Let $X$ be an irreducible complex projective variety of dimension $p$ and consider two projective embeddings $\iota\colon X\subseteq \mathbb{P}^n$ and $\iota'\colon X\subseteq \mathbb{P}^{n'}$. Denote by $\Sigma\colon \mathbb{P}^n\times\mathbb{P}^{n'}\longrightarrow \mathbb{P}^N$ the Segre embedding. Then we get another embedding $\iota''\colon X\subseteq\mathbb{P}^N$ given by $$x\longmapsto (\iota(x),\iota'(x))\longmapsto\Sigma\big((\iota(x),\iota'(x)).$$ Assume that that $\iota''\colon X\subseteq\mathbb P^N$ has degree $d$. Then there exist $(p-1)$ linear functionals $\theta''_1,\ldots,\theta''_{p-1}$ of $\mathbb{C}^{N+1}$ such that the cardinality of $$\lbrace \theta''_1=\cdots=\theta''_{p-1}=0\rbrace\cap \iota''(X)$$ is equal to $d$. I would like to check that the functional $\theta_i''$ above can be taken of the form $$\theta_i''=\theta_i\otimes\theta_i',$$ where $\theta_i$ and $\theta_i$' are linear functionals of $\mathbb{C}^{n+1}$ and $\mathbb{C}^{n'+1}$, respectively.

This is a step (not proved) of Hoyt's proof of the topological invariance of the Chow monoid.

Any help is well accepted. Thanks in advance.

cross-posted: MSE


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