A new Castelnuovo bound for two codimensional subvarieties of P

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