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Consider the intersection of the Segre variety associated to product of $n$ copies of $\mathbb P^2$, with $k$ linearly independent hyperplanes. Is it possible to drop one of the hyperplanes and obtain the same zero-set?

Alternatively, let $S$ be the set of rank $1$ tensors in $\mathbb R^3 \otimes\ldots \otimes \mathbb R^3$ (n times) which are annihilated by $T_1,\ldots, T_k \in ({\mathbb R^3})^*\otimes \ldots\otimes ({\mathbb R^3})^* $ (n times). Is it true that if $T_1, \ldots, T_k$ are linearly independent then no subset of them can define $S$?

Thanks

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