Twisted derivations and distinct sets of lines that cover the same pairs of points (Q1900012)

A well known construction of projective planes is the method of derivation. Given a projective plane \(\pi = (P,L)\) of order \(n = m^2\) and a set \(D \subseteq l \in L\) of \(m + 1\) collinear points such that for each pair \(p,q \in P \backslash l\) with \((p \vee q) \wedge l \in D\), there exists a Baer subplane of \(\pi\) containing \(p,q\) and all points of \(D\), then we replace the \(m^2 (m + 1)\) lines of \(L \backslash \{l\}\) intersecting \(l\) in a point belonging to \(D\) with the above mentioned Baer subplanes of \(\pi\). The resulting incidence structure is a projective plane on the same point set \(P\) which has all but \(m^2 (m + 1)\) lines in common with \(\pi\). Denote by \(L_1\) the set of the \(m^2 (m + 1)\) removed lines, and by \(L_2\) the set of the new lines. Then \(L_1\) and \(L_2\) cover the same pairs of points, i.e. two points \(p,q \in P\) are contained in an element of \(L_1\) if and only if they are contained in an element of \(L_2\). For \(0 \leq i \leq n + 1\) denote by \(v_i\) the number of points of \(\pi\) which are contained in exactly \(i\) elements of \(L_1\) (or \(L_2)\). These numbers are called the parameters of \(L_1\); in our case we have \(v_0 = n - m\), \(v_{m + 1} = n^2\), \(v_n = m + 1\) and \(v_i = 0\) for the remaining \(i\)'s. The author proves: Given two non-isomorphic projective planes \(\pi\), \(\pi'\) of order \(n = m^2\) on the same point set \(P\) such that \(\pi'\) is obtained from \(\pi\) by replacing \(m^2 (m + 1)\) lines of \(\pi\) such that the set of these lines has the parameters of a derivation, then this line replacement is actually a derivation. More generally, the author considers two non-isomorphic projective planes \(\pi\), \(\pi'\) on the same point set \(P\) and denotes by \(s\) the number of lines of \(\pi\) which are not lines of \(\pi'\). He assumes that the order \(n\) of \(\pi\) is a square \(n = m^2 > 81\). Under these assumptions, he proves the following theorem: \(s \geq m^2 (m + 1)\); and if \(s = m^2 (m + 1)\), then the parameters of the set of lines of \(\pi\) which are not lines of \(\pi'\) are the parameters of a derivation or the parameters of a twisted derivation, i.e. \(v_0 = (1/2)\) \((n - 2)\) \((m - 1)\), \(v_m = n + m\), \(v_{m + 1} = n^2 - nm\), \(v_{2m} = (m + 1) n/2\) and \(v_i = 0\) for the remaining \(i\)'s. The parameters of a twisted derivation are realized by the so-called twisted derivations which are described in the paper. Actually, the author proves a slightly stronger result; he considers partial projective planes and obtains his results in this situation. There are also some results if \(n\) is not a square.