$\mathbb{A}^1_k = \operatorname{Spec} k[x]$ is affine. For $k = \bar{k}$, the closed points correspond to elements $a \in k$ (via the maximal ideal $(x - a)$), plus the generic point $(0)$.

Over $\mathbb{Z}$: $\mathbb{A}^1_\mathbb{Z} = \operatorname{Spec} \mathbb{Z}[x]$. This is a $2$-dimensional scheme: the "arithmetic surface." Its points include:

• Closed points $(p, f(x))$ where $p$ is prime and $f$ is irreducible mod $p$ — points of $\mathbb{A}^1_{\mathbb{F}_p}$.
• Generic points of "horizontal curves" $(g(x))$ where $g \in \mathbb{Z}[x]$ is irreducible.
• Generic points of "vertical fibers" $(p)$.
• The generic point $(0)$.

$\mathbb{P}^1_k$ is the simplest non-affine scheme. It is obtained by gluing two copies of $\mathbb{A}^1$:

$U_0 = \operatorname{Spec} k[t] \quad \text{and} \quad U_1 = \operatorname{Spec} k[s]$

along $D(t) \cong D(s)$ via $s = 1/t$. Concretely:

• $U_0 \cap U_1 = \operatorname{Spec} k[t, 1/t]$,
• the gluing isomorphism is $k[t, 1/t] \cong k[s, 1/s]$ via $t \leftrightarrow 1/s$.

$\mathbb{P}^1$ is not affine: $\mathcal{O}(\mathbb{P}^1) = k$ (global sections are constant), but $\mathbb{P}^1$ is not a point. This shows that the functor $\Gamma : \mathbf{Sch} \to \mathbf{CRing}^{\mathrm{op}}$ is not an equivalence on non-affine schemes.

$\mathbb{P}^n_k = \operatorname{Proj} k[x_0, \ldots, x_n]$ is covered by $n+1$ affine charts:

$U_i = \operatorname{Spec} k\left[\frac{x_0}{x_i}, \ldots, \widehat{\frac{x_i}{x_i}}, \ldots, \frac{x_n}{x_i}\right] \cong \mathbb{A}^n_k, \quad i = 0, \ldots, n.$

Key properties:

• $\mathcal{O}(\mathbb{P}^n) = k$ (only constant global functions).
• $\dim \mathbb{P}^n = n$.
• $\mathbb{P}^n$ is proper over $k$ (the algebraic analogue of compact).
• $\operatorname{Pic}(\mathbb{P}^n) = \mathbb{Z}$, generated by $\mathcal{O}(1)$.

$\operatorname{Proj} k[x_0, x_1] = \mathbb{P}^1_k$. The two standard opens:

• $D_+(x_0) = \operatorname{Spec} k[x_1/x_0] = \operatorname{Spec} k[t] = \mathbb{A}^1$.
• $D_+(x_1) = \operatorname{Spec} k[x_0/x_1] = \operatorname{Spec} k[s] = \mathbb{A}^1$.

More generally, $\operatorname{Proj} k[x_0, \ldots, x_n] = \mathbb{P}^n_k$.

A projective variety $V(F) \subseteq \mathbb{P}^n$ for homogeneous $F$ of degree $d$ is:

$V(F) = \operatorname{Proj} k[x_0, \ldots, x_n]/(F).$

For example, the elliptic curve $E = V(Y^2Z - X^3 + XZ^2) \subseteq \mathbb{P}^2$ is:

$E = \operatorname{Proj} k[X, Y, Z]/(Y^2Z - X^3 + XZ^2).$

In the affine chart $Z \neq 0$: $\operatorname{Spec} k[x,y]/(y^2 - x^3 + x)$, where $x = X/Z$, $y = Y/Z$.

