Degenerations of Pascal Lines

TL;DR

This paper investigates the degenerations of Pascal lines when six points on a conic coalesce, identifying indeterminacy loci and resolving them via blow-ups to understand the geometry of degenerate cases.

Contribution

It introduces a method to analyze and resolve the indeterminacies of Pascal lines in degenerate configurations using blow-ups, extending classical Pascal's theorem.

Findings

Identified the indeterminacy locus where Pascal lines are not well-defined.

Developed a blow-up technique to define Pascal lines in degenerate cases.

Analyzed the geometry of Pascal lines in degenerate configurations.

Abstract

Let $\mathcal{K}$ denote a nonsingular conic in the complex projective plane. Pascal's theorem says that, given six distinct points $A,B,C,D,E,F$ on $\mathcal{K}$, the three intersection points $AE \cap BF, AD \cap CF, BD \cap CE$ are collinear. The line containing them is called the Pascal line of the sextuple. However, this construction may fail when some of the six points come together. In this paper, we find the indeterminacy locus where the Pascal line is not well-defined and then use blow-ups along polydiagonals to define it. We analyse the geometry of Pascals in these degenerate cases. Finally we offer some remarks about the indeterminacy of other geometric elements in Pascal's hexagrammum mysticum.