2014/07/06 by Jaydeep Chipalkatti, Chipalkatti, Jaydeep
Computer Science · Mathematics · #History and Theory of Mathematics #Mathematics and Applications #math.AC #math.AG #msc:14N05 #msc:51N35 #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.1407.1447
22 pages, with 7 diagrams
arxiv created 2014/07/06 · arxiv updated 2014/07/08
Let \mathcal K denote a smooth conic in the complex projective plane. Pascal's theorem says that, given six points A,B,C,D,E,F on \mathcal K, the three intersection points AE ∩ BF, AD ∩ CF, BD ∩ CE are collinear. This defines the Pascal line of the array [ A · B · C
F · E · D ], and one gets sixty such lines in general by permuting the points. In this paper we consider the variety Ψ of sextuples \A, …, F\, for which some of these Pascal lines coincide. We show that Ψ has two irreducible components: a five-dimensional component of sextuples in involution, and a four-dimensional component of the so-called `ricochet configurations'. This gives a complete synthetic characterisation of points in Ψ. The proof relies upon Gröbner basis techniques to solve multivariate polynomial equations.