The standard quadratic transformation $\sigma: \mathbb{P}^2 \dashrightarrow \mathbb{P}^2$ is defined by:

$\sigma([x_0 : x_1 : x_2]) = [x_1 x_2 : x_0 x_2 : x_0 x_1].$

This is a birational involution ($\sigma^2 = \mathrm{id}$). It is undefined at the three coordinate points $P_0 = [1:0:0]$, $P_1 = [0:1:0]$, $P_2 = [0:0:1]$, and it contracts the three coordinate lines $x_i = 0$ to the opposite coordinate points.

Resolving: blow up $P_0, P_1, P_2$ to get $\tilde{\mathbb{P}}^2$. On $\tilde{\mathbb{P}}^2$, $\sigma$ becomes a morphism that contracts the proper transforms of the three coordinate lines (each now a $(-1)$-curve) and blows down to $\mathbb{P}^2$.

• $\mathbb{P}^2$ and $\mathbb{P}^1 \times \mathbb{P}^1$ are birational: the map $\mathbb{P}^2 \dashrightarrow \mathbb{P}^1 \times \mathbb{P}^1$ given by projection from a point realizes this. Concretely, $\operatorname{Bl}_P(\mathbb{P}^2) \cong \mathbb{F}_1$ admits a morphism to $\mathbb{P}^1 \times \mathbb{P}^1$ by blowing down a different $(-1)$-curve.

• The Hirzebruch surfaces $\mathbb{F}_n$ for all $n \geq 0$ are rational: each is birational to $\mathbb{P}^1 \times \mathbb{P}^1 \cong \mathbb{F}_0$.

• A smooth cubic surface $S \subset \mathbb{P}^3$ is rational. Projecting from a line $\ell \subset S$ gives a birational map $S \dashrightarrow \mathbb{P}^2$. Since $S$ contains exactly $27$ lines, there are many such projections.

• A smooth quartic surface $S_4 \subset \mathbb{P}^3$ is a K3 surface and is not rational (it has $\kappa = 0$, while rational surfaces have $\kappa = -\infty$).

Let $X = \operatorname{Bl}_P(\mathbb{P}^2)$ with exceptional divisor $E$ and proper transform $H$ of a line through $P$. Then $\operatorname{Pic}(X) = \mathbb{Z}H \oplus \mathbb{Z}E$ with $H^2 = 1$, $E^2 = -1$, $H \cdot E = 0$.

The curve $E$ is a $(-1)$-curve, and blowing it down recovers $\mathbb{P}^2$. But $H - E$ is also a $(-1)$-curve (it is the proper transform of a line not through $P$... actually, $H$ itself satisfies $H^2 = 1$, so $H$ is not a $(-1)$-curve). The curve $L = H - E$ satisfies $L^2 = 1 - 1 = 0$, so it is not a $(-1)$-curve either.

On $\operatorname{Bl}_{P,Q}(\mathbb{P}^2)$ (blowup at two points), with basis $H, E_1, E_2$: the proper transform of the line through $P$ and $Q$ is $L = H - E_1 - E_2$, with $L^2 = 1 - 1 - 1 = -1$ and $L \cong \mathbb{P}^1$. So $L$ is a $(-1)$-curve that can be blown down, yielding $\mathbb{F}_1$.

A del Pezzo surface $S_d$ of degree $d$ is a blowup of $\mathbb{P}^2$ at $r = 9 - d$ points in general position (for $1 \leq d \leq 9$). The number of $(-1)$-curves is:

• $d = 9$ ($r = 0$, i.e., $\mathbb{P}^2$): $0$ exceptional curves.
• $d = 8$ ($r = 1$): $1$ curve ($E_1$).
• $d = 7$ ($r = 2$): $3$ curves ($E_1, E_2, L_{12}$).
• $d = 6$ ($r = 3$): $6$ curves.
• $d = 5$ ($r = 4$): $10$ curves.
• $d = 4$ ($r = 5$): $16$ curves.
• $d = 3$ ($r = 6$): $27$ curves --- these are the famous 27 lines on a cubic surface.
• $d = 2$ ($r = 7$): $56$ curves.
• $d = 1$ ($r = 8$): $240$ curves.

The count for degree $d \leq 6$ is $\binom{r}{1} + \binom{r}{2} + \ldots$ (exceptional divisors, proper transforms of lines, conics, etc.), reflecting the $E_r$ root system structure.

