Some remarks to a Theorem of van Geemen

In [ Ge], Bert van Geemen computed the dimension of the space of the fourth power of the theta nullwerte. In [SM2], it has been observe that all linear relations between the ${\theta }_m^4$ are consequences of the quartic Riemann relations. In this note, we want to give a new proof of these result and extend them. In a last section we treat the linear dependencies between arbitrary powers $\vartheta [m{]}^k$. We will show that $k=4$ is the only case where such dependencies can occur. For this reason, we give a slightly different title: Some remarks to a Theorem of van Geemen