I'm reading this article and I didn't understand this part in the second page of the second chapter:

Why this is true using Riemann-Roch theorem? ($D\doteqdot \inf\{\text{div}(\omega_{g-1}),\text{div}(\omega_g)\}$.
My only background in Algebraic Geometry is Fulton's algebraic curves book, I need help!
Thanks in advance.