6 - The Big Three: C, K, S

Summary

There are three central players in our subject. Although to the unsuspecting they may appear quite different, the unreasonable truth is that they're one and the same, each in different clothing. Without all the proper definitions just yet, they are

• C, an irreducible curve in ℙ²(ℂ);

• K, a field of transcendence degree 1 over ℂ;

• S, a compact Riemann surface which, for the moment, can be thought of as a nonsingular curve in ℙ³(ℂ).

Each of these three has a notion of equivalence, and there are equivalences from any one to any other.

Uniting the apparently dissimilar is nothing new to science. Uncovering unsuspected relationships is a hallmark of scientific progress. Examples:

• Descartes discovered the connection between Euclidean geometry and algebra, two huge branches of mathematics that for many centuries had led mostly separate lives. His coordinate system allowed us to translate between geometry and much of algebra. This relation eventually expanded to algebraic geometry, of which algebraic curves is a part.

• Before Newton, there was on the one hand “terrestrial physics” and on the other, “celestial physics.” His force laws and Universal Law of Gravitation united them into one physics.

• Darwin uncovered the kinship between various forms of life, and in modern times this kinship has been extended to show DNA overlap between virtually any two forms of life—a broad and enlightening unity.

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