We want to understand schemes $\mathfrak{X} \to \operatorname{Spec} \mathbb{Z}$ that is smooth proper with geometrically connected fibers. To do this, we first develop a classification of surfaces over $k = \bar{k}$ of arbitrary characteristic.
- For any such $X$, we show that there exists a birational map $X \to X^\mathrm{min}$ that is a blowup, where $X^\mathrm{min}$ is minimal in the sense that it is not a blowup.
- If $X$ is minimal then we want to show that either $X$ is ruled (i.e., $X$ is birational to $\mathbb{P}^1 \times C$) or $K_X$ is nef (i.e., for all $C \subset X$ we have $K_X.C \ge 0$).
- Once $K_X$ is nef, it is going to be semi-ample, i.e., for $n \gg 0$ the linear system $\lvert nK_X \rvert$ is basepoint free. This means that we get a map $X \to \mathbb{P}^N$ and the (stabilizing) image is called the canonical model $X^\mathrm{can}$. This is normal and possibly has smaller dimension. We can also thing of this as $X^\mathrm{can} = \operatorname{Proj}(\bigoplus_{n \ge 0} H^0(X, \mathscr{O}(nK_X)))$, where the finite generation of this graded rings is the nontrivial part. The Kodaira dimension by by definition $\kappa(X) = \dim X^\mathrm{can}$.
- The case $\kappa(X) = 2$ is the general type case, and here we don’t say much.
- The case $\kappa(X) = 1$ is called “properly elliptic” and here $\pi \colon X \to X^\mathrm{can}$ has $nK_X = \pi^\ast \mathscr{O}(1)$ and so the fibers are curves with arithmetic genus zero. Then this is an elliptic fibration.
- In the case $\kappa(X) = 0$, we have $n K_X = 0$, and actually Mumford proves (in arbitrary characteristic) that either $4K_X = 0$ or $6K_X = 0$. If $nK_X = 0$, then there is a cyclic cover $\tilde{X} \to X$ given by taking solutions to $z^n = f$ with $f \in H^0(\mathscr{O}(nK_X))$ and $K_\tilde{X} = 0$. If $\operatorname{char} k \neq 2, 3$ this is an étale $\mu_n$-cover.
So let us try to carry this.
Definition 1. A $(-1)$-curve on $X$ is an integral curve $C \subset X$ such that $C \cong \mathbb{P}^1$ and $\mathscr{N}_ {C/X} \cong \mathscr{O}(-1)$, i.e., $C^2 = -1$.
Theorem 2 (Castelnuovo). Assume $k = \bar{k}$. Let $X$ be a smooth surface and let $C \subseteq X$ be an integral curve. Then there exists $X \to X^\prime$ proper birational such that $X-C \cong X^\prime-\ast$ if and only if $C$ is a $(-1)$-curve and $X$ is the blowup at the point.
Definition 3. For $k = \bar{k}$, we say that $X$ is minimal when it doesn’t contain $(-1)$-curves.
Any sequence $X \to X_1 \to \dotsb$ of blow-downs terminates. This is because the rank of $\operatorname{NS}_ X$ (i.e., the Picard number) decreases by $1$ whenever we blow down.
Remark 4. The surface $\operatorname{Bl}_ {10} \mathbb{P}^2$ has infinitely many $(-1)$-curves. If we blow up $\mathbb{P}^2$ at two points through a line $\ell$ and then contract $\ell$, we get $\mathbb{P}^1 \times \mathbb{P}^1$. These are two non-isomorphic minimal models.
Theorem 5. If there exists one minimal model $X_1^\mathrm{min}$ on which $K_X$ is nef, then uniqueness holds, i.e., there is an isomorphism $X_1^\mathrm{min} \cong X_2^\mathrm{min}$ making the diagram commutative.
It follows that $X$ has finitely many $(-1)$-curves.
Theorem 6. If $X$ is minimal then either $X$ is ruled or $K_X$ is nef. Moreover, $K_X$ is nef if and only if $\kappa(X) \ge 0$.
First we show that $H^0(X, \mathscr{O}(nK_X)) \neq 0$ for any $n \gt 0$ then $K_X$ is nef. If not, $C.K_X \lt 0$, then we can write $D \sim nK_X$ for some effective $D$ and then $C.D \lt 0$ and so $C \subseteq D$ and $C^2 \lt 0$. By adjunction, we have $2 g(C) - 2 = (C + K_X) . C \lt 0$ and so $g(C) = 0$, i.e., $C$ is a smooth rational curve, and $C^2 = -1$ and $K_X.C = -1$. That is, $C$ is a $(-1)$-curve.
At this point, we want to show that $X$ has a “pencil” of rational curves if $K_X$ is not nef. It turns out that we can find a blowup $\operatorname{Bl} X \to X$ where there is a $\mathbb{P}^1$-bundle $\operatorname{Bl} X \to C$. Now by Tsen’s theorem any $\mathbb{P}^1$-bundle over a curve is ruled.
Theorem 7 (abundance). For $X$ a surface, if $K_X$ is nef then $\lvert nK_X \rvert$ has no basepoints for $n \gt 0$ sufficiently divisible.
Theorem 8 (Mumford). Let $X$ be a minimal surface.
- If $\lvert 12 K_X \rvert = \emptyset$ then $X$ is ruled.
- If $\lvert 12 K_X \rvert \neq \emptyset$ if and only if $K_X$ is nef, and in this case either $12 K_X = 0$ or $\lvert nK_X \rvert$ for $n \gg 0$ is basepoint free and $\kappa(X) \gt 0$.
Let us try to show that if $K_X$ is nef then $\lvert 12 K_X \rvert \neq \emptyset$. It is a general (but tricky) fact that if $K_X$ is nef then $K_X \ge 0$.
Theorem 9. If $K_X^2 = 0$ then either $2K_X = 0$ or there exists a pencil of curves of arithmetic genus $1$.
Theorem 10. If $X$ is minimal and $f \colon X \to C$ is a fibration in arithmetic genus $1$ curves, then $m K_X = f^\ast A$ for $A \subseteq C$ an effective divisor. If $K_X$ is nef and the generic fiber is smooth, then $\lvert 12 K_X \rvert \neq \emptyset$. If $K_X$ is nef and the generic fiber is singular, then either $\lvert 2K_X \rvert \neq \emptyset$ or $X$ admits an elliptic fibration.
Theorem 11. If $K_X^2 \gt 0$ then $\kappa(X) = 2$ and $\lvert 2K_X \rvert \neq \emptyset$ and $\lvert nK_X \rvert$ is basepoint free for $n \gg 0$.