Mathematics Lecture Notes

Definition5.10Canonical class

Let $X$ be a smooth projective surface over an algebraically closed field $k$ . The canonical sheaf (or dualizing sheaf) is $\omega_X = \det \Omega^1_{X/k} = \wedge^2 \Omega^1_{X/k}$ , the determinant of the cotangent bundle.

The canonical class is the corresponding divisor class:

$K_X = c_1(\omega_X) = c_1(\det \Omega^1_{X/k}) \in \operatorname{Pic}(X).$

Any divisor in this class is called a canonical divisor. The canonical class is defined up to linear equivalence: if $\omega$ is a nonzero rational $2$ -form on $X$ , then $K_X = \operatorname{div}(\omega)$ .

Definition5.11Plurigenera and Kodaira dimension

For a smooth projective surface $X$ , the $n$ -th plurigenus is

$P_n(X) = h^0(X, \omega_X^{\otimes n}) = h^0(X, \mathcal{O}_X(nK_X)), \quad n \geq 1.$

In particular, $P_1 = p_g = h^0(K_X)$ is the geometric genus.

The Kodaira dimension $\kappa(X)$ is defined as:

$\kappa(X) = -\infty$ if $P_n = 0$ for all $n \geq 1$ ,
$\kappa(X) = 0$ if the $P_n$ are bounded but not all zero,
$\kappa(X) = 1$ if $P_n$ grows linearly: $P_n \sim cn$ for some $c > 0$ ,
$\kappa(X) = 2$ if $P_n$ grows quadratically: $P_n \sim cn^2$ for some $c > 0$ .

Equivalently, $\kappa(X) = \limsup_{n \to \infty} \frac{\log P_n}{\log n}$ when $P_n \neq 0$ for some $n$ .

RemarkBirational invariance of plurigenera

The plurigenera $P_n$ are birational invariants of smooth projective surfaces. If $f: X \dashrightarrow Y$ is a birational map between smooth projective surfaces, then $P_n(X) = P_n(Y)$ for all $n \geq 1$ . In particular, the Kodaira dimension $\kappa(X)$ is a birational invariant. This is because rational maps induce isomorphisms on pluricanonical forms away from codimension- $2$ loci, and sections extend by Hartogs' theorem.

ExampleCanonical class of P^2

On $\mathbb{P}^2$ with hyperplane class $H$ , the Euler sequence $0 \to \Omega^1_{\mathbb{P}^2} \to \mathcal{O}(-1)^3 \to \mathcal{O} \to 0$ gives:

$K_{\mathbb{P}^2} = -3H.$

Since $K_{\mathbb{P}^2}$ is anti-ample, $P_n = h^0(nK) = h^0(\mathcal{O}(-3n)) = 0$ for all $n \geq 1$ . Thus $\kappa(\mathbb{P}^2) = -\infty$ .

The self-intersection is $K_{\mathbb{P}^2}^2 = (-3H)^2 = 9$ .

ExampleCanonical class of P^1 x P^1

On $X = \mathbb{P}^1 \times \mathbb{P}^1$ with fiber classes $F_1, F_2$ :

$K_X = -2F_1 - 2F_2.$

This follows from the product formula $\omega_X = \omega_{\mathbb{P}^1} \boxtimes \omega_{\mathbb{P}^1} = \mathcal{O}(-2) \boxtimes \mathcal{O}(-2)$ .

Again $K_X$ is anti-ample, so $\kappa = -\infty$ . The self-intersection is $K_X^2 = (-2F_1 - 2F_2)^2 = 2 \cdot 4 = 8$ .

ExampleCanonical class of Hirzebruch surfaces

On the Hirzebruch surface $\mathbb{F}_n = \mathbb{P}(\mathcal{O} \oplus \mathcal{O}(-n))$ over $\mathbb{P}^1$ , with negative section $C_0$ ( $C_0^2 = -n$ ) and fiber $F$ :

$K_{\mathbb{F}_n} = -2C_0 - (n+2)F.$

All Hirzebruch surfaces are rational, so $\kappa(\mathbb{F}_n) = -\infty$ and $K^2 = 8$ (for $n \geq 0$ ). Note $K^2 = (-2C_0 - (n+2)F)^2 = 4(-n) + 2 \cdot 2(n+2) = -4n + 4n + 8 = 8$ .

