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Algebra Seminar
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Tristram de Piro
University of Edinburgh

A non-standard Bezout Theorem

Friday 12th November, 12:05-12:55pm, Carslaw 157.

For projective plane curves without common components over an algebraically closed field K, defined by equations f and g, we traditionally define their intersection multiplicity I(f,g,x0) at a point x0 as length(K[x,y]/(fx0,gx0)), the equations fx0 and gx0 being obtained by translating the curves to the origin. We consider an alternative, more intuitive definition of intersection multiplicity using the techniques of non-standard analysis, originally developed for the reals. Namely, we define I'(f,g,x0) by deforming each curve generically and infinitesimally and counting the number of points of intersection in an infinitesimal neighborhood of x0. I will give a brief sketch of the language of Zariski structures, which allows the techniques of non-standard analysis to be applied rigorously in the context of algebraic geometry, allowing one to make sense of the definition I'. I will then give a proof that the two definitions are equivalent in all characteristics for the full family of projective curves in P2(K) and obtain a simple proof of Bezout's theorem as a straightforward corollary.