The connectedness of the moduli space of maps to homogeneous spaces

TL;DR

This paper proves the connectedness of the moduli space of maps to homogeneous spaces using degeneration techniques, and shows that in genus 0, this space is also rational, extending understanding of its geometric structure.

Contribution

It establishes the connectedness of the moduli space of maps to G/P and demonstrates the rationality of the genus 0 case, linking to representation theory results.

Findings

Connectedness of the moduli space of maps to G/P.

Irreducibility of genus 0 moduli space of maps.

Rationality of the genus 0 moduli space.

Abstract

We prove the connectedness of the moduli space of maps (of fixed genus and homology class) to the homogeneous space G/P by degeneration via the maximal torus action. In the genus 0 case, the irreducibility of the moduli of maps is a direct consequence of connectedness. An analysis of a related Bialynicki-Birula stratification of the map space yields a rationality result: the (coarse) moduli space of genus 0 maps to G/P is a rational variety. The rationality argument depends essentially upon rationality results for quotients of SL2 representations proven by Katsylo and Bogomolov.