# From Pascal's Theorem to $$d$$-Constructible Curves

by William Traves

Year of Award: 2014

Publication Information: The American Mathematical Monthly, vol. 120, no. 10, December 2013, pp. 901-915.

Summary (adapted from the MAA Prizes and Awards booklet for MathFest 2014): Beginning with the history of the word syzygy, the author of this paper turns to Pascal's Theorem: if six distinct points $$A, B, C, a, b, c$$ lie on a conic then the lines $$Ab, Bc,$$ and $$Ca$$ meet the lines $$aB, bC$$, and $$cA$$ in three new collinear points. Pascal's Theorem leads to deep but very natural questions about $$d$$-constructible curves. A curve $$S$$ of degree $$t$$ is $$d$$-constructible if there exist $$k= d+t$$ red lines and $$k$$ blue lines so that: the red lines meet the blue lines in $$k2$$ distinct points and $$dk$$ of these points lie on a curve $$C$$ of degree $$d$$ and the remaining $$tk$$ points lie on the curve $$S$$.

Traves grounds the reader firmly in the history and motivation of the problems in this area of algebraic geometry and leads us to an understanding of $$d$$-constructible curves and of the dimensions in which $$d$$-construction is dense. In a lucid and comprehensive exposition of the ideas stemming from Pappus’ Theorem on line arrangements, to its generalization to 6 points on a conic by Pascal, to a converse of Pascal’s Theorem by Braikenridge and Maclaurin, to Möbius’s generalization of Pascal’s theorem, the reader is introduced to work by Eisenbud, Green and Harris on the Cayley-Bacharach Theorem.