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M BRODMANN

insight into the behaviour of the linear dimension spectrum Idim {3F \ H) for a generic Ьуреф1апе Я. A detailed study of this latter problem will be given in the next section.

Before looking at the (reduced) global subdepths 0{^ \ H) (resp. <5<^>(J^ f Я)) of the restrictions ^ f Я we prove the following result:

( 3 . 10 ) LEMMA. Let у e H, where H ^P"^ is a hyperplane which is general with respect to ^. Then (^ \ H)y ç ^^ja^y, where a € xxxpd^y is a non-zero-divisor with respect to ^^.

Proof . The local vanishing ideal J y с Opd^y of Я at >^ is a proper principal ideal, hence of the form J у = aO^dy, with a 6 Шр^^,. As Я is general with respect to ^, a has to avoid all members of Ass {^y\ and hence is a non-zero divisor with respect to ^y. As (J^ \H)y^(9H^y®^y=-(9pd^ylaOpd^y®^y^^yla^y we get our claim. D

( 3 . 11 ) LEMMA. Let H я P*' be a hyperplane. Then, for the reductions duced in (2.7) we have

( ^ f Я) = (# t Я).

Proof . Immediate from the observation that the kernel of the canonical map ^ \ H-^ß \ H is of finite length hence contained in the torsion subsheaf T(^ \ Я). D

( 3 . 12 ) PROPOSITION. Let ^ Ф0 and let H^P"^ be a hyperplane which is general with respect to ^. Then

( i ) b{^ \Щ^Ь{^)-\.

( ii ) //dim (J^) > 1, then О^У^ \ H) ^ 5<«>(#-) - 1. (iii) // dim (J^^) > r + 1, then b^\^ \ H) ^ ô^'"'^\^) > 0.

Proof (i) The case ô(^) = 0 is obvious. So let о(^) > 0. Choose у e H. Then, by (3.10), depth {{^ \ Я)_,) = depth (J^^) - 1. Making у run through all closed points of Я, we get our claim.

( ii ) Apply (i) to #, thereby observing (3.11), (2.6) and (2.9)(v).

( iii ) Immediate from (i), observing the definition (2.11). D

We close this section by a lemma, which will be used later.

( 3 . 13 ) LEMMA. Let ^ Ф0 and let H^P"^ be a hyperplane which is general with respect to ^. Let x € Ass (J^). Then, any generic point у of {x}n H belongs to Ass (J^ f Я).