A new Castelnuovo bound for two codimensional subvarieties of 𝑃^{𝑟}

Abstract

Let X X be a smooth n n -dimensional projective subvariety of P r ( C ) , ( r ≥ 3 ) {\mathbb {P}^r}(\mathbb {C}),(r \geq 3) . For any positive integer k , X k,X is said to be k k -normal if the natural map H 0 ( P r , O P r ( k ) ) → H 0 ( X , O X ( k ) ) {H^0}({\mathbb {P}^r},{\mathcal {O}_{\mathbb {P}r}}(k)) \to {H^0}(X,{\mathcal {O}_X}(k)) is surjective. Mumford and Bayer showed that X X is k k -normal if k ≥ ( n + 1 ) ( d − 2 ) + 1 k \geq (n + 1)(d - 2) + 1 where d = deg ⁡ ( X ) d = \deg (X) . Better inequalities are known when n n is small (Gruson-Peskine, Lazarsfeld, Ran). In this paper we consider the case n = r − 2 n = r - 2 , which is related to Hartshorne’s conjecture on complete intersections, and we show that if k ≥ d + 1 + ( 1 / 2 ) r ( r − 1 ) − 2 r k \geq d + 1 + (1/2)r(r - 1) - 2r then X X is k k -normal and I X {I_X} , the ideal sheaf of X X in P r {\mathbb {P}^r} , is ( k + 1 ) (k + 1) -regular. About these problems Lazarsfeld developed a technique based on generic projections of X X in P n + 1 {\mathbb {P}^{n + 1}} ; our proof is an application of some recent results of Ran’s (on the secants of X X ): we show that in our case there exists a projection such generic as Lazarsfeld requires. When r ≥ 6 r \geq 6 we also give a better inequality: k ≥ d − 1 + ( 1 / 2 ) r ( r − 1 ) − ( r − 1 ) [ ( r + 4 ) / 2 ] k \geq d - 1 + (1/2)r(r - 1) - (r - 1)[(r + 4)/2] ([] means: integer part); it is obtained by refining Lazarsfeld’s technique with the help of some results of ours about k k -normality.