Why rational curves?

It is fair to say that perhaps to main object of study in algebraic geometry is a projective variety. Of course algebraic geometry is also about schemes, algebraic spaces, algebraic stacks, commutative algebra, number theory, etc, but really it started with things you can define in projective space or affine space by polynomial equations. What is very interesting is to see the relationship between the geometry of the variety and the arithmetic properties of its solution set.

For projective nonsingular curve C there is a number called the genus g which tells you a lot about the geometry of the curve. If we are over the complex numbers, then

• g = 0 implies that C is topologically a sphere
• g = 1 implies C is topologically a donut with universal covering space the complex plane
• g > 1 implies C is an oriented Riemann surface with g holes and with universal covering space a disc

It turns out that the higher the genus the more complicated the curve gets. Also if you have a nonconstant morphism C —> C' of curves then g >= g'.

A celebrated theorem of Faltings (the Mordell conjecture) says that if you have a curve over a number field, then it has only finitely many K-points.

On the other hand, a genus zero curve over a number field is always P^1 hence always has infinitely many points.

Finally, in the intermediate case, g = 1, there are examples of g = 1 curves over number fields with no K-rational points, but there always exists a finite extension K' of the number field K over which the curve acquires infinitely many K'-rational points.

This picture of splitting curves into the three classes g = 0, g = 1, and g > 1 persists in higher dimensions (except that there are more classes). The lowest of these classes (varieties of negative Kodaira dimension) contains the subclass of rationally connected varieties. A variety X (over the complex numbers for convenience) is called rationally connected if for most pairs of points x, y in X there exists a morphism

P^1 —→ X

which maps 0 to x and ∞ to y. You can try to think of rationall connected varieties as some kind of analogue of path connected topological spaces, although the analogy isn't perfect. 