In joint work with Chiodo, Eisenbud, and Schreyer, we formulate, and in some cases prove, three statements concerning the purity of the resolution of various rings one can attach to a generic curve of genus g and a torsion point of order 1 in its Jacobian. These statements can be viewed as analogues of Green's conjecture, and we verify them computationally for bounded genus. We then compute the cohomology class of the corresponding non-vanishing locus in the moduli space R_(g,l) of twisted level l curves of genus g and use this to derive results about the birational geometry of R_(g,l). For instance, we prove that R_(g,3) is a variety of general type when g>11. I will also discuss the surprising failure of the Prynn-Green conjecture for genera which are powers of 2.