Consider $\phi: k[x] \to k[x,y]$, the inclusion. This induces $f: \mathbb{A}^2_k = \text{Spec } k[x,y] \to \mathbb{A}^1_k = \text{Spec } k[x]$ the projection $(x,y) \mapsto x$. On prime ideals:

• The zero ideal $(0) \mapsto (0)$ (generic point to generic point)
• Maximal ideal $(x-a, y-b) \mapsto (x-a)$ (closed point to closed point)
• Principal prime $(f(x,y)) \mapsto (0)$ or $(x-a)$ depending on whether $f$ depends on both variables

This is the prototypical example of a dominant morphism.

Let $\phi: k[x,y] \to k[x,y]/(y^2 - x^3)$. This induces $f: V(y^2 - x^3) \hookrightarrow \mathbb{A}^2_k$ The morphism is:

• On topological spaces: inclusion of the closed subset $V(y^2-x^3)$
• On structure sheaves: the quotient map restricted to open sets

This is a closed immersion, the scheme-theoretic generalization of a closed subvariety.

In characteristic $p > 0$, consider $\phi: k[x] \to k[x]$ given by $x \mapsto x^p$ (assuming $k$ is perfect). This induces the Frobenius morphism $F: \mathbb{A}^1_k \to \mathbb{A}^1_k$ On points, if $\mathfrak{m} = (x-a)$ for $a \in k$, then $F(\mathfrak{m}) = \phi^{-1}(x-a) = (x - a^{1/p})$ So $F$ is bijective on $k$-rational points (when $k$ is algebraically closed) but not an isomorphism of schemes! The induced map on structure sheaves is not an isomorphism.

Consider the map $\mathbb{P}^1_k \to \mathbb{A}^1_k$ defined by $[x:y] \mapsto x/y$ where $y \neq 0$. On affine charts:

• On $U_0 = \text{Spec } k[t]$ (where $t = y/x$): we get the map $k[u] \to k[t]$ sending $u \mapsto 1/t$
• On $U_1 = \text{Spec } k[s]$ (where $s = x/y$): we get the map $k[u] \to k[s]$ sending $u \mapsto s$

On the overlap $U_0 \cap U_1 = \text{Spec } k[t, t^{-1}]$, both maps agree: $u \mapsto 1/t = s$.

This morphism is not defined at $[1:0]$, showing that not all rational maps extend to morphisms of schemes.

The blow-up of $\mathbb{A}^2$ at the origin is $\text{Bl}_0 \mathbb{A}^2 \subseteq \mathbb{A}^2 \times \mathbb{P}^1$, defined by the equation $x_1 y_0 = x_0 y_1$ where $(x_0, x_1) \in \mathbb{A}^2$ and $[y_0:y_1] \in \mathbb{P}^1$.

The blow-up map $\pi: \text{Bl}_0 \mathbb{A}^2 \to \mathbb{A}^2$ is the projection. On the chart $y_0 \neq 0$:

• $\text{Spec } k[x_0, x_1, t]/(x_1 - x_0 t) \cong \text{Spec } k[x_0, t]$
• The map is $k[x_0, x_1] \to k[x_0, t]$ sending $x_0 \mapsto x_0, x_1 \mapsto x_0 t$

The exceptional divisor $E = \pi^{-1}(0) \cong \mathbb{P}^1$ is the fiber over the origin.

For $\mathbb{A}^n_k = \text{Spec } k[x_1, \ldots, x_n]$, we have $\mathbb{A}^n_k(T) = \text{Hom}(T, \mathbb{A}^n_k)$ When $T = \text{Spec } R$ is affine, this equals $\text{Hom}(k[x_1,\ldots,x_n], R) \cong R^n$ So $\mathbb{A}^n_k(T)$ is the set of $n$-tuples of global sections of $\mathcal{O}_T$.

For instance, $\mathbb{A}^1_k(\mathbb{A}^1_k) = k[x]$ consists of all polynomial functions $\mathbb{A}^1 \to \mathbb{A}^1$.

For $\mathbb{P}^n_k$, the set $\mathbb{P}^n_k(T)$ consists of line bundles $\mathcal{L}$ on $T$ together with $(n+1)$ global sections $s_0, \ldots, s_n \in \Gamma(T, \mathcal{L})$ that generate $\mathcal{L}$ at every point.

When $T = \text{Spec } k$ is a point, this reduces to $k^{n+1} \setminus \{0\}$ modulo scalars, recovering classical projective space.

