The space of complete quotients

TL;DR

This paper introduces complete quotients over the projective line, showing they form smooth projective varieties that serve as modular compactifications of algebraic maps to Grassmannians, aligning with previous constructions.

Contribution

It constructs and characterizes complete quotients as smooth projective varieties, providing a new modular compactification of spaces of algebraic maps to Grassmannians.

Findings

Complete quotients form smooth projective varieties.

They coincide with previously constructed varieties.

They provide modular smooth compactifications with normal crossing boundaries.

Abstract

We introduce complete quotients over the projective line and prove that they form smooth projective varieties. The resulting parameter spaces coincide with the varieties constructed in [HLS11] and [Shao11]. Hence they provide modular smooth compactifications with normal crossing boundaries of the spaces of algebraic maps from the projective line to Grassmannian varieties.