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Given a compact Riemann surface |$X$|⁠ , we consider the line, in the space of holomorphic 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. [ABSTRACT FROM AUTHOR]