Let $K$ be the canonical line bundle of a compact Riemann surface $M$ of genus $g$. Consider the pull back of $K$ on $M \times M$ via projection on the first factor. What is the dimension of the space of global holomorphic sections of this pull back line bundle on $M \times M$?

I can calculate its Euler characteristic by Riemann–Roch (Noether’s formula), but how does one calculate $H^0$?