On varieties as hyperplane sections

Hyperplane Sections of Calabi-yau Varieties

Theorem. If W is a smooth complex projective variety with h(OW ) = 0, then a sufficiently ample smooth divisor X on W cannot be a hyperplane section of a Calabi-Yau variety, unless W is itself a Calabi-Yau. Corollary. A smooth hypersurface of degree d in P (n ≥ 2) is a hyperplane section of a Calabi-Yau variety iff n + 2 ≤ d ≤ 2n + 2. The method is to construct out of the variety W a universal ...

Reducible Hyperplane Sections I

Hyperplane Sections and Derived Categories

We give a generalization of the theorem of Bondal and Orlov about the derived categories of coherent sheaves on intersections of quadrics revealing its relation to projective duality. As an application we describe the derived categories of coherent sheaves on Fano 3-folds of index 1 and degrees 12, 16 and 18.

Hyperplane Sections of Abelian Surfaces

By a theorem of Wahl, the canonically embedded curves which are hyperplane section of K3 surfaces are distinguished by the non-surjectivity of their Wahl map. In this paper we address the problem of distinguishing hyperplane sections of abelian surfaces. The somewhat surprising result is that the Wahl map of such curves is (tendentially) surjective, but their second Wahl map has corank at least...

Hessenberg Varieties and Hyperplane Arrangements

Given a semisimple complex linear algebraic group G and a lower ideal I in positive roots of G, three objects arise: the ideal arrangement AI , the regular nilpotent Hessenberg variety Hess(N, I), and the regular semisimple Hessenberg variety Hess(S, I). We show that a certain graded ring derived from the logarithmic derivation module of AI is isomorphic to H∗(Hess(N, I)) and H∗(Hess(S, I)) , t...