Start with $X = \operatorname{Bl}_{P_1, \ldots, P_6}(\mathbb{P}^2)$, the blowup of $\mathbb{P}^2$ at $6$ general points (a cubic surface $S_3 \subset \mathbb{P}^3$).

This surface has $27$ exceptional curves ($(-1)$-curves). Pick any one, say $E_1$, and blow it down to get $\operatorname{Bl}_{P_2, \ldots, P_6}(\mathbb{P}^2)$, a del Pezzo of degree $4$ with $16$ exceptional curves. Continue:

• Blow down $E_2$: get $\operatorname{Bl}_{P_3, \ldots, P_6}(\mathbb{P}^2)$ ($10$ exceptional curves).
• Blow down $E_3$: get $\operatorname{Bl}_{P_4, P_5, P_6}(\mathbb{P}^2)$ ($6$ exceptional curves).
• Blow down $E_4, E_5, E_6$: recover $\mathbb{P}^2$ (no exceptional curves, minimal).

Alternatively, at any stage we could blow down different $(-1)$-curves, potentially arriving at $\mathbb{F}_n$ instead of $\mathbb{P}^2$. This illustrates the non-uniqueness for $\kappa = -\infty$.

A K3 surface $X$ has $K_X \sim 0$ (trivial canonical class). If $E$ were a $(-1)$-curve, then $K_X \cdot E = 0$ but also $K_X \cdot E = -1$ by adjunction, a contradiction. Therefore K3 surfaces contain no $(-1)$-curves and are automatically minimal.

The same argument applies to abelian surfaces ($K_X \sim 0$) and Enriques surfaces ($2K_X \sim 0$, so $K_X \cdot E = 0$ for any curve $E$ since $2K_X \cdot E = 0$).

The minimal surfaces with $\kappa = -\infty$ are completely classified:

Rational surfaces ($q = 0$):

• $\mathbb{P}^2$: the projective plane, $K_{\mathbb{P}^2} = -3H$, $K^2 = 9$.
• $\mathbb{F}_n = \mathbb{P}(\mathcal{O}_{\mathbb{P}^1} \oplus \mathcal{O}_{\mathbb{P}^1}(-n))$ for $n \neq 1$: Hirzebruch surfaces. Note $\mathbb{F}_0 = \mathbb{P}^1 \times \mathbb{P}^1$ and $\mathbb{F}_1 = \operatorname{Bl}_P(\mathbb{P}^2)$ (not minimal since it contains a $(-1)$-curve).

Irrational ruled surfaces ($q = g(C) \geq 1$):

• $\mathbb{P}(\mathcal{E}) \to C$ for an indecomposable rank-$2$ bundle $\mathcal{E}$ on a curve $C$ of genus $g \geq 1$. These satisfy $P_m = 0$ for all $m$ and $q = g(C) \geq 1$.

Castelnuovo's rationality criterion: A surface is rational if and only if $q = h^1(\mathcal{O}_X) = 0$ and $P_2 = h^0(2K_X) = 0$.

There are exactly four types of minimal surfaces with $\kappa = 0$:

K3 surfaces: $K_X \sim 0$, $q = 0$, $p_g = 1$, $\chi(\mathcal{O}_X) = 2$. Examples: smooth quartic in $\mathbb{P}^3$, complete intersection of a quadric and cubic in $\mathbb{P}^4$, double cover of $\mathbb{P}^2$ branched along a smooth sextic. The lattice $H^2(X, \mathbb{Z})$ is isomorphic to $U^3 \oplus E_8(-1)^2$ (rank $22$, signature $(3,19)$).

Abelian surfaces: $K_X \sim 0$, $q = 2$, $p_g = 1$, $\chi(\mathcal{O}_X) = 0$. These are $2$-dimensional abelian varieties, e.g., $E_1 \times E_2$ for elliptic curves $E_i$, or the Jacobian $\operatorname{Jac}(C)$ of a genus-$2$ curve.

Enriques surfaces: $K_X \not\sim 0$ but $2K_X \sim 0$, $q = 0$, $p_g = 0$, $\chi(\mathcal{O}_X) = 1$. An Enriques surface is the quotient of a K3 surface by a fixed-point-free involution. Over $\mathbb{C}$, the fundamental group is $\pi_1(X) \cong \mathbb{Z}/2$.

