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Let $X$ be a projective variety over an algebraically closed field $k$, $S$ be a $k$-scheme, $E$ be a coherent sheaf on $X \times_k S$, flat over $S$. We know that if $X$ is smooth then $E$ has a locally free resolution. Under what condition on $X$ (other than smooth, for example if $X$ is a reduced, reducible connected curve) does a similar result (existence of locally free resolution) still hold i.e., $E$ has a locally free resolution?

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