Images of conics under derivation
Let \(L_\infty\) be a line of the desarguesian projective plane \(PG(2,q^2)\) of order \(q^2\), \(q > 2\), so that \(AG(2,q^2) = PG(2,q^2) \setminus L_\infty\) is an affine plane of the same order. A derivation set \(\mathcal D\) of \(AG(2,q^2)\) is a set of \(q + 1\) points of \(L_\infty\) such that for any two affine points \(P\) and \(Q\) for which the ideal point of the line \(PQ\) is in \(\mathcal D\) there is a Baer subplane containing \(P\), \(Q\) and whose ideal points coincide with \(\mathcal D\). A new incidence structure \({\mathcal D} AG(2,q^2)\) is defined as follows: the points are just the points of \(AG(2,q^2)\); a line is either a line of \(AG(2,q^2)\) with ideal point not in \(\mathcal D\) or the points of a Baer subplane in \(AG(2,q^2)\) whose ideal points coincide with those of \(\mathcal D\). Incidence is the natural containment relation. Then \({\mathcal D}AG(2,q^2)\) is an affine plane of order \(q^2\) whose projective completion is the Hall plane \(\text{Hall} (q^2)\), of the same order. If \(q \geq 3\), and \(\mathcal C\) is a nondegenerate conic of \(AG(2,q^2)\), then \(\mathcal C\) is contained in a unique nondegenerate conic \(\overline {\mathcal C}\) in \(PG(2,q^2)\). Moreover, as a pointset, \(\mathcal C\) is called a derived conic in \(\text{Hall} (q^2)\). Many authors have considered the question as to when is this derived conic also an arc (no three points collinear) in \(\text{Hall} (q^2)\), and if so to determine how it extends to a complete arc. In general, given any conic \(\mathcal C\) in \(AG(2,q^2)\), the line \(L_\infty\) may be a secant, tangent or external line to \(\overline {\mathcal C}\). In the paper under review the cases considered are when \(L_\infty\) is secant to \(\overline {\mathcal C}\), \(q \geq 3\), and for \(q\) even, \(q > 2\), when \(L_\infty\) is tangent to \(\overline {\mathcal C}\). In the former case, the authors treat the case where the two points of \(\overline{\mathcal C}\) on \(L_\infty\) are in \(\mathcal D\), and they exhaust the possibilities. In the latter case, let \(P\) be the point of \(L_\infty\) in \(\overline {\mathcal C}\), and let \(N\) be the nucleus of \(\overline{\mathcal C}\), which must also be on \(L_\infty\). The authors treat completely the case where at least one of \(P\), \(N\) is in \(\mathcal D\). Finally, with the help of a computer, all possibilities when \(q = 3\) are determined. We mention a sample result that corrects a claim made in previously published work. Let \(\mathcal C\) be the nondegenerate conic \(x^2 - cy^2 = d\) in \(AG(2,q^2)\), where \(d\) is a fixed non-square in \(GF(q^2)\) and \(c \in GF(q)\), and let \(\mathcal D\) be the real derivation set for \(AG(2,q^2)\). Then \(\mathcal C\) completes to an oval in \(\text{Hall}(q^2)\) if \(c\) is a non-square in \(GF(q)\), and is a complete \((q^2 - 1)\)-arc if \(c\) is a square in \(GF(q)\). It appears to the reviewer, based on remarks made in the paper, that the general problem is now completely solved except when \(q > 3\) and \(L_\infty\) is an external line of \(\overline {\mathcal C}\).
scientific article; zbMATH DE number 896541