A Higher Dimensional Mordell Conjecture

Faltings’ long awaited proof of the Mordell conjecture completes, roughly speaking, the question of whether a given curve has only finitely many integral or rational points. Indeed, if a complete curve has genus g ≥ 2, then it has finitely many rational points; any affine curve whose projective closure is a curve of genus at least two will, a fortiori, have only finitely many integral points. A curve of genus 1 is an elliptic curve; it will have infinitely many rational points over a sufficiently large ground field, but no affine subvariety has an infinite number of integral points. Finally, a curve of genus zero is, after a base change, the projective line, which has an infinite number of rational points; affine sub-varieties omitting at most two points will have infinitely many integral points over a sufficiently large ring; but affine sub-varieties omitting at least three points will have only finitely many integral points. Thus the answer to the finiteness question is given entirely by the structure of the curve over the complex numbers.