Sectional monodromy groups of projective curves

Fix a degree (Formula presented.) projective curve (Formula presented.) over an algebraically closed field (Formula presented.). Let (Formula presented.) be a dense open subvariety such that every hyperplane (Formula presented.) intersects (Formula presented.) in (Formula presented.) smooth points. Varying (Formula presented.) produces the monodromy action (Formula presented.). Let (Formula presented.). The permutation group (Formula presented.) is called the sectional monodromy group of (Formula presented.). In characteristic 0, (Formula presented.) is always the full symmetric group, but sectional monodromy groups in characteristic (Formula presented.) can be smaller. For a large class of space curves ((Formula presented.)), we classify all possibilities for the sectional monodromy group (Formula presented.) as well as the curves with (Formula presented.). We apply similar methods to study a particular family of rational curves in (Formula presented.), which enables us to answer an old question about Galois groups of generic trinomials.