On the non-defectivity and non weak-defectivity of Segre--Veronese embeddings of products of projective spaces

Edoardo Ballico

Abstract: Fix integers $s\ge 2$ and $n\ge 1$. Set $\tilde{x_i}:=n+i-1$ if $3\le i\le s$ and $\tilde{x_2}:=\max\{3,n+1\}$. Set $\tilde{x_1}:=9$ if $n=1$ and $\tilde{x_1}=n!(n+1)-n$ if $n\ge 2$. Fix integers $x_i\ge\tilde{x_i}$, $1\le i\le s$. Here we prove that the line bundle $\mathcal{O}_{{\bf{P}}^n\times({\bf{P}}^1)^{s-1}}(x_1,\dots,x_s)$ is not weakly defective, i.e. for every integer $z$ such that $z(n+s)+1\le\binom{n+x_1}{n}\prod_{i=2}^{s}(x_i+1)$ the linear system $\vert\mathcal{I}_Z(x_1,\dots,x_s)\vert$ has dimension $\binom{n+x_1}{n}\prod_{i=2}^{s}(x_i+1)-z(n+s)-1$ and a general $T\in\vert\mathcal{I}_Z(x_1,\dots,x_s)\vert$ has an ordinary double point at each point of $Z_{red}$ as only singularities, where $Z\subset{\bf {P}}^n\times({\bf{P}}^1)^{s-1}$ is a general union of $z$ double points.

Keywords: Segre--Veronese variety; Segre--Veronese embedding; multiprojective space; products of projective spaces; weakly defective variety; zero-dimensional scheme; double point; fat point.

Classification (MSC2000): 14N05.

Full text of the article:

• Compressed PostScript file (188 kilobytes)
• PDF file (188 kilobytes)