On the non-defectivity of Segre–Veronese embeddings

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发表日期：

2024年7月27日

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Edoardo Ballico

摘要

We prove a theorem which implies that all Segre–Veronese varieties of multidegree $(d_1,\dots ,d_k)$ $(d_{1}, \dots, d_{k})$ and format $(n_1,\dots ,n_k)$ $(n_{1}, \dots, n_{k})$ with $n_1\ge \cdots \ge n_k>0$ $n_{1} \geq \dots \geq n_{k} > 0$ are not defective if $d_1\ge 3$ $d_{1} \geq 3$ , $d_2\ge 3$ $d_{2} \geq 3$ and $d_i\ge 2$ $d_{i} \geq 2$ for all $i>2$ $i > 2$ . As a particular case we prove the non-defectivity of any Segre–Veronese variety with at least 2 factors and $d_i\ge 3$ $d_{i} \geq 3$ for all i, extending to the case $k>2$ $k > 2$ a theorem of Galuppi and Oneto. Our general result also shows that many Segre–Veronese varieties with 2 factors are not secant defective if they are embedded in bidegree (x, 2), $x\ge 4$ $x \geq 4$ .