  Motivated by the study of hyperkahler structures in moduli problems and
hyperkahler implosion, we initiate the study of non-reductive hyperkahler and
algebraic symplectic quotients with an eye towards those naturally tied to
projective geometry, like cotangent bundles of blow-ups of linear arrangements
of projective space. In the absence of a Kempf-Ness theorem for non-reductive
quotients, we focus on constructing algebraic symplectic analogues of additive
quotients of affine spaces, and obtain hyperkahler structures on large open
subsets of these analogues by comparison with reductive analogues. We show that
the additive analogue naturally arises as the central fibre in a one-parameter
family of isotrivial but non-symplectomorphic varieties coming from the
variation of the level set of the moment map. Interesting phenomena only
possible in the non-reductive theory, like non-finite generation of rings,
already arise in easy examples, but do not substantially complicate the
geometry.
