Multiple Structures

TL;DR

This paper develops a systematic method for constructing complex algebraic schemes with specific properties, revealing deep connections between the schemes' structures and the underlying varieties.

Contribution

It introduces a new framework for building and classifying multiple structures on smooth projective varieties, linking geometric properties to algebraic constructions.

Findings

Reformulation of Hartshorne's conjecture in terms of multiple structures

Construction of examples of non-reduced schemes with particular degrees

Classification results for schemes with special properties like low degree

Abstract

We give a systematic approach to constructing non-reduced, locally Cohen-Macaulay schemes with reduced support a smooth projective variety. The hierarchy of such structures includes a lot of information about the underlying variety, its embeddings in projective space and the behaviour of its vector bundles. For instance, Hartshorne's conjecture on complete intersections in codimension two is reformulated in terms of existence of certain schemes of degrees two and three. There are many examples, and classifications of multiple structures with special properties (like low degree).