| Speaker: Tomasz Popiel Affiliation: The University of Auckland Time: 15:00 Thursday, 29 July, 2021 Location: 303-257 |
| A conic in a projective plane PG(2,F) is the zero locus of a (ternary) quadratic form. The quadratic forms comprise a vector space W of dimension 6, and the subspaces of PG(W) = PG(5,F) are called linear systems of conics. One- and two-dimensional systems are called pencils and nets, respectively. The problem of classifying such systems under the natural action of PGL(3,F) dates back at least to Jordan, who classified pencils over the real and complex numbers in 1906-1907. The analogous problem for finite fields of odd characteristic was solved by Dickson in 1908. Beyond that, things get a bit confusing. Campbell (1928) produced an incomplete (and in some cases erroneous) classification of pencils over finite fields of even characteristic, yet a complete classification in this case mysteriously appears in at least one well-known text book, without proof and attributed to Campbell. For nets (as opposed to pencils), things are even less clear. Wilson (1914) attempted to classify the nets of 'rank one' over finite fields of odd characteristic, where 'rank one' means that a net contains at least one repeated line. His classification, much like Campbell's, was incomplete and contained errors. I will describe some recent work with Michel Lavrauw (Istanbul) and John Sheekey (Dublin) in which we (i) complete (and correct) Wilson's classification, and (ii) calculate certain combinatorial invariants of the 'rank one' nets, which completely determine the orbit of a given net. I will also say a bit about work in progress in which we aim to complete the classification in full, i.e. to classify the nets not containing a repeated line. (This situation is apparently far more complicated and the answer seems to depend, in particular, on the number of projectively inequivalent nonsingular cubic curves in PG(2,F).) |