Foundational Issues
Qualification of what was actually proved is necessary because of the foundational difficulties. These included intensive use of birational models in dimension 3 of surfaces that can have non-singular models only when embedded in higher-dimensional projective space. That is, the theory wasn't posed in an intrinsic way. To get around that, a sophisticated theory of handling a linear system of divisors was developed (in effect, a line bundle theory for hyperplane sections of putative embeddings in projective space). Many of the modern techniques were found, in embryonic form, and in some cases the articulation of these ideas exceeded the available technical language.
Read more about this topic: Italian School Of Algebraic Geometry
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