The cohomology and intersection form of smooth hypersurfaces in weighted projective spaces are computed explicitly via Alexander duality and a covering by ordinary projective space.
Smoothing Gorenstein toric Fano 3-folds
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abstract
We introduce admissible Minkowski decomposition data (amd) for a 3-dimensional reflexive polytope P. This notion is defined purely in terms of the combinatorics of P. Denoting by X the Gorenstein toric Fano 3-fold whose fan is the spanning fan (a.k.a. face fan) of P, our first result states that amd for P determine a smoothing of X. Our second result amounts to an effective recipe for computing the Betti numbers of the smoothing.
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Cohomology of hypersurfaces of weighted projective space and the intersection form on $H^2$
The cohomology and intersection form of smooth hypersurfaces in weighted projective spaces are computed explicitly via Alexander duality and a covering by ordinary projective space.