Rational Functions in One Variable with given Ramification

We sketch the solution to a simple enumerative problem in linear series in characteristic p: given points Pi on the projective line, integers ei < p and d satisfying P i (ei − 1) = 2d − 2, how many maps from P 1 to itself are there of degree d and having ramification index precisely ei at the Pi? Limit linear series are a tool for solving this problem. I’ll explain what they are, and some of the special issues that come up in characteristic p. 1. The main question and background We work throughout over an algebraically closed field k. The material discussed is taken from [2]. The theory of limit linear series has provided a powerful tool for approaching a range of problems in characteristic 0, but people have typically avoided the situation of positive characteristic, on the grounds that inseparability is too difficult to control. We will demonstrate that while this apprehension is reasonably well-justified, it is not an absolute obstruction to obtaining results. Our guiding question is: Question 1.1. Given n general points Pi on P 1 and integers ei ≥ 2, with ∑ i(ei − 1) = 2d− 2, and ei ≤ d for all i, counted modulo automorphism of the image, how many separable maps are there from P to P of degree d which ramify to order ei at the Pi? Notation 1.2. When the answer to the above question is finite, we denote it by N({ei}i). The answer to our question can be rephrased in terms of intersections of Schubert cycles on Grassmannians. Indeed, rational maps of degree d, taken up to automorphism, correspond to 2-dimensional spaces of polynomials of degree d without common factors, so are points in G(1, d). And one checks that each intersection condition cuts out a Schubert cycle in this Grassmannian. In the case of characteristic 0, these Schubert cycles intersect transversely, so the answer to our question is given by Schubert calculus. This was first proved by Eisenbud and Harris [1, Thm. 9.1], as a key part of an argument for the Brill-Noether theorem using degeneration to rational cuspidal curves. Originally, motivation for the characteristic-p version was provided by a relationship with certain logarithmic connections with vanishing p-curvature on vector bundles on P, and from there to vector bundles on higher-genus curves which are semistable, but pull back to unstable bundles under the Frobenius map. Additionally, one can use the work discussed here to derive new existence and nonexistence results on branched covers of P in positive characteristic.