## 111.34 Morphisms

An important question is, given a morphism $\pi : X \to S$, whether the morphism has a section or a rational section. Here are some example exercises.

Exercise 111.34.1. Consider the morphism of schemes

\[ \pi : X = \mathop{\mathrm{Spec}}(\mathbf{C}[x, t, 1/xt]) \longrightarrow S = \mathop{\mathrm{Spec}}(\mathbf{C}[t]). \]

Show there does not exist a morphism $\sigma : S \to X$ such that $\pi \circ \sigma = \text{id}_ S$.

Show there does exist a nonempty open $U \subset S$ and a morphism $\sigma : U \to X$ such that $\pi \circ \sigma = \text{id}_ U$.

Exercise 111.34.2. Consider the morphism of schemes

\[ \pi : X = \mathop{\mathrm{Spec}}(\mathbf{C}[x, t]/(x^2 + t)) \longrightarrow S = \mathop{\mathrm{Spec}}(\mathbf{C}[t]). \]

Show there does not exist a nonempty open $U \subset S$ and a morphism $\sigma : U \to X$ such that $\pi \circ \sigma = \text{id}_ U$.

Exercise 111.34.3. Let $A, B, C \in \mathbf{C}[t]$ be nonzero polynomials. Consider the morphism of schemes

\[ \pi : X = \mathop{\mathrm{Spec}}(\mathbf{C}[x, y, t]/(A + Bx^2 + Cy^2)) \longrightarrow S = \mathop{\mathrm{Spec}}(\mathbf{C}[t]). \]

Show there does exist a nonempty open $U \subset S$ and a morphism $\sigma : U \to X$ such that $\pi \circ \sigma = \text{id}_ U$. (Hint: Symbolically, write $x = X/Z$, $y = Y/Z$ for some $X, Y, Z \in \mathbf{C}[t]$ of degree $\leq d$ for some $d$, and work out the condition that this solves the equation. Then show, using dimension theory, that if $d >> 0$ you can find nonzero $X, Y, Z$ solving the equation.)

Exercise 111.34.5. Consider the morphism of schemes

\[ \pi : X = \mathop{\mathrm{Spec}}(\mathbf{C}[x, y, t] /(1 + t x^3 + t^2 y^3)) \longrightarrow S = \mathop{\mathrm{Spec}}(\mathbf{C}[t]) \]

Show there does not exist a nonempty open $U \subset S$ and a morphism $\sigma : U \to X$ such that $\pi \circ \sigma = \text{id}_ U$.

Exercise 111.34.6. Consider the schemes

\[ X = \mathop{\mathrm{Spec}}(\mathbf{C}[\{ x_ i\} _{i = 1}^{8}, s, t] /(1 + s x_1^3 + s^2 x_2^3 + t x_3^3 + st x_4^3 + s^2t x_5^3 + t^2 x_6^3 + st^2 x_7^3 + s^2t^2 x_8^3)) \]

and

\[ S = \mathop{\mathrm{Spec}}(\mathbf{C}[s, t]) \]

and the morphism of schemes

\[ \pi : X \longrightarrow S \]

Show there does not exist a nonempty open $U \subset S$ and a morphism $\sigma : U \to X$ such that $\pi \circ \sigma = \text{id}_ U$.

Exercise 111.34.7. (For the number theorists.) Give an example of a closed subscheme

\[ Z \subset \mathop{\mathrm{Spec}}\left({\mathbf Z}[x, \frac{1 }{ x(x-1)(2x-1)}]\right) \]

such that the morphism $Z \to \mathop{\mathrm{Spec}}({\mathbf Z})$ is finite and surjective.

Exercise 111.34.8. If you do not like number theory, you can try the variant where you look at

\[ \mathop{\mathrm{Spec}}\left({\mathbf F}_ p[t, x, \frac{1 }{ x(x-t)(tx-1)}]\right) \longrightarrow \mathop{\mathrm{Spec}}({\mathbf F}_ p[t]) \]

and you try to find a closed subscheme of the top scheme which maps finite surjectively to the bottom one. (There is a theoretical reason for having a finite ground field here; although it may not be necessary in this particular case.)

Exercise 111.34.10. Prove there exist a $f \in \mathbf{C}[x, t]$ which is not divisible by $t - \alpha $ for any $\alpha \in \mathbf{C}$ such that there does not exist any $Z \subset \mathop{\mathrm{Spec}}(\mathbf{C}[x, t, 1/f])$ which maps finite surjectively to $\mathop{\mathrm{Spec}}(\mathbf{C}[t])$. (I think that $f(x, t) = (xt - 2)(x - t + 3)$ works. To show any candidate has the required property is not so easy I think.)

Exercise 111.34.11. Let $A \to B$ be a finite type ring map. Suppose that $\mathop{\mathrm{Spec}}(B) \to \mathop{\mathrm{Spec}}(A)$ factors through a closed immersion $\mathop{\mathrm{Spec}}(B) \to \mathbf{P}^ n_ A$ for some $n$. Prove that $A \to B$ is a finite ring map, i.e., that $B$ is finite as an $A$-module. Hint: if $A$ is Noetherian (please just assume this) you can argue using that $H^ i(Z, \mathcal{O}_ Z)$ for $i \in \mathbf{Z}$ is a finite $A$-module for every closed subscheme $Z \subset \mathbf{P}^ n_ A$.

Exercise 111.34.12. Let $k$ be an algebraically closed field. Let $f : X \to Y$ be a morphism of projective varieties such that $f^{-1}(\{ y\} )$ is finite for every closed point $y \in Y$. Prove that $f$ is finite as a morphism of schemes. Hints: (a) being finite is a local property, (b) try to reduce to Exercise 111.34.11, and (c) use a closed immersion $X \to \mathbf{P}^ n_ k$ to get a closed immersion $X \to \mathbf{P}^ n_ Y$ over $Y$.

## Comments (2)

Comment #2364 by Marco Castronovo on

Comment #2427 by Johan on