Cohomology of singular curves

Question

Suppose $X$ is a singular quasi-projective curve over the complex numbers, and $X'$ is a good nonsingular compactification of a resolution of singularities $Y\to X$. Let $D$ be the complement of $Y$ in $X'$.

Can one compute $H^*(X,\mathbf{Q})$ in terms of $H^*(X',\mathbf{Q})$ and $H^*(D,\mathbf{Q})$ and perhaps of self-products of $X'$ and $D$?

The answer should be "yes" using a combination of spectral sequences. I tried to first compute the hypercovering-to-cohomology spectral sequence for $Y\to X$, and then the weight spectral sequence for $Y\subset X'$ (and this is really just a Gysin long exact sequence), but I couldn't quite work them out.

Answer 1

You don't need all this machinery in this case (unless your goal was to understand the machinery). You have to exact sequences $$0\to W_0\to H^1(X) \to H^1(Y)\to 0$$ $$0 \to H^1(X')\to H^1(Y)\to \oplus_{p\in D} \mathbb{Q}(-1)\to H^2(Y)\to 0$$ The last sequence (which is Gysin) can be shortened (non canonically) to $$0 \to H^1(X')\to H^1(Y)\to \mathbb{Q}(-1)^{|D|-1}\to 0$$ The first sequence is easier to understand by drawing a picture and using homology. Then $W_0$ is dual to the span of loops which get destroyed when the singular points of $X$ are pulled apart in the normalization $Y$.