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

Abstract:

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 ${H^0}({\mathbb {P}^r},{\mathcal {O}_{\mathbb {P}r}}(k)) \to {H^0}(X,{\mathcal {O}_X}(k))$ 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.