Foundations of Algebraic Geometry is a book by André Weil (1946, 1962) that develops algebraic geometry over fields of any characteristic. In particular it gives a careful treatment of intersection theory by defining the local intersection multiplicity of two subvarieties.
Weil was motivated by the need for a rigorous theory of correspondences on algebraic curves in positive characteristic, which he used in his proof of the Riemann hypothesis for curves over a finite field.
Weil introduced abstract rather than projective varieties partly so that he could construct the Jacobian of a curve. (It was not known at the time that Jacobians are always projective varieties.) It was some time before anyone found any examples of complete abstract varieties that are not projective.
In the 1950s Weil's work was one of several competing attempts to provide satisfactory foundations for algebraic geometry, all of which were superseded by Grothendieck's development of schemes.
References
Raynaud, Michel (1999), "André Weil and the foundations of algebraic geometry" (PDF), Notices of the American Mathematical Society, 46 (8): 864–867, MR 1704257
van der Waerden, Bartel Leendert (1971), "The foundation of algebraic geometry from Severi to André Weil", Archive for History of Exact Sciences, 7 (3): 171–180, doi:10.1007/BF00357215, MR 1554142
Weil, André (1946), Foundations of Algebraic Geometry, American Mathematical Society Colloquium Publications, 29, Providence, R.I.: American Mathematical Society, ISBN 9780821874622, MR 0023093
Weil, André (1962), Foundations of Algebraic Geometry, American Mathematical Society Colloquium Publications, 29 (2 ed.), Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-1029-3, MR 0144898
Zariski, Oscar (1948), "Book Review: Foundations of algebraic geometry", Bulletin of the American Mathematical Society, 54 (7): 671–675, Bibcode:1948Sci...107...75W, doi:10.1090/S0002-9904-1948-09040-1, MR 1565074
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