TheoremAdjunction formula for hypersurfaces

Let $X$ be a smooth projective variety and $H \subset X$ a smooth divisor. The adjunction formula states:

$K_H = (K_X + H)|_H.$

Equivalently, $\omega_H \cong \omega_X \otimes \mathcal{O}_X(H)|_H$ . On surfaces, this specializes to: if $C$ is a smooth curve on a surface $X$ , then $2g(C) - 2 = C^2 + K_X \cdot C$ .

ExampleCanonical class of smooth hypersurfaces in P^3

Let $S \subset \mathbb{P}^3$ be a smooth surface of degree $d$ , so $S \sim dH$ where $H$ is the hyperplane class. By adjunction with $K_{\mathbb{P}^3} = -4H$ :

$K_S = (K_{\mathbb{P}^3} + S)|_S = (-4H + dH)|_S = (d - 4)H|_S.$

$d = 1$ (plane $\mathbb{P}^2$ ): $K = -3H|_S$ , anti-ample. $\kappa = -\infty$ .
$d = 2$ (quadric $\cong \mathbb{P}^1 \times \mathbb{P}^1$ ): $K = -2H|_S$ , anti-ample. $\kappa = -\infty$ .
$d = 3$ (cubic surface): $K = -H|_S$ , anti-ample. $\kappa = -\infty$ . Rational (del Pezzo).
$d = 4$ (quartic surface): $K = 0$ . This is a K3 surface. $\kappa = 0$ .
$d = 5$ (quintic surface): $K = H|_S$ , ample. $\kappa = 2$ . Surface of general type.
$d \geq 5$ : $K = (d-4)H|_S$ is ample, so $\kappa = 2$ .

The self-intersection: $K_S^2 = (d-4)^2 d$ (since $H|_S^2 = d$ ). For $d = 5$ : $K^2 = 5$ . For $d = 6$ : $K^2 = 24$ .

ExampleCanonical class of complete intersections

For a smooth complete intersection surface $S = V(F_1, F_2) \subset \mathbb{P}^4$ of degrees $d_1, d_2$ :

$K_S = (d_1 + d_2 - 5)H|_S.$

$(2, 2)$ : $K = -H|_S$ , anti-ample. This is a del Pezzo surface of degree $4$ . $\kappa = -\infty$ .
$(2, 3)$ : $K = 0$ . A K3 surface. $\kappa = 0$ .
$(2, 4)$ : $K = H|_S$ , ample. $K^2 = 8$ . Surface of general type.
$(3, 3)$ : $K = H|_S$ , ample. $K^2 = 9$ . Surface of general type.

Definition5.12Rational and ruled surfaces

A surface $X$ is rational if it is birational to $\mathbb{P}^2$ . It is ruled if it is birational to $C \times \mathbb{P}^1$ for some smooth curve $C$ . Ruled surfaces with $C = \mathbb{P}^1$ are exactly the rational surfaces.

A surface has $\kappa(X) = -\infty$ if and only if $P_n(X) = 0$ for all $n \geq 1$ , which happens if and only if $X$ is ruled (Enriques--Kodaira classification).

ExampleSurfaces with kappa = -infty

$\mathbb{P}^2$ : rational, $K = -3H$ , $K^2 = 9$ . The prototype.
$\mathbb{P}^1 \times \mathbb{P}^1$ : rational, $K = -2F_1 - 2F_2$ , $K^2 = 8$ . Birational to $\mathbb{P}^2$ via blowup/blowdown.
$\mathbb{F}_n$ : rational, $K^2 = 8$ . For $n \neq 1$ , these are minimal; $\mathbb{F}_1 \cong \operatorname{Bl}_P(\mathbb{P}^2)$ .
Del Pezzo surfaces $S_d$ (degree $d$ ): $K_{S_d} = -H$ (anticanonical embedding). $K^2 = d$ for $d = 1, \ldots, 9$ . These are $\mathbb{P}^2$ blown up at $9 - d$ general points ( $d \leq 8$ ) or $\mathbb{P}^1 \times \mathbb{P}^1$ ( $d = 8$ ) or $\mathbb{P}^2$ ( $d = 9$ ).
Ruled surfaces over curves of genus $g \geq 1$ : $\mathbb{P}(\mathcal{E}) \to C$ for a rank- $2$ bundle $\mathcal{E}$ on a curve $C$ of genus $g$ . Still $\kappa = -\infty$ , but not rational when $g \geq 1$ .

