Exercise 109.37.1. Give examples of graded rings $S$ such that

$\text{Proj}(S)$ is affine and nonempty, and

$\text{Proj}(S)$ is integral, nonempty but not isomorphic to ${\mathbf P}^ n_ A$ for any $n\geq 0$, any ring $A$.

Exercise 109.37.1. Give examples of graded rings $S$ such that

$\text{Proj}(S)$ is affine and nonempty, and

$\text{Proj}(S)$ is integral, nonempty but not isomorphic to ${\mathbf P}^ n_ A$ for any $n\geq 0$, any ring $A$.

Exercise 109.37.2. Give an example of a nonconstant morphism of schemes ${\mathbf P}^1_{\mathbf C} \to {\mathbf P}^5_{\mathbf C}$ over $\mathop{\mathrm{Spec}}({\mathbf C})$.

Exercise 109.37.3. Give an example of an isomorphism of schemes

\[ {\mathbf P}^1_{\mathbf C} \to \text{Proj}({\mathbf C}[X_0, X_1, X_2]/(X_0^2 + X_1^2 + X_2^2)) \]

Exercise 109.37.4. Give an example of a morphism of schemes $f : X \to {\mathbf A}^1_{\mathbf C} = \mathop{\mathrm{Spec}}({\mathbf C}[T])$ such that the (scheme theoretic) fibre $X_ t$ of $f$ over $t \in {\mathbf A}^1_{\mathbf C}$ is (a) isomorphic to ${\mathbf P}^1_{\mathbf C}$ when $t$ is a closed point not equal to $0$, and (b) not isomorphic to ${\mathbf P}^1_{\mathbf C}$ when $t = 0$. We will call $X_0$ the *special fibre* of the morphism. This can be done in many, many ways. Try to give examples that satisfy (each of) the following additional restraints (unless it isn't possible):

Can you do it with special fibre projective?

Can you do it with special fibre irreducible and projective?

Can you do it with special fibre integral and projective?

Can you do it with special fibre smooth and projective?

Can you do it with $f$ a flat morphism? This just means that for every affine open $\mathop{\mathrm{Spec}}(A) \subset X$ the induced ring map $\mathbf{C}[t] \to A$ is flat, which in this case means that any nonzero polynomial in $t$ is a nonzerodivisor on $A$.

Can you do it with $f$ a flat and projective morphism?

Can you do it with $f$ flat, projective and special fibre reduced?

Can you do it with $f$ flat, projective and special fibre irreducible?

Can you do it with $f$ flat, projective and special fibre integral?

What do you think happens when you replace ${\mathbf P}^1_{\mathbf C}$ with another variety over ${\mathbf C}$? (This can get very hard depending on which of the variants above you ask for.)

Exercise 109.37.5. Let $n \geq 1$ be any positive integer. Give an example of a surjective morphism $X \to {\mathbf P}^ n_{\mathbf C}$ with $X$ affine.

Exercise 109.37.6. Maps of $\text{Proj}$. Let $R$ and $S$ be graded rings. Suppose we have a ring map

\[ \psi : R \to S \]

and an integer $e \geq 1$ such that $\psi (R_ d) \subset S_{de}$ for all $d \geq 0$. (By our conventions this is not a homomorphism of graded rings, unless $e = 1$.)

For which elements $\mathfrak p \in \text{Proj}(S)$ is there a well-defined corresponding point in $\text{Proj}(R)$? In other words, find a suitable open $U \subset \text{Proj}(S)$ such that $\psi $ defines a continuous map $r_\psi : U \to \text{Proj}(R)$.

Give an example where $U \not= \text{Proj}(S)$.

Give an example where $U = \text{Proj}(S)$.

(Do not write this down.) Convince yourself that the continuous map $U \to \text{Proj}(R)$ comes canonically with a map on sheaves so that $r_\psi $ is a morphism of schemes:

\[ \text{Proj}(S) \supset U \longrightarrow \text{Proj}(R). \]What can you say about this map if $R = \bigoplus _{d \geq 0} S_{de}$ (as a graded ring with $S_ e$, $S_{2e}$, etc in degree $1$, $2$, etc) and $\psi $ is the inclusion mapping?

**Notation.** Let $R$ be a graded ring as above and let $n \geq 0$ be an integer. Let $X = \text{Proj}(R)$. Then there is a unique quasi-coherent ${\mathcal O}_ X$-module ${\mathcal O}_ X(n)$ on $X$ such that for every homogeneous element $f \in R$ of positive degree we have ${\mathcal O}_ X |_{D_{+}(f)}$ is the quasi-coherent sheaf associated to the $R_{(f)} = (R_ f)_0$-module $(R_ f)_ n$ ($ = $elements homogeneous of degree $n$ in $R_ f = R[1/f]$). See [page 116+, H]. Note that there are natural maps

\[ {\mathcal O}_ X(n_1) \otimes _{{\mathcal O}_ X} {\mathcal O}_ X(n_2) \longrightarrow {\mathcal O}_ X(n_1 + n_2) \]

Exercise 109.37.7. Pathologies in $\text{Proj}$. Give examples of $R$ as above such that

${\mathcal O}_ X(1)$ is not an invertible ${\mathcal O}_ X$-module.

${\mathcal O}_ X(1)$ is invertible, but the natural map ${\mathcal O}_ X(1) \otimes _{{\mathcal O}_ X} {\mathcal O}_ X(1) \to {\mathcal O}_ X(2)$ is NOT an isomorphism.

Exercise 109.37.8. Let $S$ be a graded ring. Let $X = \text{Proj}(S)$. Show that any finite set of points of $X$ is contained in a standard affine open.

Exercise 109.37.9. Let $S$ be a graded ring. Let $X = \text{Proj}(S)$. Let $Z, Z' \subset X$ be two closed subschemes. Let $\varphi : Z \to Z'$ be an isomorphism. Assume $Z \cap Z' = \emptyset $. Show that for any $z \in Z$ there exists an affine open $U \subset X$ such that $z \in U$, $\varphi (z) \in U$ and $\varphi (Z \cap U) = Z' \cap U$. (Hint: Use Exercise 109.37.8 and something akin to Schemes, Lemma 26.11.5.)

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