# What are the irreducible curves on the blow up of $\mathbb{P}^{2}$?

On the blow-up of $2$-dimensional complex projective space, $\mathbb{P}^2$, I know that $$Pic(\mathbb{\tilde{P}^2})=\mathbb{Z}[\tilde{H}]+\mathbb{Z}[E]$$ where $\tilde{H}$ is the blow-up of the hyperplane bundle of $\mathbb{P}^2$, and $E$ is the exceptional divisor on $\tilde{\mathbb{P}}^2$. Note that $[\cdot]$ denotes numerical equivalence class.

I already know that the set $\{[\tilde{H}], [E]\}$ form a basis for $\tilde{\mathbb{P}}^2$. but how do the irreducible curves on the blow-up look like?

They are either strict transforms of irreducible curves from $\mathbb{P}^2$, or they are the exceptional divisor $E$.
• Maybe I am getting some wires crossed: Is it true that $\{[\tilde{H}], [E]\}$ is a basis for $\tilde{\mathbb{P}}^2$? If so, does this mean that any can be written in the form of $\mathbb{Z}[\tilde{H}]+\mathbb{Z}[E]$? Mar 25, 2015 at 2:01
• This helps a lot. So... if we take an irreducible curve, say $C$, on $\mathbb{P}^2$, and its strict transform, $\tilde{C}$ in $\tilde{\mathbb{P}}^2$...then $\tilde{C}$ can be written in the form of $\mathbb{Z}[\tilde{H}]+\mathbb{Z}[E]?$. What confuses me is that if this is true, then, it appears $\tilde{C}$ won't be irreducible since it can be written in terms of the classes of $\tilde{H}$ and $E$. Mar 25, 2015 at 2:30
• Unless, an irreducible curve $\tilde{C}$ on $\tilde{\mathbb{P}}^2$ is either of the form $\mathbb{Z}[\tilde{H}]+0[E]$ or of the form $0[\tilde{H}]+\mathbb{Z}[E]$....? Mar 25, 2015 at 3:12
• I think you are confusing linear equivalence with irreducible decomposition. An irreducible curve can be linearly equivalent to a combination of $\tilde H$ and $E$. Mar 25, 2015 at 4:41