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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 \begin{align*} \varphi(y)=\operatorname{dim}_k\left(m_y / m_y^2\right) \end{align*} 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 \begin{align*} h^0(t, \mathcal{I})= \begin{cases}0 & \text { for } t \neq 0 \\ 1 & \text { for } t=0\end{cases} \end{align*} and also \begin{align*} h^1(t, \mathcal{I})= \begin{cases}0 & \text { for } t \neq 0 \\ 1 & \text { for } t=0 .\end{cases} \end{align*} 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 \(\mathcal{L} \cong \mathcal{M} \otimes f^* \mathcal{N}\). ¹

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 \(\operatorname{Pic}X \cong( \operatorname{Pic}Y) \times \mathbf{Z}\). 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 \(k\).

• 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 \(t \in T\).

• Show that \(\operatorname{Pic}(X \times T)=\) Pic \(X \times\) Pic \(T\). (Do not assume that \(T\) is reduced!) ²

  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.