By Chern S., Osserman R. (eds.)

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2] that for any x ∈ P(Fq ) (corresponding to a cubic defined over Fq ), one has Card pr −1 (x) ≡ 1 (mod q). Since hypersurface X ⊂ PFn+1 q −1 pr (x) = F(X), this proves the theorem. Acknowledgements This collaboration started after a mini-course given by the first author for the CIMPA School “II Latin American School of Algebraic Geometry and Applications” given in Cabo Frio, Brazil. We thank CIMPA for financial support and the organizers C. Araujo and S. Druel for making this event successful.

Item (a) is proved (in characteristic = 2) via the theory of Prym varieties ([4, Proposition 7]). 1. To extend (a) to all characteristics, we consider X as the reduction modulo the maximal ideal m of a smooth cubic X defined over a valuation ring of characteristic zero. There is a “difference morphism” δF(X) : F(X) × F(X) → A(F(X)), defined over k, which is the reduction modulo m of the analogous morphism δF(X ) : F(X ) × F(X ) → A F(X ) . By [4, Proposition 5], the image of δF(X ) is a divisor which defines a principal polarization ϑ on A F(X ) , hence the image of δF(X) is also a principal polarization on A(F(X)), defined over k.

In this paper we describe a technology for finding such “good” flat families of perverse sheaves of categories. This is done by deforming LG models as sheaves of categories. The main geometric outcomes of our work are: Classical Categorical W = P equality for tropical varieties “W = P” for perverse sheaves of categories Voisin theory of deformations Good flat deformations of PSC Canonical deformations and compactification HN and additional filtrations of perverse of moduli spaces sheaves of categories We will briefly discuss our procedure.

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