
Tristram de Piro
University of Edinburgh
A nonstandard Bezout Theorem
Friday 12th November, 12:0512:55pm,
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,x_{0}) at a point
x_{0} as
length(K[x,y]/(f_{x0},g_{x0})),
the equations f_{x0} and
g_{x0} being obtained by translating the
curves to the origin. We consider
an alternative, more intuitive definition of intersection multiplicity using
the techniques of nonstandard analysis, originally developed for the reals.
Namely, we define I'(f,g,x_{0}) by deforming each curve generically and
infinitesimally and counting the number of points of intersection in an
infinitesimal neighborhood of x_{0}. I will give a brief
sketch of the language
of Zariski structures, which allows the techniques of nonstandard 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 P^{2}(K) and obtain a simple proof of
Bezout's theorem as a straightforward corollary.







