# Degree of Varieties and Segre's Embedding

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