  An arc space of an affine cone over a projective toric variety
  is known to be non-reduced in general. It was demonstrated recently that the
reduced scheme structure is worth studying due to various connections
  with representation theory and combinatorics. In this paper we develop a
general machinery for the description of the reduced arc spaces of affine cones
over toric varieties. We apply our techniques to a number of classical cases
and explore some connections with representation theory of current algebras.
