The geometry of complete linear maps (Q1823287)

A complete linear map is a point of a certain compactification of the open subset \(L(E,F)'\) of the space L(E,F) of linear maps of vector spaces consisting of maps of maximal rank. Special cases are complete quadrics, complete collineations and correlations. The intersection theory on the parameter spaces of complete linear maps is used for justifications of some results in the enumerative geometry of the last century. The author gives a scheme-theoretical foundation of this theory. He generalizes an approach of J. Semple and J. Tyrrell in which a parameter space is obtained as the closure of the map \({\mathbb{P}}(L(E,F)')\to \prod_{i}(L(\bigwedge^ iE,\bigwedge^ iF)) \), \(\alpha \to (\alpha,\bigwedge^ 2\alpha,...)\). Instead of linear maps of vector spaces, the author considers a map \(E\to F\otimes L\), where E and F are vector bundles, and L is a line bundle over a scheme S.