Quotients by non-reductive algebraic group actions

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Given a suitable action on a complex projective variety X of a non-reductive affine algebraic group H, this paper considers how to choose a reductive group G containing H and a projective completion of G xₕ X which is a reductive envelope in the sense of math.AG/0703131. In particular it studies the family of examples given by moduli spaces of hypersurfaces in the weighted projective plane P(1,1,2) obtained as quotients by linear actions of the (non-reductive) automorphism group of P(1,1,2).