The key invariant: $P_{12}(X) = 0$ characterizes $\kappa = -\infty$ (Castelnuovo's rationality criterion says $P_2 = q = 0$ implies rationality).

Definition5.13Surfaces with kappa = 0

A minimal surface has $\kappa(X) = 0$ if and only if $nK_X \sim 0$ for some $n \geq 1$ (i.e., $K_X$ is a torsion element in $\operatorname{Pic}(X)$ ). The plurigenera satisfy $P_n \in \{0, 1\}$ for all $n$ , and $P_n = 1$ whenever $n$ is a multiple of the order of $K_X$ in $\operatorname{Pic}(X)$ .

ExampleK3 surfaces (kappa = 0)

A K3 surface $X$ is a smooth projective surface with $K_X \sim 0$ (trivial canonical class) and $H^1(X, \mathcal{O}_X) = 0$ (simply connected in the algebraic sense: $q = 0$ ).

Numerical invariants: $K^2 = 0$ , $p_g = 1$ , $q = 0$ , $\chi(\mathcal{O}_X) = 2$ , $c_2(X) = 24$ , $b_2 = 22$ .

Examples of K3 surfaces:

Quartic surface in $\mathbb{P}^3$ (e.g., the Fermat quartic $x^4 + y^4 + z^4 + w^4 = 0$ ).
Complete intersection $(2, 3) \subset \mathbb{P}^4$ .
Complete intersection $(2, 2, 2) \subset \mathbb{P}^5$ .
Double cover of $\mathbb{P}^2$ branched along a smooth sextic.
Kummer surface: $\operatorname{Km}(A)$ for an abelian surface $A$ (minimal resolution of $A/\{{\pm 1}\}$ ).

All K3 surfaces are deformation-equivalent (over $\mathbb{C}$ ), forming a $20$ -dimensional moduli space.

ExampleAbelian surfaces (kappa = 0)

An abelian surface $A$ is a $2$ -dimensional abelian variety: a smooth projective group variety of dimension $2$ . Equivalently, over $\mathbb{C}$ , $A \cong \mathbb{C}^2 / \Lambda$ for a lattice $\Lambda \cong \mathbb{Z}^4$ .

The canonical class is trivial: $K_A = 0$ (the group structure provides a nowhere-vanishing $2$ -form $dz_1 \wedge dz_2$ ).

Numerical invariants: $K^2 = 0$ , $p_g = 1$ , $q = 2$ , $\chi(\mathcal{O}_A) = 0$ , $c_2(A) = 0$ , $b_1 = 4$ , $b_2 = 6$ .

Examples: products $E_1 \times E_2$ of elliptic curves, Jacobians $\operatorname{Jac}(C)$ of genus- $2$ curves.

ExampleEnriques surfaces (kappa = 0)

An Enriques surface $X$ is a smooth projective surface with $K_X \not\sim 0$ but $2K_X \sim 0$ , and $q = 0$ .

The canonical class has order exactly $2$ in $\operatorname{Pic}(X)$ . The plurigenera alternate: $P_n = 0$ for $n$ odd, $P_n = 1$ for $n$ even.

Numerical invariants: $K^2 = 0$ , $p_g = 0$ , $q = 0$ , $\chi(\mathcal{O}_X) = 1$ , $c_2(X) = 12$ , $b_2 = 10$ .

Every Enriques surface admits an unramified double cover that is a K3 surface. The moduli space of Enriques surfaces (over $\mathbb{C}$ ) has dimension $10$ .

