Linear systems projections

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The notion of linear systems is fundamental in algebraic geometry as it is one of the main tools used to construct and study rational maps. The basic idea is to construct maps to projective space by using global sections of a line bundle, and note that the most important data is the space spanned by these sections. This is a linear system, though the corresponding projective space which corresponds to a set of effective divisors is sometimes the better way to view it.

In this video, we outline some of the main points of the theory. Describing what these linear systems are, how to think of them via divisors and looking at the all important example of projections away from a linear subvariety. We look at the notion of base points which determines where the rational map of a linear system is defined as well as a criterion for when the resulting map is a closed imbedding. This is explained through a simple geometric example.