Critical exponents, colines, and projective geometries (Q2703024)

The author proves that the following conjecture of \textit{J. G. Oxley} [`Matroid theory', Oxford Univ. Press (1992; Zbl 0784.05002), p. 468] is not valid.NEWLINENEWLINENEWLINEConjecture. Let \(G\) be a rank-\(n\) \(GF(q)\)-representable simple matroid with critical exponent \(n-\gamma\). If, for every coline \(X\) in \(G\), \(c(G|X;q)= c(G;q)-2=n- \gamma-2\), then \(G\) is the projective geometry \(PG(n-1,q) \).NEWLINENEWLINENEWLINE(For definitions and denotations see Oxley's book.)NEWLINENEWLINENEWLINEThe rank \(n\), the critical `co-exponent' \(\gamma\) and the order \(q\) of the field \(GF(q)\) are called the parameters of Oxley's conjecture.NEWLINENEWLINENEWLINEThe construction of counterexamples uses the fact that for a subset \(C\) of \(PG(n-1,q)\) the complement \(PG(n-1,q)\setminus C\) is a counterexample to Oxley's conjecture with parameters \(n,\gamma,q\) if and only if \(C\) is a \((\gamma,2)\)-cordon containing no \((\gamma+1)\)-dimensional subspaces (Theorem 1.2).NEWLINENEWLINENEWLINESo it is sufficient to find or construct suitable \((\gamma,2)\)-cordons. The main part of the paper is devoted to solving this problem.NEWLINENEWLINENEWLINEThe examples presented in the paper show that, for a given prime power \(q\) and a given positive integer \(\gamma\), Oxley's conjecture holds for only finitely many ranks \(n\).