The $d$-th Veronese subring of $S = k[x_0, \ldots, x_n]$ is $S^{(d)} = \bigoplus_k S_{kd}$. Then $\operatorname{Proj} S^{(d)} \cong \operatorname{Proj} S = \mathbb{P}^n$ (same scheme, different embedding). This is the scheme-theoretic version of the Veronese embedding.

The blowup of $\mathbb{A}^2$ at the origin is:

$\operatorname{Bl}_0 \mathbb{A}^2 = \operatorname{Proj} k[x,y][s,t]/(xt - ys) \subseteq \mathbb{A}^2 \times \mathbb{P}^1$

where $[s:t]$ are the homogeneous coordinates on $\mathbb{P}^1$. This is covered by two affine charts:

• $s \neq 0$: $\operatorname{Spec} k[x, t/s] = \operatorname{Spec} k[x, u]$ where $y = xu$ — the chart where $u = y/x$.
• $t \neq 0$: $\operatorname{Spec} k[y, s/t] = \operatorname{Spec} k[y, v]$ where $x = yv$ — the chart where $v = x/y$.

The exceptional divisor $E \cong \mathbb{P}^1$ is the preimage of the origin.

• $\operatorname{Spec} k[x]/(x^2)$: the double point. One point, but $\mathcal{O} = k[\varepsilon]$ with $\varepsilon^2 = 0$.
• $\operatorname{Spec} k[x,y]/(x^2, xy)$: a "fuzzy point" at the origin, non-reduced in the $x$-direction.
• The scheme-theoretic intersection $V(y) \cap V(y - x^2) = \operatorname{Spec} k[x,y]/(y, y-x^2) = \operatorname{Spec} k[x]/(x^2)$: a double point recording the tangency.

Non-reduced schemes arise naturally from:

• Intersections with multiplicity.
• Fibers of morphisms at critical values.
• Deformation theory (flat limits can acquire nilpotents).

$\operatorname{Spec}(k \times k) = \operatorname{Spec} k \sqcup \operatorname{Spec} k$: two disjoint points.

$\operatorname{Spec} k[x]/(x^2 - 1) = \operatorname{Spec} k[x]/((x-1)(x+1)) \cong \operatorname{Spec}(k \times k)$: two points $x = 1$ and $x = -1$. But over $\mathbb{F}_2$: $x^2 - 1 = (x-1)^2$, so $\operatorname{Spec} \mathbb{F}_2[x]/(x-1)^2$ is a single (double) point — connected but non-reduced!

$\operatorname{Spec} k[x_1, x_2, x_3, \ldots]$ (polynomial ring in infinitely many variables) is an affine scheme that is not Noetherian: the chain of ideals $(x_1) \subset (x_1, x_2) \subset \cdots$ does not stabilize. In practice, most schemes in algebraic geometry are locally Noetherian.

The following are all closed subschemes of $\mathbb{A}^1 = \operatorname{Spec} k[x]$ supported at the origin:

| Subscheme | Ideal | Ring | "Thickness" | |---|---|---|---| | $\operatorname{Spec} k$ | $(x)$ | $k$ | reduced point | | $\operatorname{Spec} k[x]/(x^2)$ | $(x^2)$ | $k[\varepsilon]$ | double point | | $\operatorname{Spec} k[x]/(x^3)$ | $(x^3)$ | $k[x]/(x^3)$ | triple point | | $\operatorname{Spec} k[x]/(x^n)$ | $(x^n)$ | $k[x]/(x^n)$ | $n$-fold point |

As $n \to \infty$, the limit is $\operatorname{Spec} k[[x]]$, the formal neighborhood of the origin.

In $\mathbb{A}^2$, the curves $y = x^2$ and $y = 0$:

• Set-theoretic intersection: the origin $\{(0,0)\}$.
• Scheme-theoretic intersection: $\operatorname{Spec} k[x,y]/(y, y-x^2) = \operatorname{Spec} k[x]/(x^2)$ — a double point.

