An epimorphic subgroup arising from Roberts' counterexample

TL;DR

This paper explores a new example of a homogeneous space with an epimorphic stabilizer, extending Roberts' counterexample to Hilbert's fourteenth problem, and demonstrates its properties regarding projective embeddings.

Contribution

It introduces an extension of Roberts' counterexample, showing that the extended group is epimorphic in SL(V) and analyzing the geometric implications.

Findings

The group H is epimorphic in SL(V).

The homogeneous space SL(V)/H admits no projective embeddings with small boundary.

Provides a new example of a homogeneous space with specific embedding properties.

Abstract

In 1994, based on Roberts' counterexample to Hilbert's fourteenth problem, A'Campo-Neuen constructed an example of a linear action of a 12-dimensional commutative unipotent group H_0 on a 19-dimensional vector space V such that the algebra of invariants k[V]^{H_0} is not finitely generated. We consider a certain extension H of H_0 by a one-dimensional torus and prove that H is epimorphic in SL(V). In particular, the homogeneous space SL(V)/H provides a new example of a homogeneous space with epimorphic stabilizer that admits no projective embeddings with small boundary.