ExampleBielliptic (hyperelliptic) surfaces (kappa = 0)

A bielliptic surface (also called hyperelliptic surface) is a surface of the form $X = (E_1 \times E_2)/G$ where $E_1, E_2$ are elliptic curves and $G$ is a finite group acting by translations on $E_1$ and faithfully on $E_2$ .

The canonical class satisfies $nK_X \sim 0$ where $n \in \{2, 3, 4, 6\}$ depending on $G$ .

Numerical invariants: $K^2 = 0$ , $p_g = 0$ , $q = 1$ , $\chi(\mathcal{O}_X) = 0$ , $c_2(X) = 0$ .

These are the only minimal surfaces with $\kappa = 0$ and $q = 1$ .

RemarkComplete classification of kappa = 0

Over $\mathbb{C}$ , the four classes of minimal surfaces with $\kappa = 0$ are distinguished by their invariants:

K3: $K \sim 0$ , $q = 0$ , $p_g = 1$ , $\chi(\mathcal{O}) = 2$ .
Abelian: $K \sim 0$ , $q = 2$ , $p_g = 1$ , $\chi(\mathcal{O}) = 0$ .
Enriques: $2K \sim 0$ , $K \not\sim 0$ , $q = 0$ , $p_g = 0$ , $\chi(\mathcal{O}) = 1$ .
Bielliptic: $nK \sim 0$ ( $n = 2, 3, 4, 6$ ), $q = 1$ , $p_g = 0$ , $\chi(\mathcal{O}) = 0$ .

In all four cases, $K^2 = 0$ and $c_2 = 12\chi(\mathcal{O}_X)$ (by Noether's formula).

Definition5.14Elliptic surfaces

A smooth projective surface $X$ has $\kappa(X) = 1$ if and only if it admits an elliptic fibration $f: X \to C$ (a morphism to a curve whose general fiber is an elliptic curve) and the canonical class $K_X$ is not torsion.

For such surfaces, $K_X$ is numerically equivalent to a positive rational multiple of the fiber class. The plurigenera grow linearly: $P_n \sim cn$ for some positive constant $c$ as $n \to \infty$ .

ExampleSurfaces with kappa = 1

Elliptic surface over $\mathbb{P}^1$ with $\chi(\mathcal{O}) \geq 3$ : For instance, the surface $y^2 = x^3 + t^{12} + 1$ in a suitable weighted projective bundle over $\mathbb{P}^1_t$ gives $\kappa = 1$ .
Product $E \times C$ where $E$ is elliptic and $g(C) \geq 2$ : $K = \pi_1^* K_E + \pi_2^* K_C = \pi_2^* K_C$ . Then $P_n = h^0(C, nK_C) = (2n - 1)(g - 1)$ for $n \geq 2$ , growing linearly. So $\kappa = 1$ .
Dolgachev surfaces: certain logarithmic transforms of rational elliptic surfaces, giving $p_g = 0$ , $q = 0$ , $\kappa = 1$ .

Note: surfaces with $\kappa = 1$ always have $K^2 = 0$ (since $K$ is nef and $K^2 = 0$ for a numerically fiber-type class).

Definition5.15Surface of general type

A smooth projective surface $X$ has $\kappa(X) = 2$ (is of general type) if and only if, for the minimal model $X_{\min}$ , the canonical class $K_{X_{\min}}$ is nef and big:

Nef: $K \cdot C \geq 0$ for every curve $C \subset X_{\min}$ .
Big: $K^2 > 0$ .

Equivalently, $P_n$ grows quadratically: $P_n \sim \frac{1}{2}K^2 n^2$ as $n \to \infty$ (by Riemann--Roch and vanishing). These are the "generic" surfaces, analogous to curves of genus $\geq 2$ .

