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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.

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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 $Y$.

Donu Arapura
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  • Thank you! This is very helpful. (Yes, I wanted to understand the machinery on the simplest nontrivial case) Is $W_0$ a pure Hodge structure of weight zero, since it is dual to the span of finitely many loops? –  Jul 23 '23 at 17:15
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    Yes, $W_0$ is Hodge structure of weight $0$ – Donu Arapura Jul 24 '23 at 16:07
  • Hi Donu, could you possibly comment on this? https://mathoverflow.net/questions/452509/how-much-of-the-axiom-of-choice-do-you-need-in-mathematics/452648#comment1171148_452648 – Mikhail Katz Aug 15 '23 at 16:15
  • Hi Misha (it's been a while). Are you are asking me to comment on the Bhatt-Mathew paper -- I'm afraid I haven't really looked at it -- or the assertion that ultraproducts appear "frequently" in AG? Concerning the latter I would probably suggest only "occasionally". – Donu Arapura Aug 15 '23 at 20:10
  • I was unable to follow Bhatt-Mathew, but was wondering whether there is a basic case where the use of ultraproducts in AG occurs, that one could understand without mastering all the high-level terminology... – Mikhail Katz Aug 16 '23 at 14:23
  • Algebraic geometers often reduce results about complex algebraic varieties to the corresponding statements over finite fields. This can be done using ultrapoducts, although there are other techniques which are probably used more often. The Ax-Grothendieck theorem is one rather simple and striking example of this. – Donu Arapura Aug 16 '23 at 20:01