Consider the line $x = 0$ in $\mathbb{A}^2_k$. There are multiple closed subschemes with the same underlying space:

• $\text{Spec } k[x,y]/(x)$: the reduced line (simple line)
• $\text{Spec } k[x,y]/(x^2)$: the doubled line (fat line)
• $\text{Spec } k[x,y]/(x^n)$: the $n$-fold thickened line

Each gives a different closed immersion into $\mathbb{A}^2_k$, corresponding to ideals $(x), (x^2), (x^n)$.

In $\mathbb{A}^2_k$, consider the curves $C_1: y = x^2$ and $C_2: y = 0$. Their scheme-theoretic intersection is $C_1 \cap C_2 = \text{Spec } k[x,y]/(x^2, y) = \text{Spec } k[x]/(x^2)$ which is a non-reduced scheme supported at the origin. This captures the tangency: the curves meet with multiplicity 2.

In contrast, the set-theoretic intersection is just the origin as a reduced point.

Let $k$ be a field and consider $f: \text{Spec } k[t] \to \text{Spec } k[x,y]/(y^2 - x^3)$ induced by $x \mapsto t^2, y \mapsto t^3$. This is the normalization morphism.

Check finiteness:

• $k[t]$ is finitely generated as a $k[x,y]/(y^2-x^3)$-algebra (generator: $t$)
• Is $k[t]$ finitely generated as a module? The subring $k[t^2, t^3] \subseteq k[t]$ has $[k[t] : k[t^2,t^3]] = \infty$ as a module

So $f$ is of finite type but NOT finite. The normalization is finite for curves, but this requires proving integrality.

Let $k \subseteq L$ be a finite field extension. Then $f: \text{Spec } L \to \text{Spec } k$ is a finite morphism. Indeed, $L$ is a finite-dimensional $k$-vector space, hence a finitely generated $k$-module.

The map is surjective on points (the unique point of $\text{Spec } L$ maps to the unique point of $\text{Spec } k$) but can have degree $> 1$ in the sense of extension degree $[L:k]$.

Consider the morphism $f: \mathbb{A}^1_k \to \mathbb{A}^1_k$ induced by $k[x] \to k[y]$ sending $x \mapsto y^n$. This is the $n$-th power map, a finite flat morphism of degree $n$.

Finiteness: $k[y]$ is a free $k[x]$-module with basis $\{1, y, \ldots, y^{n-1}\}$ via $x = y^n$.

The fiber over $x = a \neq 0$ consists of $n$ points (the $n$-th roots of $a$), while the fiber over $x = 0$ is a single reduced point.

A line bundle $\mathcal{L}$ on $X$ gives an affine morphism $\mathbb{V}(\mathcal{L}) = \text{Spec}_X \text{Sym}(\mathcal{L}) \to X$ This is the total space of the line bundle viewed as a scheme.

For $\mathcal{L} = \mathcal{O}_X$, we get $\mathbb{A}^1_X = X \times \mathbb{A}^1 \to X$, the projection from the affine line bundle.

Let $X = \mathbb{P}^1_k$ and $\mathcal{A} = \mathcal{O}_{\mathbb{P}^1} \oplus \mathcal{O}_{\mathbb{P}^1}(-1)$ with the natural algebra structure. Then $\text{Spec}_{\mathbb{P}^1} \mathcal{A} \to \mathbb{P}^1$ is an affine morphism whose fibers are varying: over $\mathbb{P}^1 \setminus \{[0:1]\}$, the fiber is $\mathbb{A}^2$, but the total space is not a vector bundle.

The projection $\mathbb{A}^2_k \to \mathbb{A}^1_k$ given by $(x,y) \mapsto x$ is dominant. The generic point $(0) \in \text{Spec } k[x]$ pulls back to $(0) \in \text{Spec } k[x,y]$.

In classical terms, the image of the projection map contains a dense open subset.

A proper closed immersion is never dominant. For instance, $V(x) \hookrightarrow \mathbb{A}^2_k$ has image the closed set $V(x)$, which is not dense. The generic point $(0)$ of $\mathbb{A}^2$ is not in $V(x)$.

For any scheme $Y$, the projection $\mathbb{P}^n_Y \to Y$ is proper. This is the fundamental example of a proper morphism.

In particular, $\mathbb{P}^n_k \to \text{Spec } k$ is proper, generalizing the compactness of projective space.

The structure morphism $\mathbb{A}^1_k \to \text{Spec } k$ is not proper (though it is separated and of finite type). The closed subset $V(x-t) \subseteq \mathbb{A}^1_k \times \text{Spec } k[\epsilon]/(\epsilon^2)$ projects to the open point $(0) \in \text{Spec } k[\epsilon]/(\epsilon^2)$, which is not closed.

