Given a compact Riemann surface $X$, we consider the line, in the space of sections of $2\Theta$ on $J^0(X)$, orthogonal to all the sections that vanish at the origin. This line produces a natural meromorphic bidifferential on $X\times X$ with a pole of order two on the diagonal. This bidifferential is extensively investigated. In particular we show that it produces a projective structure on $X$ which is different from the standard ones.