  A family of covariant non-linear projections from the space of SO(10) Weyl
spinors onto the space of pure SO(10) Weyl spinors is presented. The Jacobian
matrices of these projections are related to a linear projector which was
previously discussed in pure spinor string literature and which maps the
antighost to its gauge invariant part. Only one representative of the family
leads to a Hermitian Jacobian matrix and can itself be derived from a scalar
potential. Comments on the SO(1,9) case are given as well as on the
non-covariant version of the projection map. The insight is applied to the
ghost action of pure spinor string theory, where the constraints on the fields
can be removed using the projection, while introducing new gauge symmetries.
This opens the possibility of choosing different gauges which might help to
clarify the origin of the pure spinor ghosts. Also the measure of the pure
spinor space is discussed from the projection point of view. The appendix
contains the discussion of a toy model which served as a guideline for the pure
spinor case.