Geometrically, $\mathbb{A}^1$ is not compact.

For a variety $X$ over $k$, the structure morphism $X \to \text{Spec } k$ encodes how $X$ is defined over $k$. The $k$-rational points of $X$ are the sections of this morphism: $X(k) = \text{Hom}_k(\text{Spec } k, X)$

In characteristic $p > 0$, the absolute Frobenius $F: X \to X$ is defined by:

• Identity on topological spaces
• On structure sheaves: $a \mapsto a^p$

For $X = \text{Spec } A$, this comes from the ring homomorphism $A \to A, a \mapsto a^p$ (which is a homomorphism since $(a+b)^p = a^p + b^p$ in characteristic $p$).

Properties:

• $F$ is a homeomorphism but not an isomorphism of schemes
• $F$ is purely inseparable of degree $p^{\dim X}$
• The differential $dF = 0$ (Frobenius kills all differentials)

For a scheme $X$ over $\mathbb{F}_p$, the relative Frobenius is the unique morphism $F_{X/\mathbb{F}_p}: X \to X^{(p)}$ factoring the absolute Frobenius $F_{\mathrm{abs}} : X \to X$ through the base change $X^{(p)} \to X$, where $X^{(p)} = X \times_{\operatorname{Spec} \mathbb{F}_p, F} \operatorname{Spec} \mathbb{F}_p$ is the Frobenius twist.

The relative Frobenius is a morphism over $\mathbb{F}_p$ (unlike the absolute Frobenius, which is not). For an elliptic curve $E / \mathbb{F}_p$, the degree of $F_{E/\mathbb{F}_p}$ is $p$, and $\#E(\mathbb{F}_p) = \deg(1 - F)$.

Let $X = V(y^2 - x^2(x+1)) \subseteq \mathbb{A}^2_k$ be the node curve. The normalization is $\nu: \tilde{X} = \mathbb{A}^1_k \to X$ given by $t \mapsto (t^2-1, t(t^2-1))$.

On the ring level: $k[x,y]/(y^2 - x^2(x+1)) \subseteq k[t]$ where the left side is the coordinate ring of $X$. The map is finite since $k[t]$ is integral over the coordinate ring (both $x$ and $y$ satisfy monic polynomial relations).

The normalization is bijective on underlying sets except at the node, where the two branches separate.

Consider $\mathbb{A}^n_k \to \text{Spec } k$ and base change by $\text{Spec } L \to \text{Spec } k$ for a field extension $L/k$. The fiber product is $\mathbb{A}^n_k \times_{\text{Spec } k} \text{Spec } L = \mathbb{A}^n_L$ which is affine $n$-space over $L$. This is the process of extending scalars.

For $f: X \to Y$ and a point $y \in Y$ corresponding to $\text{Spec } k(y) \to Y$, the fiber is $X_y = X \times_Y \text{Spec } k(y)$ This is the scheme-theoretic fiber, generalizing the classical fiber of a map.

For $\mathbb{A}^2_k \to \mathbb{A}^1_k$ projecting $(x,y) \mapsto x$, the fiber over $a \in k$ is $\mathbb{A}^2_k \times_{\mathbb{A}^1_k} \text{Spec } k = V(x-a) \cong \mathbb{A}^1_k$ the vertical line through $x = a$.

The automorphism group $\text{Aut}(\mathbb{A}^1_k)$ consists of affine transformations $x \mapsto ax + b$ with $a \in k^*$ and $b \in k$.

For $\mathbb{A}^2_k$, the automorphism group is much larger and includes the group of tame automorphisms (generated by affine and triangular maps) and the wild automorphisms (e.g., the Nagata automorphism).

The automorphism group $\text{Aut}(\mathbb{P}^n_k) \cong \text{PGL}_{n+1}(k)$ consists of projectivized linear transformations: $[x_0 : \cdots : x_n] \mapsto [a_{00}x_0 + \cdots + a_{0n}x_n : \cdots : a_{n0}x_0 + \cdots + a_{nn}x_n]$ where $(a_{ij}) \in \text{GL}_{n+1}(k)$, considered up to scalar multiplication.

Consider the projection from the point $p = [0:0:1] \in \mathbb{P}^2_k$ to a line $\ell \cong \mathbb{P}^1_k$. This gives a rational map $\pi: \mathbb{P}^2_k \dashrightarrow \mathbb{P}^1_k$ defined everywhere except at $p$. On the affine chart $z \neq 0$, this is given by $(x,y) \mapsto x/y$ or similar.

This is a classical example of a birational map that is not a morphism.