Notes on varieties of codimension 3 in \(\mathbb{P}^ N\) (Q1340667)

In this article the author describes several methods for the construction of smooth subvarieties of codimension 3, in projective spaces or other ambient spaces. These methods include liaison (thus generalizing a typical two-codimensional approach) of 3-folds in \(\mathbb{P}^6\), sections of reflexive sheaves and Pfaffians of twisted skew-symmetric vector bundle morphisms, where a Bertini-type argument gives smoothness if the dimension of the ambient space is \(< 10\). This kind of construction is new; for more results about Pfaffians and how far this construction can actually be generalized, see also the recent works by \textit{Charles Walter}. -- As applications, new families of three-codimensional subvarieties are given in \(\mathbb{P}^6\), \(\mathbb{P}^8\) and \(\mathbb{P}^9\).