  Inside the projectivized $k$-th Hodge bundle, we construct a collection of
divisors obtained by imposing vanishing at a Brill-Noether special point. We
compute the classes of the closures of such divisors in two ways, using
incidence geometry and restrictions to various families, including pencils of
curves on K3 surfaces and pencils of Du Val curves. We also show the
extremality and rigidity of the closure of the incidence divisor consisting of
smooth pointed curves together with a canonical or 2-canonical divisor passing
through the marked point.
