The Space of Morphisms on Projective Space

Abstract

The theory of moduli of morphisms on P^n generalizes the study of rational maps on P^1. This paper proves three results about the space of morphisms on P^n of degree d > 1, and its quotient by the conjugation action of PGL(n+1). First, we prove that this quotient is geometric, and compute the stable and semistable completions of the space of morphisms. This strengthens previous results of Silverman, as well as of Petsche, Szpiro, and Tepper. Second, we bound the size of the stabilizer group in PGL(n+1) of every morphism in terms of only n and d. Third, we specialize to the case where n = 1, and show that the quotient space is rational for all d > 1; this partly generalizes a result of Silverman about the case d = 2.