Let $f:T^2\to Y$ be a resolution of singularities where $Y$ is a torus with two "pinched" points (or, if you prefer, two copies of $\mathbb{P}^1$ meeting at two points). I'm interested in using the Leray spectral sequence to calculate the cohomology of the constant sheaf on $Y$ . My goal is to better understand spectral sequences and this looks like a nice example to me.

To begin with, $Rf_*\mathbb{Q}_X$ has cohomology sheaves $R^0f_*\mathbb{Q}_X = \mathbb{Q}_Y$ and $R^1f_*\mathbb{Q}_X = \mathcal{S}_{\{a,b\}}$, the skyscraper sheaf with stalk $\mathbb{Q}$ at the pinched points $a,b$.

The $E_2$ page is \begin{align*} \begin{matrix} H^2(Y; \mathbb{Q}_Y) & 0 & 0\\ H^1(Y; \mathbb{Q}_Y) & 0& 0\\ H^0(Y; \mathbb{Q}_Y) & H^0(Y;\mathcal{S}_{\{a,b\}})=\mathbb{Q}^{\oplus 2} & 0 \end{matrix} \end{align*} and the $E_3 (=E_\infty)$ page is \begin{align*} \begin{matrix} H^2(Y; \mathbb{Q}_Y)/ \text{im}(d_2^{0,1}) & 0 &0 \\ H^1(Y; \mathbb{Q}_Y) & 0& 0\\ H^0(Y; \mathbb{Q}_Y) & \text{ker}(d_2^{0,1}) & 0 \end{matrix} \end{align*}

Using the fact that the cohomology $H^n(T^2)$ is filtered by these objects, we find $H^0(T^2) = E_\infty^{0,0} = E_2^{0,0} = H^0(Y)$ and which obviously makes sense, as well as filtrations $$E_3^{0,1}\hookrightarrow H^1(X)$$ where $H^1(X) / \text{ker}(d^{0,1}_2) = E^{1,0}_3 = H^1(Y)$ and $$E_3^{0,2}\hookrightarrow (?) \hookrightarrow H^2(X)$$ where $H^2(X) / (?) = E^{2,0}_3 = H^2(Y)/\text{im}(d^{0,1}_2)$ and $(?)/E^{0,2}_3 = E^{1,1}_3$. Since $E^{1,1}_3 = 0 = E^{0,2}_3$, this implies $H^2(Y)/\text{im}(d^{0,1}_2) = H^2(X)$.

How can I get my hands on the differential $d_2^{0,1}$ in order to finish this calculation?