Monodromy of projective curves

The uniform position principle states that, given an irreducible non- degenerate curve $C \subset \mathbb{P} ^r (\mathbb{C} )$, a general $(r-2)$-plane $L \subset \mathbb{P} ^r$ is uniform; that is, projection from $L$ induces a rational map $C \dashrightarrow \mathbb{P} ^{1}$ 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 $x \in \mathbb{P} ^r$ such that projection from $x$ induces a map $C \dashrightarrow \mathbb{P} ^{r-1}$ 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 $\mathbb{P} ^3$, 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.