For projective space $\mathbb{P}^n_k$, the canonical sheaf is: $\omega_{\mathbb{P}^n} = \mathscr{O}_{\mathbb{P}^n}(-n-1)$

Computation: Cover $\mathbb{P}^n$ by affine charts $U_i = \mathrm{Spec}(k[x_0/x_i, \ldots, \widehat{x_i/x_i}, \ldots, x_n/x_i])$. On $U_0$, set $t_j = x_j/x_0$ for $j = 1, \ldots, n$. Then: $\Omega_{U_0/k} = \bigoplus_{j=1}^n \mathscr{O}_{U_0} \cdot dt_j$

The top form is: $\omega|_{U_0} = \mathscr{O}_{U_0} \cdot dt_1 \wedge \cdots \wedge dt_n$

On the overlap $U_0 \cap U_1$ with coordinates $s_j = x_j/x_1$, we have $t_1 = 1/s_0$ and $t_j = s_j/s_0$ for $j \geq 2$. The Jacobian gives: $dt_1 \wedge \cdots \wedge dt_n = (-1)^{n-1} \cdot s_0^{-(n+1)} \cdot ds_1 \wedge \cdots \wedge ds_n$

Since $s_0 = x_0/x_1$ has degree $0$, the factor $s_0^{-(n+1)}$ corresponds to twisting by $\mathscr{O}(-n-1)$.

Consequence: The canonical divisor $K_{\mathbb{P}^n} = (-n-1)H$ where $H$ is a hyperplane.

Let $C$ be a smooth projective curve of genus $g$ over $k$. Then: $\omega_C = \Omega_{C/k}$ is a line bundle of degree $2g - 2$.

For $\mathbb{P}^1$: We have $\omega_{\mathbb{P}^1} = \mathscr{O}_{\mathbb{P}^1}(-2)$ and $\deg(K_{\mathbb{P}^1}) = -2 = 2(0) - 2$.

For an elliptic curve $E$: We have $\omega_E \cong \mathscr{O}_E$ and $\deg(K_E) = 0 = 2(1) - 2$.

For a curve of genus $g \geq 2$: The canonical divisor $K_C$ is ample when $g \geq 2$, giving the canonical embedding: $\phi_K: C \to \mathbb{P}^{g-1}, \quad p \mapsto [s_0(p) : \cdots : s_{g-1}(p)]$ where $s_0, \ldots, s_{g-1}$ is a basis of $H^0(C, \omega_C)$.

Let $X \subseteq \mathbb{P}^{n+1}$ be a smooth hypersurface of degree $d$. By adjunction: $\omega_X = (\omega_{\mathbb{P}^{n+1}} \otimes \mathscr{O}_{\mathbb{P}^{n+1}}(d))|_X = \mathscr{O}_{\mathbb{P}^{n+1}}(-n-2+d)|_X = \mathscr{O}_X(d-n-2)$

Special cases:

• Plane curve ($n = 1$): $\omega_C = \mathscr{O}_C(d-3)$
  • For $d = 3$ (cubic): $\omega_C \cong \mathscr{O}_C$ (elliptic curve)
  • For $d = 4$ (quartic): $\omega_C \cong \mathscr{O}_C(1)$ (genus 3)
• Surface in $\mathbb{P}^3$ ($n = 2$): $\omega_S = \mathscr{O}_S(d-4)$
  • For $d = 4$ (quartic): $\omega_S \cong \mathscr{O}_S$ (K3 surface)
  • For $d = 5$ (quintic): $\omega_S \cong \mathscr{O}_S(1)$
• Threefold in $\mathbb{P}^4$ ($n = 3$): $\omega_X = \mathscr{O}_X(d-5)$
  • For $d = 5$ (quintic): $\omega_X \cong \mathscr{O}_X$ (Calabi-Yau threefold)

Let $X \subseteq \mathbb{P}^n$ be a smooth complete intersection of hypersurfaces of degrees $d_1, \ldots, d_r$. Then $\dim(X) = n - r$ and: $\omega_X = \mathscr{O}_X\left(\sum_{i=1}^r d_i - n - 1\right)$

Derivation: Apply adjunction repeatedly. If $X = H_1 \cap \cdots \cap H_r$ where $H_i$ has degree $d_i$: $K_X = (K_{\mathbb{P}^n} + d_1 + \cdots + d_r)|_X = (-n-1 + d_1 + \cdots + d_r)|_X$

Examples:

• Curve in $\mathbb{P}^3$ as $(d_1, d_2)$ complete intersection: $\omega_C = \mathscr{O}_C(d_1 + d_2 - 4)$ The genus is $g = \frac{1}{2}d_1 d_2 (d_1 + d_2 - 4) + 1$.

