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Cohomology of desingularization of moduli space of vector bundles. (English) Zbl 0702.14007
Let X be a smooth projective non-hyperelliptic curve of genus $$g\geq 3$$ over $${\mathbb{C}}$$, and let $$M_ 0$$ be the moduli space of rank 2 semistable vector bundles on X with trivial determinant. The aim of this paper is to show that $$H^ 3(M'_ 0,{\mathbb{Z}})$$ is a free group of rank $$2g,$$ where $$M'_ 0$$ is the canonical desingularization of $$M_ 0$$ constructed by Narasimhan and Ramanan [cf. M. S. Narasimhan and S. Ramanan, “Geometry of Hecke cycles. I” in C. P. Ramanujam, A tribute, Collect. Publ. of C. P. Ramanujam and Pap. in his Mem., TATA Inst. Fund. Res. Stud. Math. 8, 291-345 (1978; Zbl 0427.14002)].
The motivation of this result comes from the fact that it is known that $$M_ 0$$ is unirational, but it is unknown whether $$M_ 0$$ is rational or not. On the other hand, the torsion subgroup of $$H^ 3(M'_ 0,{\mathbb{Z}})$$ (which is a birational invariant) is of interest to distinguish between the rationality and non-rationality of $$M'_ 0$$.
Reviewer: L.Bădescu

##### MSC:
 14D20 Algebraic moduli problems, moduli of vector bundles 14F45 Topological properties in algebraic geometry 14M20 Rational and unirational varieties 14E15 Global theory and resolution of singularities (algebro-geometric aspects) 14H10 Families, moduli of curves (algebraic) 14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
Zbl 0427.14002
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##### References:
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