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By a conical resolution I mean a resolution of singularities $\pi:X\rightarrow Y$ where $X$ and $Y$ are endowed with an action of $\mathbb G_m$ (commuting with $\pi$) such that $Y$ is contracted to a single point by the action (everything is over $\mathbb C$).

Does anybody know a particular example when $H_1(X,\mathbb Z)$ has non-trivial $p$-torsion for some (preferably small) prime $p$? Or may be it is obvious somehow that there is always no torsion?

Thanks!

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    $\begingroup$ Let $V$ be a smooth projective variety with nontrivial $p$-torsion (e.g. a Godeaux surface...). Let $Y$ be an affine cone over it, and $X$ the blow up of the vertex. $\endgroup$
    – Donu Arapura
    Commented Dec 14, 2016 at 17:28
  • $\begingroup$ Oh, that's very simple and works - thanks a lot $\endgroup$
    – user42024
    Commented Dec 14, 2016 at 17:34

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