• Surface in $\mathbb{P}^4$ as $(d_1, d_2)$ complete intersection: $\omega_S = \mathscr{O}_S(d_1 + d_2 - 5)$

• Calabi-Yau threefold: $(3, 3)$ complete intersection in $\mathbb{P}^5$ has $\omega_X \cong \mathscr{O}_X$.

For a smooth projective curve $C$ of genus $g$: $H^0(C, \mathscr{L}) \cong H^1(C, \mathscr{L}^\vee \otimes \omega_C)^\vee$ $H^1(C, \mathscr{L}) \cong H^0(C, \mathscr{L}^\vee \otimes \omega_C)^\vee$

Application: For $\mathscr{L} = \mathscr{O}_C$: $H^0(C, \mathscr{O}_C) \cong H^1(C, \omega_C)^\vee$ $H^1(C, \mathscr{O}_C) \cong H^0(C, \omega_C)^\vee$

Since $h^0(C, \mathscr{O}_C) = 1$ and $h^1(C, \mathscr{O}_C) = g$, we get $h^0(C, \omega_C) = g$ and $h^1(C, \omega_C) = 1$.

By Riemann-Roch: $\deg(\omega_C) = 2g - 2$.

Let $X \subseteq \mathbb{P}^{n+1}$ be a hypersurface of degree $d$ (possibly singular). Then $X$ is Gorenstein and: $\omega_X = \mathscr{O}_X(d - n - 2)$

This formula holds even when $X$ is singular, though we cannot derive it from the adjunction formula in the same way.

Example: The cubic surface $X \subseteq \mathbb{P}^3$ defined by $x_0^3 + x_1^3 + x_2^3 + x_3^3 = 0$ has at most isolated singularities and: $\omega_X = \mathscr{O}_X(3 - 4) = \mathscr{O}_X(-1)$

Note that $\omega_X$ is still invertible despite potential singularities.

For a normal variety $X$, there is a notion of canonical singularities and log-canonical singularities defined using discrepancies.

Let $\pi: \widetilde{X} \to X$ be a resolution of singularities with $\widetilde{X}$ smooth. Write: $K_{\widetilde{X}} = \pi^* K_X + \sum a_i E_i$ where $E_i$ are exceptional divisors and $a_i \in \mathbb{Q}$ are discrepancies.

• $X$ has canonical singularities if $a_i \geq 0$ for all $i$
• $X$ has terminal singularities if $a_i > 0$ for all $i$
• $X$ has log-canonical singularities if $a_i \geq -1$ for all $i$

Examples:

• Smooth points are terminal
• Nodes $(xy = z^2)$ are canonical
• Cusps $(x^2 = y^3)$ are log-canonical but not canonical

For a smooth projective curve $C$ of genus $g$:

• $g = 0$ ($C \cong \mathbb{P}^1$): $\omega_C = \mathscr{O}_{\mathbb{P}^1}(-2)$ is anti-ample, so $P_m(C) = 0$ for all $m > 0$. Thus $\kappa(C) = -\infty$.

• $g = 1$ (elliptic curve): $\omega_C \cong \mathscr{O}_C$ is trivial, so $P_m(C) = 1$ for all $m > 0$. The maps $\phi_{|mK_C|}$ are all empty. Thus $\kappa(C) = 0$.

• $g \geq 2$: $\omega_C$ is ample with $\deg(\omega_C) = 2g - 2 > 0$. For $m \geq 2$: $P_m(C) = h^0(C, \omega_C^{\otimes m}) = (2m-1)(g-1)$ by Riemann-Roch. The canonical map $\phi_{|K_C|}: C \to \mathbb{P}^{g-1}$ is an embedding (except for hyperelliptic curves where it's a degree-2 cover). Thus $\kappa(C) = 1$.

For a smooth projective surface $S$:

$\kappa(S) = -\infty$: Ruled surfaces, rational surfaces ($\mathbb{P}^2$, $\mathbb{P}^1 \times \mathbb{P}^1$, del Pezzo surfaces)

$\kappa(S) = 0$: Surfaces with numerically trivial canonical class

• K3 surfaces: $\omega_S \cong \mathscr{O}_S$, $P_m(S) = 1$ for all $m > 0$
• Abelian surfaces
• Enriques surfaces (with $2K_S \sim 0$)
• Hyperelliptic surfaces

$\kappa(S) = 1$: Properly elliptic surfaces (elliptic fibrations over curves with $K_S$ not nef)

$\kappa(S) = 2$: Surfaces of general type. The canonical ring: $R(S) = \bigoplus_{m \geq 0} H^0(S, \omega_S^{\otimes m})$ is finitely generated and $\mathrm{Proj}(R(S))$ gives the canonical model.

