2011/02/07 by Xander Faber, Faber, Xander · 1 citation
Mathematics · #11S15 #14H05 #Algebraic Geometry (math.AG) #FOS: Mathematics #Mathematical Analysis and Transform Methods #Mathematical and Theoretical Analysis #Number Theory (math.NT) #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.1102.1432
openalex publication_date 2011/02/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given a nonconstant holomorphic map f: X -> Y between compact Riemann surfaces, one of the first objects we learn to construct is its ramification divisor Rf, which describes the locus at which f fails to be locally injective. The divisor Rf 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 = Cp. 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. This article initiates a detailed study of the ramification locus for self-maps f: P1 -> P1. This simplest first case has the benefit of being approachable by concrete (and often combinatorial) techniques.