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Let $X$ be a normal projective surface with at most rational singularites (in finitely many points). Let $\pi:\tilde{X} \to X$ be the blow up of $X$ at finitely many singular points. The question is whether the Picard number of $\tilde{X}$ is at least $1$ more than the Picard number of $X$?

I know that this is true if $X$ is smooth. I have been trying to find a good reference for Neron-Severi group on singular surfaces without much sucess. If some one could suggest a reference for this result, that will be very helpful as well.

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The relative Picard group of a minimal resolution of a Du Val singularity is well understood.

Let $(X,p)$ be a normal surface singularity and $\epsilon:\widetilde{X}\rightarrow X$ be a minimal resolution. Let $Pic(\widetilde{X}/X) = Pic(\widetilde{X})/\epsilon^{*}Pic(X)$ be the relative Picard group.

If $p\in X$ is a canonical singularity (Du Val), and $E$ is the exceptional divisor, then $$Pic(\widetilde{X})\cong \epsilon^*Pic(X)\oplus H^{2}(E,\mathbb{Z}).$$

Furthermore $E = \bigcup_{i=1}^{n}E_i$ is a connected union of $(-2)$-curves intersecting transversally in at most one point. Therefore we have

$$Pic(\widetilde{X}/X)\cong H^{2}(E,\mathbb{Z}) = \bigoplus_{i=1}^n H^{2}(E_i,\mathbb{Z}) = \mathbb{Z}^n.$$

For instance:

  • If $p\in X$ is an $A_n$ singularity $(xy-z^{n+1} = 0)$ then $Pic(\widetilde{X}/X)\cong \mathbb{Z}^n$. Furthermore $Cl(X)\cong\mathbb{Z}/(n+1)\mathbb{Z}$. If $X$ is a quadric cone then the Picard number of $\widetilde{X}$ is two. The Picard group if generated by the strict transform a line through the vertex and the exceptional divisor.

You could take a look to these papers:

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    $\begingroup$ Dear F_L, in the example of canonical singularities you gave, we observe that the rank of $\mathrm{Pic}(\widetilde{X}/X)$ coincides with the number of irreducible components of $E$. Is that always the case (regardless the type of singularities of $X$) ? $\endgroup$
    – Watson
    Commented Sep 23, 2020 at 16:03
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    $\begingroup$ (At least is it true that the Picard number of $\tilde{X}$ is equal to the Picard number of $X$ plus the number of irreducible components of the exceptional divisor $E$ above $p$ ?) $\endgroup$
    – Watson
    Commented Sep 23, 2020 at 16:07

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