This question is a follow-up to this question which I asked on MSE.
Let $f: X \rightarrow Y$ be a surjective morphism of schemes, and $\mathscr{F}$ a coherent sheaf on $Y$. Are there conditions we can put on $X$ and $Y$ that imply that the global pullback map $H^0(Y, \mathscr{F}) \rightarrow H^0(X, f^* \mathscr{F})$ is injective?
If $f$ is flat or $\mathscr{F}$ is locally free, this map is always injective. In addition, this answer to the linked question on MSE indicates a proof that, at least when the schemes are noetherian and we either assume that $Y$ is regular or that $X$ and $Y$ are both normal, the same is true when $f$ is any finite map (of course, by miracle flatness, this is only interesting when $X$ is not Cohen-Macaulay or $Y$ is not regular). In the comments and edits to this question, we found counterexamples when $X \rightarrow Y$ is the normalization of a curve and when $X$ is smooth but not connected. Both of these are, to some degree, related to the phenomenon of $f$ "breaking apart infinitesimal neighborhoods". (Both of the modules considered were infinitesimal thickenings of points).
Can this sort of behavior occur when $X, Y$ are connected smooth varieties (ideally over $\mathbb{C}$)? It seems to me like the natural place to find a counterexample is when $f$ is a birational morphism and $\mathscr{F}$ has support on the image of the exceptional locus, but I have not been able to construct one along these lines.
Best Answer
Begin with $Y$ equal to the affine plane, $\text{Spec}\ R$, for $R=k[s,t]$. Let $\overline{f}:\overline{X}\to Y$ be the blowing up of $Y$ at the ideal $\mathfrak{m} = \langle s,t \rangle$. Define $I\subset \mathfrak{m}$ to be the ideal $\langle s,t^2 \rangle$. Define $\mathcal{G}$ to be $\widetilde{R/\mathfrak{m}}$, define $\mathcal{F}$ to be $\widetilde{R/I}$, and define $p:\mathcal{F}\to \mathcal{G}$ to be the natural surjection. The pullback $\overline{f}^*\mathcal{G}$ is the structure sheaf of the exceptional divisor $E$. The pullback $$\overline{f}^*p:\overline{f}^*\mathcal{F}\to \overline{f}^*\mathcal{G},$$ is an isomorphism except at a single point $q$ on $E$.
Define $X$ to be the open complement of $\{q\}$ in $\overline{X}$. Define $f:X\to Y$ to be the restriction of $\overline{f}$ to $X$. Since we remove a single closed point from the positive dimensional fiber $E$ of $\overline{f}$, the morphism $f$ is still surjective. On $X$, the natural homomorphism $$f^*p:f^*\mathcal{F}\to f^*\mathcal{G},$$ is an isomorphism. Therefore, the pullback map $H^0(Y,\mathcal{F}) \to H^0(X,f^*\mathcal{F})$ factors through the map $$H^0(p):H^0(Y,\mathcal{F})\to H^0(Y,\mathcal{G}).$$ This map has a one-dimensional kernel.