TITLE: Applications of non-reductive GIT
QUESTION [7 upvotes]: Geometric invariant theory works well when the algebraic group $G$ acting on a variety is reductive.  There has been recent work by Doran and Kirwan here and here to find a canonical method of constructing GIT quotients for non-reductive groups.  My question is what are potential applications for their work?  One specific application they mention is constructing moduli of hypersurfaces in toric varieties.  I would be interested in knowing of other applications.

REPLY [6 votes]: On 80's Atiyah conference Kirwan spoke about one application. Namely she stated in her talk that there is an application to Green-Griffits conjecture.
You can download the talk here http://www.maths.ed.ac.uk/~aar/atiyah80.htm
and the slides are here http://www.icms.org.uk/downloads/GandP/Kirwan.pdf
I am not sure if this was written down somewhere.