Bielliptic (hyperelliptic) surfaces: $K_X \not\sim 0$ but $nK_X \sim 0$ for some $n \in \{2, 3, 4, 6\}$, $q = 1$, $p_g = 0$. These are quotients $(E_1 \times E_2)/G$ where $G$ is a finite group acting by translations on $E_1$ and faithfully on $E_2$.

A minimal surface with $\kappa = 1$ admits a unique elliptic fibration $\pi: X \to C$ over a smooth curve $C$, whose general fiber is an elliptic curve.

Invariants: $K_X^2 = 0$ (since $K_X$ is nef and $\kappa = 1$, Kodaira's formula gives $K_X \sim_\mathbb{Q}$ a sum along fibers), with the canonical bundle formula $K_X \sim \pi^*(K_C \otimes L) + \sum_i a_i F_i$ involving the discriminant divisor and singular fibers.

Kodaira's classification of singular fibers: the singular fibers can be of type $I_n$ (cycle of $n$ rational curves), $II$ (cuspidal rational curve), $III$ (two tangent rational curves), $IV$ (three concurrent rational curves), $I_n^*$, $II^*$, $III^*$, $IV^*$ (the starred types correspond to extended Dynkin diagrams $\tilde{D}_{n+4}$, $\tilde{E}_8$, $\tilde{E}_7$, $\tilde{E}_6$).

Concrete example: the Fermat pencil $x^3 + y^3 + z^3 = t \cdot xyz$ in $\mathbb{P}^2 \times \mathbb{P}^1$. The general fiber is a smooth cubic (elliptic), with singular fibers at $t^3 = 27$ (nodal cubics, type $I_1$) and $t = \infty$ (the triangle $xyz = 0$, type $I_3$). After resolving base points, this gives an elliptic surface with $\kappa = 1$.

A surface of general type is a surface with $\kappa = 2$. For a minimal surface of general type, $K_X$ is nef and big ($K_X^2 > 0$).

Key inequalities for minimal surfaces of general type:

• Noether's inequality: $K_X^2 \geq 2p_g - 4$ (equivalently, $\chi(\mathcal{O}_X) \leq \frac{1}{2}K_X^2 + 2$).
• Bogomolov--Miyaoka--Yau inequality: $K_X^2 \leq 9\chi(\mathcal{O}_X) = 9(1 - q + p_g)$.
• Equality $K_X^2 = 9\chi(\mathcal{O}_X)$ holds iff $X$ is a ball quotient $\mathbb{B}^2/\Gamma$ (Yau).

Examples:

• Smooth quintic in $\mathbb{P}^3$: $K_X = \mathcal{O}_X(1)$, $K_X^2 = 5$, $p_g = 4$, $q = 0$, $\chi = 5$.
• Product $C_1 \times C_2$ of curves with $g(C_i) \geq 2$: $K^2 = 8(g_1 - 1)(g_2 - 1)$, $\chi = (g_1 - 1)(g_2 - 1)$.
• Godeaux surface: $K^2 = 1$, $p_g = 0$, the simplest surface of general type. Constructed as the quotient of the Fermat quintic $x_0^5 + x_1^5 + x_2^5 + x_3^5 = 0$ by the $\mathbb{Z}/5$-action $x_i \mapsto \zeta^i x_i$ ($\zeta = e^{2\pi i/5}$).
• Barlow surface: a simply connected surface of general type with $p_g = 0$, $K^2 = 1$.
• Beauville surface: a surface isogenous to a higher product $C_1 \times C_2 / G$.

• Linear maps: $[x:y:z] \mapsto [ax + by + cz : dx + ey + fz : gx + hy + iz]$, degree $1$.

• Standard Cremona $\sigma$: degree $2$. Resolves to $3$ blowups followed by $3$ blowdowns.

• Degree-$n$ Cremona maps: for example, $[x:y:z] \mapsto [x^n : y^n : z^n]$ is not birational (it has degree $n^2$). A degree-$n$ birational map is $[x:y:z] \mapsto [F_0 : F_1 : F_2]$ with $\deg F_i = n$ and a rational inverse.

• De Jonquieres transformation of degree $n$: preserves the pencil of lines through $[1:0:0]$. In affine coordinates $(x, y)$: $(x, y) \mapsto (x, \frac{a(x)y + b(x)}{c(x)y + d(x)})$ for polynomials $a, b, c, d$ with $ad - bc \neq 0$. These form a subgroup $\operatorname{dJ} \subset \operatorname{Cr}_2$.

