The Berkovich Ramification Locus for Rational Functions

Algebra Seminar
Thursday, March 31, 2011 - 15:00
1 hour (actually 50 minutes)
Skiles 006
University of Georgia
Given a nonconstant holomorphic map f: X \to Y between compact Riemann surfaces, one of the first objects we learn to construct is its ramification divisor R_f, which describes the locus at which f fails to be locally injective. The divisor R_f is a finite formal linear combination of points of X that is combinatorially constrained by the Hurwitz formula. Now let k be an algebraically closed field that is complete with respect to a nontrivial non-Archimedean absolute value. For example, k = C_p. Here the role of a Riemann surface is played by a projective Berkovich analytic curve. As these curves have many points that are not algebraic over k, some new (non-algebraic) ramification behavior appears for maps between them. For example, the ramification locus is no longer a divisor, but rather a closed analytic subspace. The goal of this talk is to introduce the Berkovich projective line and describe some of the topology and geometry of the ramification locus for self-maps f: P^1 \to P^1.