By Fedor Bogomolov, Brendan Hassett, Yuri Tschinkel

Based at the Simons Symposia held in 2015, the complaints during this quantity specialize in rational curves on higher-dimensional algebraic types and purposes of the speculation of curves to mathematics difficulties. there was major growth during this box with significant new effects, that have given new impetus to the research of rational curves and areas of rational curves on K3 surfaces and their higher-dimensional generalizations. One major fresh perception the ebook covers is the concept that the geometry of rational curves is tightly coupled to houses of derived different types of sheaves on K3 surfaces. The implementation of this concept resulted in proofs of long-standing conjectures referring to birational houses of holomorphic symplectic forms, which in flip may still yield new theorems in mathematics. This lawsuits quantity covers those new insights intimately.

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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.