## III.12: The Semicontinuity Theorem


### III.12.1.  #to-work

Let $Y$ be a scheme of finite type over an algebraically closed field $k$. Show that the function
\[
\varphi(y)=\operatorname{dim}_k\left(m_y / m_y^2\right)
\]
is upper semicontinuous of the set of closed points $Y$.

### III.12.2.  #to-work

Let $\left\{X_t\right\}$ be a family of hypersurfaces of the same degree in $\mathbf{P}_k^n$. Show that for each $i$, the function $h^i\left(X_t, \mathcal{O}_{X_t}\right)$ is a constant function of $t$.

### III.12.3.  #to-work

Let $X_1 \subseteq \mathbf{P}_k^4$ be the **rational normal quartic curve** (which is the 4-uple embedding of $\mathbf{P}^1$ in $\mathbf{P}^4$ ). Let $X_0 \subseteq \mathbf{P}_k^3$ be a nonsingular rational quartic curve, such as the one in (I, Ex. 3.18b). 

Use (9.8.3) to construct a flat family $\left\{X_t\right\}$ of curves in $\mathbf{P}^4$, parametrized by $T=\mathbf{A}^1$, with the given fibres $X_1$ and $X_0$ for $t=1$ and $t=0$.

Let $\mathcal{I} \subseteq \mathcal{O}_{\mathbf{P}^4 \times T}$ be the ideal sheaf of the total family $X \subseteq \mathbf{P}^4 \times T$. Show that $\mathcal{I}$ is flat over $T$. 

Then show that
\[
h^0(t, \mathcal{I})= \begin{cases}0 & \text { for } t \neq 0 \\ 1 & \text { for } t=0\end{cases}
\]
and also
\[
h^1(t, \mathcal{I})= \begin{cases}0 & \text { for } t \neq 0 \\ 1 & \text { for } t=0 .\end{cases}
\]
This gives another example of cohomology groups jumping at a special point.

### III.12.4.  #to-work

Let $Y$ be an integral scheme of finite type over an algebraically closed field $k$. Let $f: X \rightarrow Y$ be a flat projective morphism whose fibres are all integral schemes. Let $\mathcal{L}, \mathcal{M}$ be invertible sheaves on $X$, and assume for each $y \in Y$ that $\mathcal{L}_y \cong \mathcal{M}_y$ on the fibre $X_y$. 

Then show that there is an invertible sheaf $\mathcal{N}$ on $Y$ such that $\mathcal{L} \cong \mathcal{M} \otimes f^* \mathcal{N}$.[^hint.3.12.4]

[^hint.3.12.4]: 
Hint: Use the results of this section to show that $f_*\left(\mathcal{L} \otimes \mathcal{M}^{-1}\right)$ is locally free of rank 1 on $Y$.

### III.12.5.  #to-work

Let $Y$ be an integral scheme of finite type over an algebraically closed field $k$. Let $\mathcal{E}$ be a locally free sheaf on $Y$, and let $X=\mathbf{P}(\mathcal{E})$ -- see $(II, \S 7)$. 

Then show that $\Pic X \cong( \Pic Y) \times \mathbf{Z}$. This strengthens (II, Ex. 7.9).


### III.12.6. \* #to-work

Let $X$ be an integral projective scheme over an algebraically closed field $k$, and assume that $H^1\left(X, \mathcal{O}_X\right)=0$. Let $T$ be a connected scheme of finite type over $k$.

a. If $\mathcal{L}$ is an invertible sheaf on $X \times T$, show that the invertible sheaves $\mathcal{L}_t$ on $X=X \times\{t\}$ are isomorphic, for all closed points $t \in T$.

b. Show that $\operatorname{Pic}(X \times T)=$ Pic $X \times$ Pic $T$. (Do not assume that $T$ is reduced!)[^hint.3.12.6]


    Cf. (IV, Ex. 4.10) and (V, Ex. 1.6) for examples where $\operatorname{Pic}(X \times T) \neq \operatorname{Pic} X \times$ Pic T. 


[^hint.3.12.6]: 
Hint: Apply (12.11) with $i=0,1$ for suitable invertible sheaves on $X \times T$.