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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.

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About the Author: (From the MathFest 2014 MAA Prizes and Awards Booklet)

Will Traves was born in Toronto, Canada and completed an undergraduate degree at Queen's University. He received his Ph.D. from the University of Toronto under the supervision of Karen Smith and Mark Spivakovsky, neither of whom were on the faculty at Toronto at the time. After a short post-doc at U.C. Berkeley with Bernd Sturmfels, he joined the faculty at the U.S. Naval Academy in 1999, where he is currently the chair of the Mathematics Department. Will is a brown dot Project NExT fellow and helped found Section NExT, the faculty development program of the MD/DC/VA section of the MAA. Will's mathematical interests are very broad and include pure and applied mathematics, operations research, and statistics. He enjoys games of all kinds, including chess and backgammon.

Subject classification(s): Geometry and Topology | Algebraic Geometry
Publication Date: 
Wednesday, August 13, 2014

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