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The classic theory of correspondences between smooth algebraic curves can be found in André Weil's Foundations of algebraic geometry. However, this reference works in a pre-modern algebraic geometry way. My question is:

Do you know about a modern reference for the theory of correspondences for smooth complete algebraic curves over a field?

There are modern references for the general theory of correspondences, but they require Serre's Tor-formula and all modern intersection theory machinery. I am looking for something that addresses the specific case of correspondences between smooth algebraic curves, and therefore that does not rely on unnecessary general machinery.

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Griffiths and Harris, Principles of Algebraic Geometry.

See in particular Chapter 2 (Riemann Surfaces and Algebraic Curves), Section 5 (Correspondences).

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