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X 0 such that . 0 / TX 0 =k is ample. 10], but our needs are a little different so we spell it out. 9] a smoothing € W Y=T ! Xk0 over some smooth k-curve T with special fibre exists. 1]—applied to € TX 0 =P1 . €t / TX 0 =P1 to the generic k k fibre Yt is ample, while 0 jYt is finite, thus . 1. There is a map S0 W P1S ! X 0 to the smooth locus of X 0 =S such that . S0 / TX 0 =S is ample relative to S. Proof. 8] to 0 to obtain O 0 W P1O ! XO 0 , specialising to 0 , in order to conclude from the existence S of the Hilbert-scheme/S and the fact that being ample is open.

On the other hand, we have the usual exact sequence, _ 0 ! €; Symm N€jF ˝ L˝n / ! FmC1 ; L˝n / ! €; Symk N€jF ˝ L˝n / Ä CnrC1 kD0 where the last inequality may involve a slightly different constant, but nevertheless only depends on L as required. 2. Evidently the role of the analytic topology is only for convenience of exposition, since the above is really a proposition about formal schemes. 2 Cleaning Up We will continue to concentrate on the example of the previous section. 1, we have obtained an algebraic variety W of dimension r C 1, fibred over B by , together with a section s of , such that every fibre of W over B projects to a F -invariant sub-variety of X through the corresponding point of C.

X/ ! C/ the generic rank. We may now state, Main Theorem. C/. Here we use reduction modulo p, but only to resolve the problem for a foliation such that X=F exists as a scheme quotient over C, so in fact, (b) Notations and hypothesis as above then the minimal degree of the rational curve connecting any two points in Vx is effectively computable. In particular there is a rational curve Lx 3 x tangent to F such that for any nef. C/ C The bound, cf. 1, however, on the degree of rational curves connecting any two points in item (a) may be much worse.

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Complex Analysis and Algebraic Geometry by Grauert G. (ed.)

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Categories: Algebraic Geometry