Graduate Student Seminar, 4:30pm November 11, 2002, Hayes-Healy 231


Sindi Sabourin


Hilbert Functions of Points


We first define the projective plane as the extension of the affine plane where we demand that parallel lines meet at infinity. We study finite sets of points in the projective plane and we introduce the Hilbert function as an algebraic tool that provides geometric information about the set of points. The information is encoded as a sequence of dimensions of certain vector spaces. We show how Hilbert functions can be used to solve the following classical geometry problem: Given a hexagon whose vertices lie on a conic, then the three points of intersection in the projective plane of pairs of lines which extend opposite sides of the hexagon are collinear.

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