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.


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.

| cite | improve this answer | |

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.