Multiple Structures
  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). 
Jon Eivind Vatne