A K3 surface is a smooth projective surface $S$ with: $\omega_S \cong \mathscr{O}_S \quad \text{and} \quad H^1(S, \mathscr{O}_S) = 0$

Properties:

• $\kappa(S) = 0$
• $K_S = 0$ (canonical divisor is numerically trivial)
• $h^{2,0}(S) = 1$, $h^{1,0}(S) = 0$
• $\chi(\mathscr{O}_S) = 2$

Examples:

1. Smooth quartic surfaces in $\mathbb{P}^3$: By adjunction, $\omega_S = \mathscr{O}_S(4-4) = \mathscr{O}_S$.
2. Complete intersection of a quadric and cubic in $\mathbb{P}^4$: $\omega_S = \mathscr{O}_S(2+3-5) = \mathscr{O}_S$.
3. Double cover of $\mathbb{P}^2$ branched over a smooth sextic curve.
4. Kummer surface: Quotient of an abelian surface by involution, then blow up 16 singular points.

A Calabi-Yau threefold is a smooth projective threefold $X$ with: $\omega_X \cong \mathscr{O}_X \quad \text{and} \quad H^i(X, \mathscr{O}_X) = 0 \text{ for } i = 1, 2$

Properties:

• $\kappa(X) = 0$
• $K_X = 0$
• $\chi(\mathscr{O}_X) = 0$ (from Serre duality)

Examples:

1. Smooth quintic threefold in $\mathbb{P}^4$: By adjunction, $\omega_X = \mathscr{O}_X(5-5) = \mathscr{O}_X$.
2. Complete intersection $(3,3)$ in $\mathbb{P}^5$: $\omega_X = \mathscr{O}_X(3+3-6) = \mathscr{O}_X$.
3. Complete intersection $(2,4)$ in $\mathbb{P}^5$: $\omega_X = \mathscr{O}_X(2+4-6) = \mathscr{O}_X$.
4. Complete intersection $(2,2,3)$ in $\mathbb{P}^6$: $\omega_X = \mathscr{O}_X(2+2+3-7) = \mathscr{O}_X$.

The quintic threefold has Hodge numbers $h^{1,1}(X) = 1$ and $h^{2,1}(X) = 101$, leading to a moduli space of dimension 101.

For a smooth projective curve $C$ of genus $g \geq 2$:

The canonical ring is: $R(C) = \bigoplus_{m \geq 0} H^0(C, \omega_C^{\otimes m})$

Structure:

• $R(C)_0 = k$
• $R(C)_1 = H^0(C, \omega_C)$ has dimension $g$
• For $m \geq 2$: $\dim R(C)_m = (2m-1)(g-1)$ by Riemann-Roch

Generation: $R(C)$ is generated in degrees 1 and 2. More precisely:

• If $C$ is non-hyperelliptic: $R(C)$ is generated in degree 1, i.e., by $H^0(C, \omega_C)$
• If $C$ is hyperelliptic: $R(C)$ requires generators in degree 2

Canonical embedding: The canonical map: $\phi_K: C \to \mathbb{P}(H^0(C, \omega_C)^\vee) \cong \mathbb{P}^{g-1}$ embeds $C$ as a curve of degree $2g-2$ (unless $C$ is hyperelliptic, in which case it's a 2:1 cover of $\mathbb{P}^1$).

Let $S$ be a smooth projective surface of general type (i.e., $\kappa(S) = 2$).

Finite generation: The canonical ring $R(S) = \bigoplus_{m \geq 0} H^0(S, \omega_S^{\otimes m})$ is finitely generated.

Canonical model: The canonical model is: $S_{\text{can}} = \mathrm{Proj}(R(S))$ This is a normal projective variety with at worst canonical singularities. If $K_S$ is nef and big, then the canonical map: $\phi_K: S \to S_{\text{can}}$ contracts all $(-2)$-curves on $S$.

Example - Surface of degree $d \geq 5$ in $\mathbb{P}^3$: For a smooth surface $S \subseteq \mathbb{P}^3$ of degree $d \geq 5$:

• $\omega_S = \mathscr{O}_S(d-4)$ is ample
• $P_m(S) = h^0(S, \omega_S^{\otimes m}) = h^0(\mathbb{P}^3, \mathscr{O}(m(d-4)))$ for large $m$
• The canonical ring is generated by sections of $\omega_S$ and $\omega_S^{\otimes 2}$

Let $f: C \to D$ be a finite morphism of smooth projective curves of degrees $d$. Write: $K_C = f^* K_D + R$ where $R$ is the ramification divisor.

Formula: If $f$ has ramification index $e_p$ at $p \in C$ (meaning $f$ is locally $t \mapsto t^{e_p}$), then: $R = \sum_{p \in C} (e_p - 1) \cdot p$

Riemann-Hurwitz formula: Taking degrees: $2g_C - 2 = d(2g_D - 2) + \deg(R)$

Example - Double cover: If $f: C \to \mathbb{P}^1$ is a double cover branched over $2r$ points, then $d = 2$ and $\deg(R) = 2r$. Thus: $2g_C - 2 = 2(-2) + 2r \implies g_C = r - 1$

For $r = 4$: $g_C = 3$ (hyperelliptic curve of genus 3).

