Let $X$ be a smooth projective variety, $\mathcal{E}$ a torsion-free coherent sheaf on $X$ and $\mathfrak{d}$ a linear system of divisors in $X$.
I would like to show (at least when $X$ is a surface) the upper semicontinuity of the map $$ D\in\mathfrak{d}\longmapsto h^0(D,\mathcal{E}{\restriction_D})\,.$$ I considered the incidence locus $Z\subset X\times\mathfrak{d}$ and the projections $p:Z\to X$ and $q:Z\to\mathfrak{d}$. If $\mathcal{E}$ is locally free, then $p^*\mathcal{E}$ is flat over $\mathfrak{d}$, so by the semicontinuity theorem $$ h^0(D,\mathcal{E}{\restriction_D})=h^0(q^{-1}(D),p^*\mathcal{E}{\restriction_{q^{-1}(D)}})$$ is indeed upper semicontinuous in $D$. But what if $\mathcal{E}$ is only torsion-free? Is $p^*\mathcal{E}$ still flat over $\mathfrak{d}$?
