On quadratic and higher normality of small codimension projective varieties

Details On quadratic and higher normality of small codimension projective varieties On quadratic and higher normality of small codimension projective varieties

2000-02-17

arxiv.org/abs/math/0002140v2

Mathematics

Algebraic Geometry

12 pages

Scientific paper

Ran proved that smooth codimension 2 varieties in ${\bf P}^{m+2}$ are $j$-normal if $(j+1)(3j-1)\le m-1$, in this paper we extend this result to small codimension projective varieties. Let $X$ be a r codimension subvariety of $\pro$, we prove that if the set $\Sigma_{(j+1)}$ of $(j+1)$-secants to $X$ through a generic external point is not empty, $2(r+1)j\leq m-r$ and $(j+1)((r+1)j-1)\leq m-1$ then $X$ is $j$-normal. If $X$ is given by the zero locus of a section of a rank $r$ vector bundle $E$ on $\pro$, we prove that $\textrm{deg} \Sigma_{j+1}=\frac{1}{(j+1)!}\prod_{i=0}^{j}c_{r}(E(-i))$. Moreover we get a new simple proof of Zak's theorem on linear normality if $m\ge 3r$. Finally we prove that if $c_{r}(N(-2))\neq 0$ and $6r\le m-4$ then $X$ is 2-normal.

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