Suppose you have two integral Gorenstein projective curves $X$ and $Y$ over a field, and suppose further that we have an isomorphism $$\bigoplus_{n=0}^\infty H^0(X,\omega_X^{\otimes n}) \cong \bigoplus_{n=0}^\infty H^0(Y,\omega_Y^{\otimes n}) $$ of canonical rings. Can we then conclude that $X$ and $Y$ has the same arithmetic genus?
I think this is easy, it is claimed at the end of section 2.3 in the paper Fourier-Mukai partners of singular genus one curves. For some reason this isn't clear to me.
Hint: look at duality for schemes of dimension 1 plus the definition of arithmetic genus via Euler characteristic.