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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?

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They are either strict transforms of irreducible curves from $\mathbb{P}^2$, or they are the exceptional divisor $E$.

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  • $\begingroup$ 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 $\mathbb{Z}[\tilde{H}]+\mathbb{Z}[E]$? $\endgroup$
    – Alvey
    Mar 25, 2015 at 2:01
  • $\begingroup$ You mean a basis for the Picard group? Yes, that is correct. Every divisor is linearly equivalent to some combination of the pullback of a line, and the exceptional divisor. $\endgroup$ Mar 25, 2015 at 2:16
  • $\begingroup$ 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 $E$. $\endgroup$
    – Alvey
    Mar 25, 2015 at 2:30
  • $\begingroup$ 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 $0[\tilde{H}]+\mathbb{Z}[E]$....? $\endgroup$
    – Alvey
    Mar 25, 2015 at 3:12
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    $\begingroup$ I think you are confusing linear equivalence with irreducible decomposition. An irreducible curve can be linearly equivalent to a combination of $\tilde H$ and $E$. $\endgroup$ Mar 25, 2015 at 4:41

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