## 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.[^hint.3.11.3]

[^hint.3.11.3]: 
See (10.9.1).
Hints: Use (11.5), (Ex. 5.7) and (7.9).


### 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 $\hat{X}$ be the formal completion of $X$ along $Y$ (II, $\S$ ). Prove that the natural map $\Pic \hat{X} \rightarrow \Pic Y$ is an isomorphism.[^hint.3.11.5]

[^hint.3.11.5]: 
Hint: Use (II, Ex. 9.6), and then study the maps $\Pic X_{n+1} \rightarrow \Pic X_n$ for each $n$ using (Ex. 4.6) and (Ex. 5.5).


### III.11.6.  #to-work

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

a. If $\mathcal{F}$ is a locally free sheaf on $X$, show that the natural map
\[
H^0(X, \mathcal{F}) \rightarrow H^0(\hat{X}, \hat{\mathcal{F}})
\]
is an isomorphism.

b. Show that the following conditions are equivalent:

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

    (ii) For each locally free sheaf $\mathcal{F}$ on $\hat{X}$, there is an integer $n_0$ such that $\mathcal{F}(n)$ is generated by global sections for all $n \geqslant n_0$.[^3.6.11.hint.b]


c. Show that the conditions (i) and (ii) of (b) imply that the natural map $\Pic X \rightarrow \Pic \hat{X}$ is an isomorphism.[^3.11.6.note]

[^3.6.11.hint.b]: 
Hint: For (ii) $\Rightarrow$ (i), show that one can find sheaves $\mathcal{E}_0, \mathcal{E}_1$ on $X$, which are direct sums of sheaves of the form $\mathcal{O}\left(-q_i\right)$, and an exact sequence $\hat{\mathcal{E}}_1 \rightarrow \hat{\mathcal{E}}_0 \rightarrow \tilde{\mcf} \rightarrow 0$ on $\hat{X}$. Then apply (a) to the sheaf $\sheafhom\left(\mathcal{E}_1, \mathcal{E}_0\right)$.

[^3.11.6.note]: 
Note. In fact, (i) and (ii) always hold if $N \geqslant 3$. This fact, coupled with (Ex. 11.5) leads to Grothendieck's proof $[SGA \, 2]$ of the Lefschetz theorem which says that if $Y$ is a hypersurface in $\mathbf{P}_k^N$ with $N \geqslant 4$, then $\Pic Y \cong \mathbf{Z}$, and it. is generated by $\mathcal{O}_Y(1)$. See Hartshorne $[ 5, Ch. IV]$ for more details.


### III.11.7.  #to-work

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

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

b. Conclude that there is an invertible sheaf $\mcl$ on $\hat{X}$ which is not algebraizable.

c. Conclude also that there is a locally free sheaf $\mathcal{F}$ on $\hat{X}$ 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 $y$.