From a Pascal construction to the Burkhardt quartic

Abstract

We continue the study of Pascal-type residual constructions in projective four-space. Starting from two $k$-tuples of hyperplanes in $\mathbb P^4$ such that the $k$ diagonal intersection planes are contained in a hyperplane, one obtains a residual hypersurface of degree $k-1$ containing the remaining $k^2-k$ planes. In this work we consider the case $k=5$, where the twenty residual planes are contained in a quartic threefold. A balanced specialization of this construction is projectively equivalent to the celebrated Burkhardt quartic. In this model the twenty residual planes form one half of the forty Jacobi planes on the Burkhardt quartic. We reveal their incidence structure as governed by the directed complete graph on five vertices. The forty nodes naturally forced by these planes split as $30+10$, and the Burkhardt specialization adds five further nodes. We also write down the complementary twenty Jacobi planes explicitly and describe all forty Steiner hyperplanes in Pascal coordinates.