f(t)X_1 + g(t)X_2 + X_3 = 0\). The conics \(C_t = K \cap \pi_t\) are irreducible and \(F(f,g) = \{C_t |t \in GF (q)\}\) is a flock of \(K\) iff \((g(t) - g(s))^2 + 4(t - s) (f(t) - f(s))\) is a non-square in \(GF (q)\) for all distinct \(t\) and \(s\). Conversely, for each flock \(F\) of \(K\) there are two functions \(f\) and \(g\) of \(GF (q)\) into itself such that \(F = F(f,g)\). If for a given flock \(F = F(f,g)\) there exists a polynomial \(h(t) \in GF (q) [t]\) such that \(g^2(t) + 4tf(t) = nh^2(t)\) \((n\) a non-square) holds as a polynomial identity, then \(F\) is called a non-hyperelliptic flock. The main result of this paper is: Theorem: (a) If \(F\) is a non-hyperelliptic semifield flock, then \(F\) is isomorphic to a linear flock, a Kantor flock constructed via a Knuth semified, or a Ganley flock. (b) If \(F\) is a nonhyperelliptic likeable flock, then \(F\) is isomorphic to a Fisher-Thas-Walker flock or a Kantor likeable flock. [While it is not typical to bring up such matters in a review such as this, I will make an exception to correct the unfortunate type preceding the proof of Theorem 1(b). ``We are not able to \(\ldots\)'' should read ``We are now able to \(\ldots\)''].
scientific article; zbMATH DE number 683264