Hyperplane sections of Calabi-Yau varieties

TL;DR

This paper investigates conditions under which smooth divisors on certain varieties can be hyperplane sections of Calabi-Yau varieties, revealing restrictions based on the variety's properties and degrees.

Contribution

It establishes new criteria for when a smooth divisor can be a hyperplane section of a Calabi-Yau variety, especially relating to the properties of the ambient variety and degree conditions.

Findings

A smooth divisor on a variety with h^1(O_W)=0 cannot be a hyperplane section of a Calabi-Yau unless the variety is Calabi-Yau.

A hypersurface in projective space is a hyperplane section of a Calabi-Yau iff its degree satisfies specific bounds.

The method involves analyzing a universal family of varieties and their singularities to derive these restrictions.

Abstract

Theorem: If W is a smooth complex projective variety with h^1 (O-script_W) = 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 (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 family of all varieties Z for which X is a hyperplane section with normal bundle K_X, and examine the "bad" singularities of such Z. A motivation is to show many curves cannot be divisors on a K-3 surface.