  The uniform position principle states that, given an irreducible
nondegenerate curve C in the projective r-space $P^r$, a general (r-2)-plane L
is uniform, that is, projection from L induces a rational map from C to $P^1$
whose monodromy group is the full symmetric group. In this paper we show the
locus of non-uniform (r-2)-planes has codimension at least two in the
Grassmannian for a curve C with arbitrary singularities. This result is optimal
in $P^2$. For a smooth curve C in $P^3$ that is not a rational curve of degree
three, four or six, we show any irreducible surface of non-uniform lines is a
Schubert cycle of lines through a point $x$, such that projection from $x$ is
not a birational map of $C$ onto its image.
