On a Pascal-type theorem for lines in ℙ 3

Abstract

Abstract We prove a three-dimensional Pascal-type theorem for configurations of lines on a smooth quadric surface in ℙ 3 . Starting from two triples of skew lines in the two rulings of the quadric, we show that the associated grid of nine intersection points determines natural triples of planes whose residual intersection lines are coplanar. This construction recovers the classical Pascal line after taking a plane section of the quadric. We further study the six diagonal planes associated with permutations of the grid points, showing that they split into two pencils with skew polar axes, and describe a canonical involutive mirror construction on the grid. We work over the field of complex numbers ℂ.