  For a compact Riemann surface $X$ of any genus $g$, let $L$denote the line
bundle $K_{X\times X}\otimes {\cal O}_{X\times X}(2\Delta)$ on $X\times X$,
where $K_{X\times X}$ is the canonical bundle of $X\times X$ and $\Delta$ is
the diagonal divisor. We show that $L$ has a canonical trivialisation over the
nonreduced divisor $2\Delta$. Our main result is that the space of projective
structures on $X$ is canonically identified with the space of all
trivialisations of $L$ over $3\Delta$ which restrict to the canonical
trivialisation of $L$ over $2\Delta$ mentioned above. We give a direct
identification of this definition of a projective structure with a definition
of Deligne.We also describe briefly the origin of this work in the study of the
so-called "Sugawara form" of the energy-momentum tensor in a conformal quantum
field theory.