ExampleSurfaces of general type

Quintic in $\mathbb{P}^3$ ( $d = 5$ ): $K = H|_S$ , $K^2 = 5$ , $\chi(\mathcal{O}) = 5$ , $p_g = 4$ , $q = 0$ .
Sextic in $\mathbb{P}^3$ ( $d = 6$ ): $K = 2H|_S$ , $K^2 = 24$ , $p_g = 20$ .
Complete intersection $(2, 4) \subset \mathbb{P}^4$ : $K = H|_S$ , $K^2 = 8$ .
Complete intersection $(3, 3) \subset \mathbb{P}^4$ : $K = H|_S$ , $K^2 = 9$ .
Product $C_1 \times C_2$ with $g(C_i) \geq 2$ : $K = \pi_1^*K_{C_1} + \pi_2^*K_{C_2}$ , $K^2 = 2(2g_1 - 2)(2g_2 - 2) = 8(g_1 - 1)(g_2 - 1)$ .
Godeaux surface: $p_g = q = 0$ , $K^2 = 1$ . The simplest surface of general type. Classically constructed as $\operatorname{Fermat quintic}/(\mathbb{Z}/5)$ .
Barlow surface: $p_g = q = 0$ , $K^2 = 1$ . A simply connected surface of general type -- showing that $\pi_1$ cannot distinguish rational surfaces from general type in all cases.
Beauville surface: $p_g = q = 0$ , $K^2 = 8$ . A quotient $(C_1 \times C_2)/G$ for suitable curves and group $G$ .

TheoremAbundance theorem for surfaces

Let $X$ be a smooth projective minimal surface with $\kappa(X) \geq 0$ . Then $K_X$ is semiample: there exists $n > 0$ such that the linear system $|nK_X|$ is base-point-free.

In particular, $|nK_X|$ defines a morphism $\phi_{|nK_X|}: X \to \mathbb{P}^{P_n - 1}$ , called the pluricanonical map. The image has dimension equal to $\kappa(X)$ .

Concretely:

$\kappa = 0$ : $\phi_{|nK|}$ maps $X$ to a point (since $P_n \leq 1$ ).
$\kappa = 1$ : $\phi_{|nK|}$ factors through the elliptic fibration $X \to C$ .
$\kappa = 2$ : $\phi_{|nK|}$ is birational onto its image for $n \gg 0$ (in fact, $n \geq 5$ suffices by Bombieri's theorem).

RemarkAbundance in higher dimensions

The abundance conjecture -- that nef canonical divisors on minimal varieties are always semiample -- is proven for surfaces (Mumford, Kawamata, and others) and threefolds (Miyaoka, Kawamata), but remains open in dimension $\geq 4$ . It is one of the central conjectures in birational geometry, closely related to the Minimal Model Program.

Theorem5.2Noether's formula

For any smooth projective surface $X$ :

$\chi(\mathcal{O}_X) = \frac{1}{12}(K_X^2 + c_2(X)),$

where $c_2(X) = e(X)$ is the topological Euler characteristic (the second Chern number of $T_X$ ).

Writing out: $1 - q + p_g = \frac{1}{12}(K^2 + e(X))$ , so $K^2 + e(X) = 12\chi(\mathcal{O}_X)$ .

ExampleNoether's formula: verifications

$\mathbb{P}^2$ : $K^2 = 9$ , $e = 3$ (Betti numbers $1, 0, 1, 0, 1$ ), $\chi(\mathcal{O}) = 1$ . Check: $9 + 3 = 12$ ✓.
$\mathbb{P}^1 \times \mathbb{P}^1$ : $K^2 = 8$ , $e = 4$ (Betti numbers $1, 0, 2, 0, 1$ ), $\chi(\mathcal{O}) = 1$ . Check: $8 + 4 = 12$ ✓.
K3 surface: $K^2 = 0$ , $\chi(\mathcal{O}) = 2$ . Noether gives $e = 24$ , so $b_2 = 22$ .
Abelian surface: $K^2 = 0$ , $\chi(\mathcal{O}) = 0$ . Noether gives $e = 0$ , consistent with $b_0 = 1, b_1 = 4, b_2 = 6$ giving $e = 1 - 4 + 6 - 4 + 1 = 0$ .
Enriques surface: $K^2 = 0$ , $\chi(\mathcal{O}) = 1$ . Noether gives $e = 12$ , so $b_2 = 10$ .
Quintic in $\mathbb{P}^3$ : $K^2 = 5$ , $\chi(\mathcal{O}) = 5$ (since $p_g = 4, q = 0$ ). Noether gives $e = 55$ .

