For a degeneration of curves, Tan introduced certain Chern invariants that measure the failure of semistability; these invariants are completely determined by the topological type of the fiber germ. In this talk, I will describe an unexpected connection between these Chern invariants and positivity properties of divisors on the moduli space of stable curves. In particular, assuming the Morsification conjecture, I will obtain a lower bound for the slope of the Chern invariants by applying the Cornalba–Harris criterion for ampleness on the moduli space of stable curves. If time permits, I will also discuss a possible approach to the slope conjecture for effective divisors on the moduli space of stable curves.