Self maps of P¹ with prescribed ramification in characteristic p

Original · EN

Using limit linear series and a result controlling degeneration from separable maps to inseparable maps, we give a formula for the number of self-maps of the projective line with ramification to order eᵢ at general points Pᵢ, in the case that all eᵢ are less than the characteristic. Unlike the case of characteristic 0, the answer is not given by Schubert calculus, nor is the number of maps always finite for distinct Pᵢ, even in the tamely ramified case. However, finiteness for general Pᵢ, obtained by exploiting the relationship to branched covers, is a key part of the argument.