The Space of Morphisms on Projective Space

TL;DR

This paper studies the moduli space of morphisms on projective space P^n, proving its geometric quotient, analyzing stabilizer groups, and showing rationality of the quotient for P^1, thus extending previous results in the field.

Contribution

It establishes the geometric nature of the quotient space of morphisms on P^n, bounds stabilizer groups, and proves rationality for P^1 case, advancing the understanding of these moduli spaces.

Findings

The quotient space is geometric and its stable and semistable completions are computed.

Stabilizer groups are bounded solely in terms of n and d.

The quotient space for P^1 is rational for all degrees d > 1.

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.