Monodromy of projective curves

The uniform position principle states that, given an irreducible non-degenerate curve C in P^r (projective r-space over the complex numbers), a general(r-2)-plane L in P^r is uniform, that is, projection from L induces a rational map C -> P^1 whose monodromy group is the full symmetric group. In this paper we first show the locus of non-uniform r-2-planes has codimension at least two in the Grassmannian. This result is sharp because, if there is a point x in P^r such that projection from x induces a map C -> P^{r-1} that is not birational onto its image, then the Schubert cycle \sigma(x) of (r-2)-planes through x is contained in the locus of non-uniform (r-2)-planes. For a smooth curve C in P^3, we show that any irreducible surface of non-uniform lines is a cycle \sigma(x) as above, unless C is a rational curve of degree three, four, or six.