On some fibrations of $\overline{M}_{0,n}$

TL;DR

This paper investigates fiber type morphisms of the moduli space _{0,n} using Kapranov's description, aiming to determine when such morphisms factor through forgetful maps, with results for small n, low-dimensional targets, and certain rational maps.

Contribution

It advances understanding of the factorization of dominant morphisms from _{0,n} by establishing conditions under which they factor through forgetful morphisms, extending previous work.

Findings

Proves factorization for n  7

Establishes factorization when dim X  2

Provides examples of forgetful morphisms with high genus fibers

Abstract

The paper is a second step in the study of $\overline{M}_{0,n}$ started in arXiv:1006.0987 [math.AG]. We study fiber type morphisms of this moduli space via Kapranov's beautiful description. Our final goal is to understand if any dominant morphism $f: \overline{M}_{0,n} \to X$ with positive dimensional fibers factors through forgetful morphisms. We prove that this is true if either $n \leq 7$ or $\rm {dim} X \leq 2$ or the rational map induced on $P^{n-3}$ has linear general fibers. Along the way we give examples of forgetful morphisms whose fibers are connected curves of arbitrarily high positive genus, for $n>>0$.