The curves $y = x$ and $y = 0$:

• Scheme-theoretic intersection: $\operatorname{Spec} k[x,y]/(y, y-x) = \operatorname{Spec} k[x]/(x) = \operatorname{Spec} k$ — a simple point.

Scheme-theoretic intersection detects tangency. This is the correct notion for Bézout's theorem.

The scheme $E = \operatorname{Proj} \mathbb{Z}[X,Y,Z]/(Y^2Z - X^3 + XZ^2)$ is an elliptic curve over $\operatorname{Spec} \mathbb{Z}$. Its fibers:

• Generic fiber $E_\mathbb{Q}$: the elliptic curve $y^2 = x^3 - x$ over $\mathbb{Q}$.
• Fiber at $(p)$: $E_{\mathbb{F}_p} = E \times_{\operatorname{Spec}\mathbb{Z}} \operatorname{Spec} \mathbb{F}_p$ — the reduction mod $p$.

The number of $\mathbb{F}_p$-points $\#E(\mathbb{F}_p) = p + 1 - a_p$ where $|a_p| \leq 2\sqrt{p}$ (Hasse bound). The sequence $\{a_p\}$ encodes the $L$-function of $E$, connecting geometry to number theory (Birch–Swinnerton-Dyer, modularity theorem, ...).

For a $k$-scheme $X$ and a field extension $K/k$, the base change is:

$X_K = X \times_{\operatorname{Spec} k} \operatorname{Spec} K.$

For $X = V(x^2 + y^2 + 1) \subseteq \mathbb{A}^2_\mathbb{R}$:

• Over $\mathbb{R}$: $X(\mathbb{R}) = \emptyset$ (no real points).
• Over $\mathbb{C}$: $X_\mathbb{C} = V(x^2 + y^2 + 1) \subseteq \mathbb{A}^2_\mathbb{C}$ is a smooth affine curve with many points.

The scheme $X$ itself doesn't change — we just see more points after base change.

A group scheme over $S$ is a scheme $G \to S$ with morphisms $\mu : G \times_S G \to G$ (multiplication), $e : S \to G$ (identity), $\iota : G \to G$ (inverse) satisfying the group axioms.

Examples:

• $\mathbb{G}_m = \operatorname{Spec} \mathbb{Z}[t, t^{-1}]$: the multiplicative group. $\mathbb{G}_m(R) = R^*$ for any ring $R$.
• $\mathbb{G}_a = \operatorname{Spec} \mathbb{Z}[t]$: the additive group. $\mathbb{G}_a(R) = (R, +)$.
• $GL_n = \operatorname{Spec} \mathbb{Z}[x_{ij}, 1/\det]$: the general linear group. $GL_n(R) = GL_n(R)$.
• $\mu_n = \operatorname{Spec} \mathbb{Z}[t]/(t^n - 1)$: the $n$-th roots of unity. Over a field of char $p \mid n$, this is non-reduced (e.g., $\mu_p = \operatorname{Spec} \mathbb{F}_p[t]/(t-1)^p$).
• Elliptic curves with their group law.

$\operatorname{Spec} \mathbb{Z}[x]$ is a $2$-dimensional scheme, an "arithmetic surface." Its closed points correspond to pairs $(p, \alpha)$ where $p$ is prime and $\alpha \in \overline{\mathbb{F}_p}$ (up to Galois conjugacy).

The scheme $\operatorname{Spec} \mathbb{Z}[i] = \operatorname{Spec} \mathbb{Z}[x]/(x^2 + 1)$ is a closed subscheme: the "arithmetic curve" of the Gaussian integers. Its points:

• $(1 + i)$ above $p = 2$ (ramified: $2 = -i(1+i)^2$).
• $(a + bi)$ and $(a - bi)$ above $p \equiv 1 \pmod{4}$ (split).
• $(p)$ above $p \equiv 3 \pmod{4}$ (inert).

This decomposition pattern is governed by quadratic reciprocity!