In $\mathbb{P}^2$, let $C = V(F)$ and $D = V(G)$ be curves of degrees $d$ and $e$. If they meet transversally:

$C \cdot D = de.$

This is Bezout's theorem. Concretely:

• A line and a conic ($d = 1, e = 2$): $1 \cdot 2 = 2$ intersection points.
• Two conics ($d = e = 2$): $2 \cdot 2 = 4$ intersection points.
• A line and a cubic ($d = 1, e = 3$): $1 \cdot 3 = 3$ intersection points.
• Two cubics ($d = e = 3$): $3 \cdot 3 = 9$ points — this gives the Cayley–Bacharach theorem.

When curves meet with higher multiplicity, the intersection number accounts for it via local intersection multiplicities:

$C \cdot D = \sum_{P \in C \cap D} (C \cdot D)_P, \quad (C \cdot D)_P = \dim_k \mathcal{O}_{X,P}/(f, g),$

where $f, g$ are local equations for $C, D$ at $P$.

• Two lines meeting at a point: $(C \cdot D)_P = 1$.
• Line $y = 0$ tangent to parabola $y = x^2$ at origin: local equations $y$ and $y - x^2$. The ring $k[x,y]_{(x,y)}/(y, y - x^2) \cong k[x]/(x^2)$ has dimension $2$. So $(C \cdot D)_O = 2$.
• Cusp $y^2 = x^3$ and line $y = 0$ at origin: $k[x,y]_{(x,y)}/(y, y^2 - x^3) \cong k[x]/(x^3)$, dimension $3$.

In $\mathbb{P}^2$, a curve $C$ of degree $d$ has $C^2 = d^2$:

• A line $L$: $L^2 = 1$. Geometrically: perturbing $L$ to another line $L'$, they meet in $1$ point.
• A conic: $C^2 = 4$.
• A cubic: $C^2 = 9$.

In $\mathbb{P}^2$, all self-intersections are positive. This is a special feature of $\mathbb{P}^2$.

On the blowup $\tilde{X} = \operatorname{Bl}_P(\mathbb{P}^2)$, the exceptional divisor $E$ (the fiber over the blown-up point) satisfies:

$E^2 = -1.$

This is the prototype of a negative curve. More generally:

• On a ruled surface $\mathbb{F}_n = \mathbb{P}(\mathcal{O} \oplus \mathcal{O}(-n))$ over $\mathbb{P}^1$: the section $C_0$ with $C_0^2 = -n$.
• On a K3 surface: any smooth rational curve $C \cong \mathbb{P}^1$ has $C^2 = -2$ (by adjunction: $-2 = 2g(C) - 2 = C^2 + K \cdot C = C^2 + 0$).

On $X = \mathbb{P}^1 \times \mathbb{P}^1$, the Picard group is $\operatorname{Pic}(X) \cong \mathbb{Z}^2$, generated by:

• $F_1 = \mathbb{P}^1 \times \{pt\}$ (a "horizontal" fiber),
• $F_2 = \{pt\} \times \mathbb{P}^1$ (a "vertical" fiber).

The intersection pairing is:

• $F_1 \cdot F_2 = 1$ (they meet in one point).
• $F_1^2 = 0$ (parallel horizontal fibers don't meet).
• $F_2^2 = 0$ (parallel vertical fibers don't meet).

A divisor of bidegree $(a, b)$ means $D \sim aF_1 + bF_2$. Then:

• $D^2 = 2ab$ (since $(aF_1 + bF_2)^2 = 2ab \cdot F_1 \cdot F_2 = 2ab$).
• $D \cdot D' = ab' + a'b$ for $D' \sim a'F_1 + b'F_2$.
• A curve of bidegree $(2, 2)$ is an elliptic curve ($g = 1$ by adjunction).

Let $A$ be an abelian surface (a $2$-dimensional abelian variety). If $L$ is an ample line bundle with $L^2 = 2d$, then $L$ defines a polarization of type $(1, d)$.

