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Is it necessary to read SGA VI to understand "Intersection Theory" by William Fulton?

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No, SGA VI it is neither necessary nor sufficient!
Beware that whatever your prerequisites Fulton's book is incredibly difficult .
I would advise you to concentrate on chapter one and to take your time for digesting the rich but concisely explained material there.

A great strategy would be to simultaneously read Eisenbud-Harris's online treatise, which is very user friendly and accompanied by a mind-boggling collection of beautiful geometric illustrations of the theory.
Finally, as a road map, here is Kiritchenko's elementary Introduction to Intersection Theory.

Historical remark
The main impetus for intersection theory was to make Schubert's calculus (a very prophetic theory much in advance to its contemporary algebraic theory techniques) rigorous: this was exactly Hilbert's fifteenth problem.
Intersection theory was at the heart of 20th century algebraic geometry and Weil's notorious Foundations of Algebraic Geometry were essentially devoted to providing rigorous foundations in all characteristics for intersection theory.
Scheme theory, including Fulton's important results, provided a a definitive and satisfactory frame for the modern foundations of intersection theory.
His book "Intersection Theory" is the distillation of more than a century's research in this fundamental and difficult field.
It is thus not surprising that the resulting compact tome is necessarily so concise and hard to read .

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