TheoremBogomolov--Miyaoka--Yau inequality

For any smooth projective surface $X$ of general type (over $\mathbb{C}$ ):

$K_X^2 \leq 3c_2(X) = 3e(X).$

Using Noether's formula $K^2 + e = 12\chi(\mathcal{O})$ , this is equivalent to:

$K_X^2 \leq 9\chi(\mathcal{O}_X).$

Equality $K^2 = 3c_2$ (equivalently $K^2 = 9\chi$ ) holds if and only if the universal cover of $X$ is the complex $2$ -ball $\mathbb{B}^2 \subset \mathbb{C}^2$ , i.e., $X$ is a ball quotient.

TheoremNoether's inequality

For a minimal surface of general type:

$K_X^2 \geq 2\chi(\mathcal{O}_X) - 6.$

Equivalently, $K^2 \geq 2p_g - 4$ when $q = 0$ . This provides a lower bound complementing the BMY upper bound.

ExampleGeography of surfaces of general type

The geography problem asks: which pairs $(K^2, \chi(\mathcal{O}))$ are realized by minimal surfaces of general type? The constraints are:

$K^2 > 0$ and $\chi(\mathcal{O}) > 0$ (general type, minimal).
BMY inequality: $K^2 \leq 9\chi(\mathcal{O})$ (the "BMY line").
Noether inequality: $K^2 \geq 2\chi(\mathcal{O}) - 6$ (the "Noether line").
Bogomolov--Miyaoka--Yau: $c_1^2 \leq 3c_2$ .
$K^2$ and $\chi$ are positive integers with some congruence conditions.

The "geography" of surfaces is the region in the $(K^2, \chi)$ -plane between the Noether line and the BMY line:

$2\chi - 6 \leq K^2 \leq 9\chi.$

On the BMY line ( $K^2 = 9\chi$ ): ball quotients. The simplest is the fake projective plane with $K^2 = 9, \chi = 1$ ( $p_g = q = 0$ ). Exactly $100$ fake projective planes exist (Prasad--Yeung, Cartwright--Steger).
On the Noether line ( $K^2 = 2\chi - 6$ ): these include many families of fibered surfaces.
$K^2 = 1, \chi = 1$ : Godeaux surfaces ( $p_g = q = 0$ ).
$K^2 = 2, \chi = 1$ : Campedelli surfaces ( $p_g = q = 0$ ).
$K^2 = 1, \chi = 2$ : numerical Godeaux with $p_g = 1$ , no known examples with $q = 0$ -- an open problem.
Most lattice points in the region are realized by some surface.

TheoremCanonical class of a blowup

Let $\pi: \tilde{X} \to X$ be the blowup of a smooth surface $X$ at a point $P$ , with exceptional divisor $E$ . Then:

$K_{\tilde{X}} = \pi^* K_X + E.$

Consequently: $K_{\tilde{X}}^2 = K_X^2 - 1$ (since $E^2 = -1$ and $\pi^*K_X \cdot E = 0$ ). Each blowup decreases $K^2$ by $1$ .

ExampleBlowups of P^2

Starting from $\mathbb{P}^2$ with $K^2 = 9$ and blowing up $r$ points in general position:

$K^2 = 9 - r, \quad K_{\tilde{X}} = -3H + E_1 + \cdots + E_r.$

$r = 0$ : $K^2 = 9$ , this is $\mathbb{P}^2$ .
$r = 1$ : $K^2 = 8$ , this is $\mathbb{F}_1$ .
$r \leq 8$ : del Pezzo surfaces $S_{9-r}$ , with $-K$ ample.
$r = 9$ : $K^2 = 0$ , a rational elliptic surface. $-K$ defines an elliptic pencil $|-K|$ .
$r = 10$ : $K^2 = -1$ . No longer del Pezzo; the surface becomes increasingly complicated.