• Principally polarized ($d = 1$): $L^2 = 2$. The theta divisor $\Theta$ satisfies $\Theta^2 = 2$.
• $(1,2)$-polarization: $L^2 = 4$.
• By the Nakai–Moishezon criterion, $L$ is ample iff $L^2 > 0$ and $L \cdot C > 0$ for all curves $C$.

• $\mathbb{P}^2$: $\rho = 1$. $\operatorname{NS}(\mathbb{P}^2) = \mathbb{Z} \cdot H$, $H^2 = 1$.
• $\mathbb{P}^1 \times \mathbb{P}^1$: $\rho = 2$. Intersection matrix $\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$, signature $(1, 1)$.
• Blowup $\operatorname{Bl}_P(\mathbb{P}^2)$: $\rho = 2$. Basis $\{H, E\}$ with $H^2 = 1$, $E^2 = -1$, $H \cdot E = 0$. Matrix $\operatorname{diag}(1, -1)$, signature $(1,1)$.
• K3 surface: $\rho$ ranges from $1$ to $20$. A general K3 has $\rho = 1$; the maximum $\rho = 20$ is achieved by "singular K3 surfaces" (e.g., the Fermat quartic $x^4 + y^4 + z^4 + w^4 = 0$ over $\overline{\mathbb{F}_p}$ for some $p$).
• Abelian surface: $\rho$ ranges from $1$ to $4$. Over $\mathbb{C}$: $\rho = 1$ generically; $\rho = 4$ for products $E_1 \times E_2$ with CM.

On $\operatorname{Bl}_P(\mathbb{P}^2)$ with basis $H, E$ ($H^2 = 1, E^2 = -1, H \cdot E = 0$):

Is $D = 2H - E$ ample?

• $D^2 = 4 - 1 = 3 > 0$ ✓
• $D \cdot H = 2 > 0$ ✓
• $D \cdot E = 0 - (-1) = 1 > 0$ ✓
• For any curve $C \sim aH - bE$ with $a > 0$: $D \cdot C = 2a + b$. If $C$ is effective and irreducible, then $C \cdot H = a > 0$ and $C^2 = a^2 - b^2 \geq -1$ (for smooth rational curves). One checks $D \cdot C > 0$ for all such $C$.

So $D = 2H - E$ is ample. Geometrically, $|2H - E|$ maps $\operatorname{Bl}_P(\mathbb{P}^2)$ to $\mathbb{P}^2$ via conics through $P$, embedding the blowup as a cubic surface in $\mathbb{P}^3$... actually, $|2H - E|$ is a $g^2_?$; more precisely, $2H - E$ embeds $\operatorname{Bl}_P(\mathbb{P}^2)$ as a degree-$3$ surface in $\mathbb{P}^3$ (a cubic surface is the blowup of $\mathbb{P}^2$ at $6$ points, not $1$ — let me correct: $|3H - E|$ gives a cubic in $\mathbb{P}^3$).

• Degree-$d$ curve in $\mathbb{P}^2$: $K_{\mathbb{P}^2} = -3H$, $C \sim dH$, so $2g - 2 = d^2 - 3d$, giving $g = \frac{(d-1)(d-2)}{2}$. Recovers the genus-degree formula.

• Curve of bidegree $(a,b)$ on $\mathbb{P}^1 \times \mathbb{P}^1$: $K = -2F_1 - 2F_2$, so $2g - 2 = 2ab + (-2b - 2a) = 2ab - 2a - 2b$, giving $g = (a-1)(b-1)$.

• Smooth rational curve on a K3: $K_{K3} = 0$, so $2g - 2 = C^2$. For $g = 0$: $C^2 = -2$. Every $(-2)$-curve on a K3 is a smooth rational curve.

• Exceptional curve on blowup: $E \cong \mathbb{P}^1$, $g = 0$, $K_{\tilde{X}} \cdot E = -1$ (by $K_{\tilde{X}} = \pi^*K_X + E$). So $-2 = E^2 + (-1)$, giving $E^2 = -1$ ✓.

• Canonical curve on a surface of general type: If $C \in |K_X|$ is smooth, then $2g(C) - 2 = K_X^2 + K_X^2 = 2K_X^2$, so $g(C) = K_X^2 + 1$.