I would like to know what is the second homology of a del Pezzo surface (X), that is blow-ups of $\mathbb{CP}^2$ in up to 9 points, and to be more precise what are the dimensions of the positive and negative definite subspaces. I.e. for the second Betti number $b_2(X)$ what is $b_+(X)$ and $b_{-}(X)$ such that $$b_2(X) = b_+(X) + b_-(X).$$
In other words I want to know the full information of the intersection form $Q$ of these surfaces.
I have tried to find information online and I was not successful in finding it. I would assume that such information are classified or listed somewhere. Any reference would help.
When you blowup a point on a complex surface, the exceptional divisor is a curve of self-intersection $-1$. Therefore $b_-(\operatorname{Bl}_p(X)) = b_-(X) + 1$. As $\operatorname{Bl}_p(X)$ is diffeomorphic to $X\#\overline{\mathbb{CP}^2}$, we know that $b_2(\operatorname{Bl}_p(X)) = b_2(X) + b_2(\overline{\mathbb{CP}^2}) = b_2(X) + 1$, so we must have $b_+(\operatorname{Bl}_p(X)) = b_+(X)$.
Therefore, if $Y$ is the surface obtained by blowing up $\mathbb{CP}^2$ at $k$ points, then $b_+(Y) = b_+(\mathbb{CP}^2) = 1$ and $b_-(Y) = b_-(\mathbb{CP}^2) + k = k$. It follows that $Y$ has intersection form
$$\begin{bmatrix}1 & 0\\0 & -I_k\end{bmatrix}$$
where $I_k$ is the $k\times k$ identity matrix.