Flat family of torsion-free coherent sheaves over $\mathbb{A}^1$

Question

Let $k$ be an algebraically closed field of characteristic zero. We consider $f:\mathfrak{X}\to \mathbb{A}^1$ an smooth projective morphism of pure relative dimension $n$, where $\mathfrak{X}$ is an smooth algebraic variety.

Suppose that $\mathcal{F}$ is a torsion-free coherent sheaf on $\mathfrak{X}$ which is flat over $\mathbb{A}^1$. Then, $f_*\mathcal{F}\cong \text{H}^0(\mathfrak{X},\mathcal{F})^{\sim}$ is a torsion-free coherent sheaf on the smooth curve $\mathbb{A}^1$, which is therefore locally free. Even better, algebraic vector bundles over the affine line are trivial.

My question is the following: Suppose that $\text{H}^0(\mathfrak{X}_t,\mathcal{F}_t)$ are all isomorphic and that the map $f_*\mathcal{F}\otimes \kappa(t) \to \text{H}^0(\mathfrak{X}_t,\mathcal{F}_t)$ is an isomorphism, for every closed point $t\neq 0$.

What can we say about $h^0(\mathfrak{X}_0,\mathcal{F}_0)$ in this particular situation or, equivalently (by Mumford "Abelian varieties" II.5 Corollary 2), can we expect $f_*\mathcal{F}\otimes \kappa(0)\to \text{H}^0(\mathfrak{X}_0,\mathcal{F}_0)$ to be an isomorphism ? Maybe an injection ("by semi-continuity") ? Under additional hypotheses ?

Thank you in advance for any comment.

Answer 1

Let $i:Spec(k) \to A^1$ be the embedding if the point zero, $j:X_0 \to X$ the embedding of its fiber, and $p:X_0 \to Spec(k)$ the restriction of $f$. Then we have a Cartesian diagram. A flat base change implies that $$ Li^*\circ Rf_* \cong Rp_* \circ Lj^*, $$ an equality of derived functors. Since $i$ and $j$ are divisorrial embeddings, one has only $L_0$ and $L_1$ nontrivial. Therefore, the spectral sequence for the left hand side gives in degree zero an exact sequence $$ 0 \to i^*f_*(F) \to H^0(Li^*(Rf_*(F)) \to L_1i^*(R^1f_*(F)) \to 0, $$ and the spectral sequence for the right hand side gives $$ 0 \to R^1p_*(L_1j^*(F)) \to H^0(Li^*(Rp_*(F)) \to p_*(j^*(F)) \to R^2p_*(L_1j^*(F)). $$

If $F$ is torsion free then $L_1j^*(F) = 0$, hence $H^0$ equals $p_*(F_0)$. On the other hand, the first term of the first sequence is $f_*(F) \otimes k$. So, the required injectivity follows.