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.