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.



On Neron-Severi group of normal projective surfaces and blow up
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:


* <http://www.journalofsing.org/volume4/brevik-nollet.pdf>
* file:///home/alex/Downloads/902-3334-1-PB.pdf