For del Pezzo surfaces ( $r \leq 8$ ), $K$ is anti-ample and $-K$ embeds the surface as a degree- $(9-r)$ surface in $\mathbb{P}^{9-r}$ .

Definition5.16Irregularity and geometric genus

For a smooth projective surface $X$ :

Geometric genus: $p_g = h^0(X, K_X) = h^{0,2} = h^{2,0}$ (over $\mathbb{C}$ , by Hodge theory).
Irregularity: $q = h^1(X, \mathcal{O}_X) = h^{0,1}$ (dimension of $\operatorname{Pic}^0(X) = \operatorname{Alb}(X)$ ).
Holomorphic Euler characteristic: $\chi(\mathcal{O}_X) = 1 - q + p_g$ .
Chern numbers: $c_1^2 = K_X^2$ and $c_2 = e(X) =$ topological Euler number.
Relation: $K^2 + c_2 = 12\chi(\mathcal{O}_X)$ (Noether's formula).

These satisfy further constraints: $p_g \geq 0$ , $q \geq 0$ , and for surfaces of general type, $\chi(\mathcal{O}_X) > 0$ .

ExampleTable of invariants for classical surfaces

Summary of the canonical class and invariants:

Rational surfaces: $K^2 = 9 - r$ (for $\operatorname{Bl}_r \mathbb{P}^2$ ), $q = 0$ , $p_g = 0$ , $\chi(\mathcal{O}) = 1$ .

K3 surfaces: $K \sim 0$ , $K^2 = 0$ , $q = 0$ , $p_g = 1$ , $\chi(\mathcal{O}) = 2$ , $c_2 = 24$ .

Abelian surfaces: $K \sim 0$ , $K^2 = 0$ , $q = 2$ , $p_g = 1$ , $\chi(\mathcal{O}) = 0$ , $c_2 = 0$ .

Enriques surfaces: $2K \sim 0$ , $K^2 = 0$ , $q = 0$ , $p_g = 0$ , $\chi(\mathcal{O}) = 1$ , $c_2 = 12$ .

Bielliptic surfaces: $nK \sim 0$ , $K^2 = 0$ , $q = 1$ , $p_g = 0$ , $\chi(\mathcal{O}) = 0$ , $c_2 = 0$ .

Quintic surface ( $\mathbb{P}^3$ ): $K = H|_S$ , $K^2 = 5$ , $q = 0$ , $p_g = 4$ , $\chi(\mathcal{O}) = 5$ , $c_2 = 55$ .

Product $C_{g_1} \times C_{g_2}$ ( $g_i \geq 2$ ): $K^2 = 8(g_1-1)(g_2-1)$ , $q = g_1 + g_2$ , $p_g = g_1 g_2$ , $c_2 = 4(g_1-1)(g_2-1)$ . BMY ratio: $K^2/c_2 = 2$ .

TheoremBombieri's theorem

Let $X$ be a minimal surface of general type. Then the $n$ -th pluricanonical map $\phi_{|nK_X|}: X \to \mathbb{P}^{P_n - 1}$ is a birational morphism onto its image for all $n \geq 5$ .

Moreover:

$|2K_X|$ is base-point-free (for $K^2 \geq 2$ ; if $K^2 = 1$ then $|3K|$ is base-point-free).
$|5K_X|$ is very ample (gives a birational embedding) in all cases.
The image $\phi_{|nK|}(X)$ is a canonical model with at worst rational double point singularities.

ExampleGrowth of plurigenera

By Riemann--Roch and Kawamata--Viehweg vanishing, for $n \geq 2$ on a minimal surface of general type:

$P_n = h^0(nK_X) = \frac{n(n-1)}{2}K^2 + \chi(\mathcal{O}_X).$

Explicit values for a quintic surface ( $K^2 = 5$ , $\chi = 5$ ):

$P_1 = p_g = 4$ .
$P_2 = 5 + 5 = 10$ .
$P_3 = 15 + 5 = 20$ .
$P_4 = 30 + 5 = 35$ .
$P_5 = 50 + 5 = 55$ .

