MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Suppose I have a (2,4) complete intersection 3-fold $X\subset\mathbb P_{\mathbb C}^5$ with 118 nodal ($A_1$) singularities (if it simplifies things then assume it's the intersection of the Grassmanian $G(2,4)$ with a quartic). It is known (for example see Theorem 4.3 and Corollary 4.4 in Dimca's "Singularities and Topology of Hypersurfaces") that $H_1(X,\mathbb Z)=H_1(\mathbb P^3,\mathbb Z)=0$ and that $H_4(X,\mathbb Z)$ is torsion free.

Does anyone know if taking a crepant resolution $\tilde{X}$ of these 118 nodes could introduce torsion into $H_1$ or $H_4$ of the resolution? Specifically, does it introduce 2-torsion?

share|cite|improve this question
up vote 2 down vote accepted

If I am not mistaken, the mapping $\tilde X\to X$ blows down a disjoint union of 2-dimensional spheres, so it follows from the homology sequence of pair that it induces isomorphisms of both $H_1$'s and $H_4$'s (with integer coefficients).

share|cite|improve this answer
That is certainly what i was thinking just intuitively. – HNuer Feb 21 '13 at 19:26

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.