By S. Lang

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**Additional info for Diophantine geometry**

**Example text**

C/ C The bound, cf. 1, however, on the degree of rational curves connecting any two points in item (a) may be much worse. C/ C replaced by the minimum slope of TF jC , but this is not proved. C/. C/ < r, but well definedness will emerge in the course of the proof. One should also note that there is no need to suppose TF is saturated in TX provided there is closure under Lie bracket. Without closure under bracket one can of course find a foliation G corresponding to the minimal sub-sheaf TG of TX closed under the same.

To this end, and much as before, let x1 ; : : : ; xn be coordinates on M and write, @i D n X jD1 aij @ ; @xj 1 Ä i Ä r; 1ÄjÄn where r is the rank of the foliation. It goes without saying that the @i leave the ideal IX of X in M invariant, and induce our given weakly regular foliations F . x/ around our point of study x. n r/ matrix of meromorphic functions A0 , such that, h : i A D I :: A0 : Now with these notations let us pause to consider the case of r D s and the Frobenius theorem in this context.

KF / which at every point is as above. nKF //. The issue is therefore whether Sn is zero or not. If not this contradicts the ampleness of TF on taking Z to be our curve, so what we’ll show is that if Sn is zero for all n, the singularity is not canonical. It is wholly sufficient to prove this at a singular point, so say notations as above with x the origin. If all the symmetric functions vanish then the matrix Œaij is nilpotent. Linear changes of coordinates conjugate the matrix, so we can suppose: n 1 X @D xi iDk @ Cı @xiC1 where ı 2 m2 TX;x .