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III.11: The Theorem on Formal Functions

III.11.1. #to_work

Show that the result of \((11.2)\) is false without the projective hypothesis. For example, let \(X=\mathbf{A}_k^n\), let \(P=(0, \ldots, 0)\), let \(U=X-P\), and let \(f: U \rightarrow X\) be the inclusion. Then the fibres of \(f\) all have dimension 0 , but \(R^{n-1} f_* \mathcal{O}_U \neq 0\).

III.11.2. #to_work

Show that a projective morphism with finite fibres (= quasi-finite (II, Ex. 3.5)) is a finite morphism.

III.11.3. Improved Bertini’s Theorem. #to_work

Let \(X\) be a normal, projective variety over an algebraically closed field \(k\). Let \(D\) be a linear system (of effective Cartier divisors) without base points, and assume that \(D\) is not composite with a pencil, which means that if \(f: X \rightarrow \mathbf{P}_k^n\) is the morphism determined by \(\mathrm{D}\), then \(\operatorname{dim} f(X) \geqslant 2\).

Then show that every divisor in \(\mathrm{D}\) is connected. ¹

III.11.4. Principle of Connectedness. #to_work

Let \(\left\{X_t\right\}\) be a flat family of closed subschemes of \(\mathbf{P}_k^n\) parametrized by an irreducible curve \(T\) of finite type over \(k\). Suppose there is a nonempty open set \(U \subseteq T\), such that for all closed points \(t \in U, X_t\) is connected. Then prove that \(X_t\) is connected for all \(t \in T\).

III.11.5. * #to_work

Let \(Y\) be a hypersurface in \(X=\mathbf{P}_k^N\) with \(N \geqslant 4\). Let \(\widehat{X}\) be the formal completion of \(X\) along \(Y\) (II, \(\S\) ). Prove that the natural map \(\operatorname{Pic}\widehat{X} \rightarrow \operatorname{Pic}Y\) is an isomorphism. ²

III.11.6. #to_work

Again let \(Y\) be a hypersurface in \(X=\mathbf{P}_k^N\), this time with \(N \geqslant 2\).

• If \(\mathcal{F}\) is a locally free sheaf on \(X\), show that the natural map \begin{align*} H^0(X, \mathcal{F}) \rightarrow H^0(\widehat{X}, \widehat{\mathcal{F}}) \end{align*} is an isomorphism.

• Show that the following conditions are equivalent:

  • For each locally free sheaf \(\mathcal{F}\) on \(\widehat{X}\), there exists a coherent sheaf \(\mathscr{F}\) on \(X\) such that \(\mathcal{F} \cong \widehat{\mathscr{F}}\) (i.e., \(\mathcal{F}\) is algebraizable);

  • For each locally free sheaf \(\mathcal{F}\) on \(\widehat{X}\), there is an integer \(n_0\) such that \(\mathcal{F}(n)\) is generated by global sections for all \(n \geqslant n_0\). ³

• Show that the conditions (i) and (ii) of (b) imply that the natural map \(\operatorname{Pic}X \rightarrow \operatorname{Pic}\widehat{X}\) is an isomorphism. ⁴

III.11.7. #to_work

Now let \(Y\) be a curve in \(X=\mathbf{P}_k^2\).

• Use the method of (Ex. 11.5) to show that Pic \(\widehat{X} \rightarrow\) Pic \(Y\) is surjective, and its kernel is an infinite-dimensional vector space over \(k\).

• Conclude that there is an invertible sheaf \({\mathcal{L}}\) on \(\widehat{X}\) which is not algebraizable.

• Conclude also that there is a locally free sheaf \(\mathcal{F}\) on \(\widehat{X}\) so that no twist \(\mathcal{F}(n)\) is generated by global sections. Cf. (II, 9.9.1)

III.11.8. #to_work

Let \(f: X \rightarrow Y\) be a projective morphism, let \(\mathcal{F}\) be a coherent sheaf on \(X\) which is flat over \(Y\), and assume that \(H^i\left(X_y, \mathcal{F}_y\right)=0\) for some \(i\) and some \(y \in Y\). Then show that \(R^i f_*(\mathcal{F})\) is \(0\) in a neighborhood of \(y\).