C-OVO'S
regularity was first defined by Munr~oan [ 141, who attributes the idea to CABTEEXUOVO [4], for coherent sheaves on projective spaces. This regularity particularly provides upper bounds for the degrees of the syzygies of free resolutions of homogeneous ideals of a polynomial ring (see Section 2). Following PETRI'S approach I171 to the etudy of the idml associated to a special divieor one has found beautiful results on equations defining arithmetically CO~FMAOAULAY subvarieties of P" (see, for example, [Xi}, [B], [If 1251 or 1121). The aim of this paper is to describe an extension of the upper bonnds for the degree of these defining equations to the minimal ayzy@;ies. Our Theorem 1,2 and 3 gim the bounds for this CASTBLNUOVO regularity for Iocally COZEX-MAC~LAY aubschernes of I ? . We will s b t e our bounds in terms of degree, codimension and of the length and annihilators of local cohomology. The firat example e f &tion 8 sheds some light on the problem to extend upper bounds for the degree of defining equations to C m o v o ' s regularity.
Moreover, we will prove speeirrl case^ of a conof EISESBUDGOTO [5], p. 93 and of a conjecture of 3~ylF.E-[2}, p. 3 (see our proposition in Section 6). We
will state a consequence of this propition.
Corollary. Let X be
a pre4-l r d w d subdmne of c. Let I ( X ) be the defining ided of S = Qx,,, . . ., 2 . 3 w h e E ia an d g e h M y closed field of arbitray ckra.cte&tic. Let A be the lwmaogeneous k+ebra rS/I(X) with RRm-dim A =: d 2 2. Let P be the i r r e h n t ideal of A . Assunte firad one of the f M n g d & m is satisfied: (a) I # d = 3 &en P-@,(B) = 0 (b) A is BUCESBAUM T h n we h v e the folbwhg CAST~LNVOVO bun& (see 0217 notdm in Scdiun 2) : (ii) If X is i dand non-dqenede theta reg (I(x)) 5 degree (X)codim (X) + 1.
(c) d e . p h A Z d -1 6) reg fm) I w e e (X) In Section 6 we conclude by studying some examples and add some remarks. 20*