Let $X$ and $Y$ be smooth projective varieties with projections $p_X: X \times Y \to X$ and $p_Y: X \times Y \to Y$. Then: $\omega_{X \times Y} = p_X^* \omega_X \otimes p_Y^* \omega_Y$

Proof: Since $\Omega_{X \times Y/k} = p_X^* \Omega_{X/k} \oplus p_Y^* \Omega_{Y/k}$, taking top exterior power: $\omega_{X \times Y} = \bigwedge^{n+m} \Omega_{X \times Y/k} = p_X^* \omega_X \otimes p_Y^* \omega_Y$

Example: For $\mathbb{P}^n \times \mathbb{P}^m$: $\omega_{\mathbb{P}^n \times \mathbb{P}^m} = \mathscr{O}(-n-1, -m-1)$

Let $\pi: \widetilde{X} \to X$ be the blow-up of a smooth variety $X$ at a smooth subvariety $Y$ of codimension $r$, with exceptional divisor $E \cong \mathbb{P}(\mathscr{N}_{Y/X})$. Then: $K_{\widetilde{X}} = \pi^* K_X + (r-1) E$

Proof: The conormal exact sequence gives: $0 \to \mathscr{O}_E(-E) \to \Omega_{\widetilde{X}/k}|_E \to \Omega_{E/k} \to 0$

Computing determinants and using adjunction yields the formula.

Example - Blow-up at a point: If $X$ is a smooth surface and $\pi: \widetilde{X} \to X$ is the blow-up at a point $p$ with exceptional curve $E \cong \mathbb{P}^1$: $K_{\widetilde{X}} = \pi^* K_X + E$

Since $E^2 = -1$, we have: $K_{\widetilde{X}} \cdot E = (\pi^* K_X + E) \cdot E = 0 + (-1) = -1$

Let $X$ be a smooth complete toric variety corresponding to a fan $\Sigma$ in $N_\mathbb{R}$ where $N \cong \mathbb{Z}^n$. For each ray $\rho \in \Sigma(1)$ with primitive generator $v_\rho \in N$, let $D_\rho$ be the corresponding toric divisor.

The canonical divisor is: $K_X = -\sum_{\rho \in \Sigma(1)} D_\rho$

Example - $\mathbb{P}^n$: The fan has $n+1$ rays generating the standard basis plus $-(e_1 + \cdots + e_n)$. Thus: $K_{\mathbb{P}^n} = -(D_0 + \cdots + D_n) = -(n+1)H$ where $H$ is the hyperplane class, confirming $\omega_{\mathbb{P}^n} = \mathscr{O}_{\mathbb{P}^n}(-n-1)$.

Example - Hirzebruch surface $\mathbb{F}_a$: This is a $\mathbb{P}^1$-bundle over $\mathbb{P}^1$. The canonical divisor is: $K_{\mathbb{F}_a} = -2C_0 - (a+2)f$ where $C_0$ is the zero section and $f$ is the fiber class.

Let $A$ be an abelian variety of dimension $g$ over $k$. Then: $\omega_A \cong \mathscr{O}_A$

Proof: Since $A$ is a group variety, for any point $a \in A$ the translation map $t_a: A \to A$ is an automorphism. Thus: $t_a^* \omega_A \cong \omega_A$

But $t_a^*$ acts as identity on $H^0(A, \omega_A)$, and the space of translation-invariant sections of any line bundle on $A$ is at most 1-dimensional. Since $h^0(A, \omega_A) = 1$ (by Serre duality and $h^g(A, \mathscr{O}_A) = 1$), we have $\omega_A \cong \mathscr{O}_A$.

Consequence: Abelian varieties have Kodaira dimension 0, and all plurigenera are $P_m(A) = 1$ for $m > 0$.

Smooth projective surfaces are classified by Kodaira dimension:

$\kappa = -\infty$:

• Rational surfaces ($\mathbb{P}^2$, rational ruled surfaces)
• Ruled surfaces over curves of genus $g \geq 1$

$\kappa = 0$:

• K3 surfaces ($\omega_S \cong \mathscr{O}_S$, $q = 0$)
• Abelian surfaces
• Enriques surfaces ($2K_S \sim 0$, $q = 0$)
• Hyperelliptic surfaces (quotients of abelian surfaces)

$\kappa = 1$:

• Properly elliptic surfaces (elliptic fibrations with $\chi(\mathscr{O}_S) = 0$)

$\kappa = 2$:

• Surfaces of general type (canonical model exists)

The minimal models in each class are characterized by $K_S$ being nef.