Foundations of Algebraic Geometry

The Foundations of Algebraic Geometry is a book by André Weil that explores algebraic geometry over fields of any characteristic. This work gives a detailed look at intersection theory by explaining how to measure the local intersection multiplicity of two subvarieties.

Weil wrote this book because he needed a strong theory for studying correspondences on algebraic curves in positive characteristic. He used this theory in his important proof of the Riemann hypothesis for curves over finite fields.

In his book, Weil used abstract methods rather than projective varieties so he could build the Jacobian of a curve. At that time, it wasn’t known whether Jacobians are always projective varieties. Later, examples were found of complete abstract varieties that are not projective.

During the 1950s, Weil’s work was one of several efforts to create a solid base for algebraic geometry. Eventually, Grothendieck’s work on schemes provided a new and powerful foundation that went beyond all these earlier attempts.