Results in Mathematics

, 74:115 | Cite as

On a Resolution of Singularities with Two Strata

  • Vincenzo Di GennaroEmail author
  • Davide Franco


Let X be a complex, irreducible, quasi-projective variety, and \(\pi :{\widetilde{X}}\rightarrow X\) a resolution of singularities of X. Assume that the singular locus \({\text {Sing}}(X)\) of X is smooth, that the induced map \(\pi ^{-1}({\text {Sing}}(X))\rightarrow {\text {Sing}}(X)\) is a smooth fibration admitting a cohomology extension of the fiber, and that \(\pi ^{-1}({\text {Sing}}(X))\) has a negative normal bundle in \({\widetilde{X}}\). We present a very short and explicit proof of the Decomposition Theorem for \(\pi \), providing a way to compute the intersection cohomology of X by means of the cohomology of \({\widetilde{X}}\) and of \(\pi ^{-1}({\text {Sing}}(X))\). Our result applies to special Schubert varieties with two strata, even if \(\pi \) is non-small. And to certain hypersurfaces of \({\mathbb {P}}^5\) with one-dimensional singular locus.


Projective variety smooth fibration resolution of singularities derived category intersection cohomology Decomposition Theorem Poincaré polynomial Betti numbers Schubert varieties 

Mathematics Subject Classification

Primary 14B05 Secondary 14E15 14F05 14F43 14F45 14M15 32S20 32S60 58K15 



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Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità di Roma “Tor Vergata”RomeItaly
  2. 2.Dipartimento di Matematica e Applicazioni “R. Caccioppoli”Università di Napoli “Federico II”NapoliItaly

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