Intersection Theory and Diophantine Approximation

A keystone in the classical theory of diophantine approximation is the construction of an auxilliary polynomial. The polynomial is constructed so that it is forced (for arithmetic reasons) to vanish at certain approximating points and this contradicts an upper bound on the order of vanishing obtained by other (usually geometric) techniques; the contradiction then allows one to prove finiteness results such as Roth’s Theorem and the Schmidt Subspace Theorem. More recently, this technique has been successfully employed by Faltings and Vojta [F1, F2, V3, V4, V5] to prove the Mordell Conjecture as well as more general finiteness results. Intersection theory often enters these diophantine arguments in the following fashion: since the auxilliary polynomial is constructed in order to have high order of vanishing at the relevant approximating point x, the same holds for all derivatives of suitably small order. Intersecting the zero sets of these derivatives then yields a contradiction in one of two ways. First, on the arithmetic side, the arithmetic Bézout theorem controls the height of the intersection in terms of the height of the auxilliary polynomial; on the other hand, one knows a priori that x is contained in the intersection because the polynomial has large index at x. Hence Bézout gives an upper bound for the height of x. One obtains a contradiction if this bound is smaller than the actual height of x. Second, on the geometric side, the intersection of derivatives will, as a geometric cycle, contain a certain class supported at x. But Bézout’s theorem bounds the total intersection degree from above. Again if this bound is smaller than the actual degree of the class at x a contradiction is obtained. Faltings-Wustholz’ proof of the Schmidt subspace theorem uses the former method while the Dyson Lemma approach to Roth’s Theorem addresses the latter. The goal of this paper is to give an elementary proof of Dyson’s Lemma based entirely on intersection theory. Esnault and Viehweg’s original proof [EV] employs sophistocated cohomological techniques and covering constructions and is very involved. In [N1] I partially simplified the proof of Esnault and Viehweg by using Faltings’ product theorem to replace the intricate weak positivity arguments of [EV]. On the other hand, [N1] still proceeds via the same method, namely first showing that some line bundle is nef and then using cohomological vanishing theorems to derive the theorem. The importance of replacing these