110.39 Morphisms from surfaces to curves

Exercise 110.39.1. Let $R$ be a ring. Let $R \to k$ be a map from $R$ to a field. Let $n \geq 0$. Show that

$\mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathrm{Spec}}(R)}(\mathop{\mathrm{Spec}}(k), \mathbf{P}^ n_ R) = (k^{n + 1} \setminus \{ 0\} )/k^*$

where $k^*$ acts via scalar multiplication on $k^{n + 1}$. From now on we denote $(x_0 : \ldots : x_ n)$ the morphism $\mathop{\mathrm{Spec}}(k) \to \mathbf{P}^ n_ k$ corresponding to the equivalence class of the element $(x_0, \ldots , x_ n) \in k^{n + 1} \setminus \{ 0\}$.

Exercise 110.39.2. Let $k$ be a field. Let $Z \subset \mathbf{P}^2_ k$ be an irreducible and reduced closed subscheme. Show that either (a) $Z$ is a closed point, or (b) there exists an homogeneous irreducible $F \in k[X_0, X_1, X_2]$ of degree $> 0$ such that $Z = V_{+}(F)$, or (c) $Z = \mathbf{P}^2_ k$. (Hint: Look on a standard affine open.)

Exercise 110.39.3. Let $k$ be a field. Let $Z_1, Z_2 \subset \mathbf{P}^2_ k$ be irreducible closed subschemes of the form $V_{+}(F)$ for some homogeneous irreducible $F_ i \in k[X_0, X_1, X_2]$ of degree $> 0$. Show that $Z_1 \cap Z_2$ is not empty. (Hint: Use dimension theory to estimate the dimension of the local ring of $k[X_0, X_1, X_2]/(F_1, F_2)$ at $0$.)

Exercise 110.39.4. Show there does not exist a nonconstant morphism of schemes $\mathbf{P}^2_{\mathbf{C}} \to \mathbf{P}^1_{\mathbf{C}}$ over $\mathop{\mathrm{Spec}}(\mathbf{C})$. Here a constant morphism is one whose image is a single point. (Hint: If the morphism is not constant consider the fibres over $0$ and $\infty$ and argue that they have to meet to get a contradiction.)

Exercise 110.39.5. Let $k$ be a field. Suppose that $X \subset \mathbf{P}^3_ k$ is a closed subscheme given by a single homogeneous equation $F \in k[X_0, X_1, X_2, X_3]$. In other words,

$X = \text{Proj}(k[X_0, X_1, X_2, X_3]/(F)) \subset \mathbf{P}^3_ k$

as explained in the course. Assume that

$F = X_0 G + X_1 H$

for some homogeneous polynomials $G, H \in k[X_0, X_1, X_2, X_3]$ of positive degree. Show that if $X_0, X_1, G, H$ have no common zeros then there exists a nonconstant morphism

$X \longrightarrow \mathbf{P}^1_ k$

of schemes over $\mathop{\mathrm{Spec}}(k)$ which on field points (see Exercise 110.39.1) looks like $(x_0 : x_1 : x_2 : x_3) \mapsto (x_0 : x_1)$ whenever $x_0$ or $x_1$ is not zero.

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