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We study the set ${\mathcal P}_S$ consisting of all branched holomorphicprojective structures on a compact Riemann surface $X$ of genus $g \geq 1$ andwith a fixed branching divisor $S:= \sum_{i=1}^d n_i\cdot x_i$, where $x_i \inX$. Under the hypothesis that $n_i=1$, for all $i$, with $d$ a positive eveninteger such that $d \neq 2g-2$, we show that ${\mathcal P}_S$ coincides with asubset of the set of all logarithmic connections with singular locus $S$,satisfying certain geometric conditions, on the rank two holomorphic jet bundle$J^1(Q)$, where $Q$ is a fixed holomorphic line bundle on $X$ such that$Q^{\otimes 2}= TX\otimes {\mathcal O}_X(S)$. The space of all logarithmicconnections of the above type is an affine space over the vector space $H^0(X,K^{\otimes 2}_X \otimes {\mathcal O}_X(S))$ of dimension $3g-3+d$. We concludethat ${\mathcal P}_S$ is a subset of this affine space that has codimenison $d$at a generic point.

Branched projective structures on a Riemann surface and logarithmic connections