• The Cremona group is not simple: Cantat and Lamy (2013) proved that $\operatorname{Cr}_2(\mathbb{C})$ is not simple, using the action on an infinite-dimensional hyperbolic space.

The standard Cremona $\sigma: \mathbb{P}^2 \dashrightarrow \mathbb{P}^2$ factors as:

1. Three blowups: Blow up $P_0 = [1:0:0]$, $P_1 = [0:1:0]$, $P_2 = [0:0:1]$ to obtain $Z = \operatorname{Bl}_{P_0, P_1, P_2}(\mathbb{P}^2)$. This introduces exceptional divisors $E_0, E_1, E_2$ with $E_i^2 = -1$, and the proper transforms $\ell_{01}, \ell_{02}, \ell_{12}$ of the coordinate lines now satisfy $\ell_{ij}^2 = -1$.

2. Three blowdowns: Contract $\ell_{01}, \ell_{02}, \ell_{12}$ (the proper transforms) to get $\mathbb{P}^2$ again, where $E_0, E_1, E_2$ become the new coordinate lines.

The intermediate surface $Z$ is the del Pezzo surface $S_6$ (degree $6$, isomorphic to $\mathbb{P}^2$ blown up at $3$ non-collinear points), and it admits two different contractions to $\mathbb{P}^2$.

Consider $\mathbb{F}_0 = \mathbb{P}^1 \times \mathbb{P}^1$ and $\mathbb{F}_2$. These are birational (both rational) but not isomorphic. A birational map $\mathbb{F}_0 \dashrightarrow \mathbb{F}_2$ factors through one blowup and one blowdown:

1. Blow up a point $P \in \mathbb{F}_0$: get $\operatorname{Bl}_P(\mathbb{F}_0) \cong \operatorname{Bl}_{Q_1, Q_2}(\mathbb{P}^2)$ (blowup of $\mathbb{P}^2$ at two points), which is $\mathbb{F}_1$ with an extra blowup.
2. The proper transform of the fiber through $P$ becomes a $(-1)$-curve. Blowing it down yields $\mathbb{F}_2$.

More generally, the elementary transformation $\operatorname{elm}_P: \mathbb{F}_n \dashrightarrow \mathbb{F}_{n \pm 1}$ at a point $P$ is a blowup followed by blowdown of the proper transform of the fiber.

Consider a smooth 3-fold $X$ containing a curve $C \cong \mathbb{P}^1$ with normal bundle $N_{C/X} \cong \mathcal{O}(-1) \oplus \mathcal{O}(-1)$. Then $K_X \cdot C = -1$ (by adjunction on the 3-fold). The contraction $X \to Y$ that contracts $C$ to a point creates an ordinary double point (node) singularity.

The flip $X \dashrightarrow X^+$ replaces $C \cong \mathbb{P}^1$ with a different $\mathbb{P}^1$ having normal bundle $\mathcal{O}(-1) \oplus \mathcal{O}(-1)$ but with $K_{X^+} \cdot C^+ = +1$. This phenomenon has no analogue for surfaces.

The classical example: the Atiyah flop on the conifold $xy = zw$ in $\mathbb{C}^4$. The two small resolutions (blowing up the ideal $(x, z)$ vs. $(x, w)$) are related by a flop.

Rational surfaces ($\mathbb{P}^2$, $\mathbb{F}_n$, del Pezzo surfaces): $\kappa = -\infty$, $q = 0$, $P_m = 0$ for all $m$, $\pi_1 = 0$. But $K^2$ varies: $K^2_{\mathbb{P}^2} = 9$, $K^2_{\mathbb{F}_n} = 8$, $K^2_{S_d} = d$ for del Pezzo of degree $d$.

K3 surfaces: $\kappa = 0$, $q = 0$, $P_1 = 1$, $P_m = 1$ for all $m \geq 1$. All K3 surfaces are diffeomorphic (over $\mathbb{C}$) and have $\pi_1 = 0$.

Abelian surfaces: $\kappa = 0$, $q = 2$, $P_1 = 1$, $P_m = 1$ for all $m$. The fundamental group $\pi_1 \cong \mathbb{Z}^4$ distinguishes them from K3 surfaces.

Quintic surface $S_5 \subset \mathbb{P}^3$ (general type): $\kappa = 2$, $q = 0$, $K^2 = 5$, $p_g = P_1 = 4$, $\chi(\mathcal{O}) = 5$, $\pi_1 = 0$.