For a Godeaux surface ( $K^2 = 1$ , $\chi = 1$ ):

$P_1 = 0$ (since $p_g = 0$ , $q = 0$ ).
$P_2 = 1 + 1 = 2$ .
$P_3 = 3 + 1 = 4$ .
$P_4 = 6 + 1 = 7$ .
$P_5 = 10 + 1 = 11$ .

Note that $P_1$ can differ from the Riemann--Roch prediction because vanishing may fail for $n = 1$ .

RemarkMinimal models and canonical class

The minimal model program (MMP) for surfaces proceeds by contracting $(-1)$ -curves (smooth rational curves $E$ with $E^2 = -1$ , equivalently $K \cdot E = -1$ by adjunction). Each contraction increases $K^2$ by $1$ .

A smooth surface is minimal if it contains no $(-1)$ -curves. For a minimal surface:

$\kappa = -\infty$ : $X$ is $\mathbb{P}^2$ or a ruled surface $\mathbb{P}(\mathcal{E}) \to C$ (Castelnuovo--Enriques).
$\kappa \geq 0$ : $K_X$ is nef, and the canonical model exists by the abundance theorem.

The canonical class governs the entire MMP: $(-1)$ -curves are exactly the irreducible curves $E$ with $K \cdot E < 0$ and $E^2 < 0$ , and their contraction "improves" $K$ toward nefness.

ExampleRunning the MMP on a blowup

Let $X = \operatorname{Bl}_{P_1, P_2}(\mathbb{P}^2)$ , the blowup of $\mathbb{P}^2$ at two points, with $K_X = -3H + E_1 + E_2$ , $K^2 = 7$ .

The $(-1)$ -curves are $E_1, E_2$ , and the strict transform $L$ of the line through $P_1, P_2$ (with $L \sim H - E_1 - E_2$ , $L^2 = -1$ ).

Contract $E_1$ : get $\operatorname{Bl}_{P_2}(\mathbb{P}^2) = \mathbb{F}_1$ , $K^2 = 8$ .
Contract the remaining $(-1)$ -curve on $\mathbb{F}_1$ : get $\mathbb{P}^2$ , $K^2 = 9$ .

Alternatively, contract $L$ first: get a surface with $K^2 = 8$ , which is $\mathbb{P}^1 \times \mathbb{P}^1$ (since contracting $L$ resolves to the other ruling). Then $\mathbb{P}^1 \times \mathbb{P}^1$ is already minimal.

The MMP is not unique, but the minimal model for $\kappa = -\infty$ is always $\mathbb{P}^2$ or a Hirzebruch surface.

RemarkThe canonical class as the master invariant

The canonical class $K_X$ organizes the entire theory of algebraic surfaces:

Classification: The Kodaira dimension $\kappa(X) \in \{-\infty, 0, 1, 2\}$ determines the coarse type: rational/ruled, K3/abelian/Enriques/bielliptic, elliptic, or general type.

Numerical constraints: Noether's formula $K^2 + c_2 = 12\chi$ and the BMY inequality $K^2 \leq 3c_2$ bound the geography of surfaces.

Birational geometry: The MMP contracts curves with $K \cdot E < 0$ to reach a minimal model where $K$ is nef, and abundance ensures that $|nK|$ eventually defines the Iitaka fibration.

Pluricanonical maps: For general type surfaces, $|5K|$ is birational (Bombieri), and $P_n = \frac{n(n-1)}{2}K^2 + \chi(\mathcal{O})$ for $n \geq 2$ .

Math Notes

Canonical Class $K_X$

Definition of the canonical class

Canonical class of basic surfaces

Adjunction formula and canonical class of hypersurfaces

Kodaira dimension $-\infty$ : rational and ruled surfaces

Kodaira dimension $0$ : surfaces with torsion canonical class

Kodaira dimension $1$ : elliptic fibrations

Kodaira dimension $2$ : surfaces of general type

Abundance for surfaces

Numerical invariants and Noether's formula

Geography of surfaces: the BMY inequality

The canonical class under birational transformations

Invariants derived from $K_X$

Pluricanonical maps and Bombieri's theorem

The canonical class and the minimal model program

Summary

Canonical Class KXK_XKX​