Effectivity of Farkas classes and the Kodaira dimensions of M_{22} and M_{23}.

We develop new methods to study tropicalizations of linear series and show linear independence of algebraic sections. Using these methods, we prove two outstanding cases of the strong maximal rank conjecture and verify that certain divisor classes on M_{22} and M_{23} identified by Farkas are represented by effective divisors. This completes Farkas's program to show that the moduli spaces of curves of genus 22 and 23 are of general type.