Dimension of the secant variety $\sigma_2 (\mathbb P^1\times\mathbb P^1\times\mathbb P^1\times\mathbb P^1)$

Question

I am trying to compute the dimension of the secant variety $\sigma_2 (\mathbb P^1\times\mathbb P^1\times\mathbb P^1\times\mathbb P^1)$, for this I want to use the Terracini Lemma, that says that $$ T_p\sigma_2 (\mathbb P^1\times\mathbb P^1\times\mathbb P^1\times\mathbb P^1)= \langle T_x(\mathbb P^1\times\mathbb P^1\times\mathbb P^1\times\mathbb P^1),T_y(\mathbb P^1\times\mathbb P^1\times\mathbb P^1\times\mathbb P^1)~\rangle. $$ The expected dimension is $9$.

What I tried to do was to utilize the Segre embedding $$\sigma:\mathbb P^1\times\mathbb P^1\times\mathbb P^1\times\mathbb P^1\rightarrow \mathbb P(\mathbb C^2\otimes \mathbb C^2\otimes \mathbb C^2\otimes \mathbb C^2), $$ and now this space corresponds to the projetivization of rank 1 tensors in $\mathbb C^2\otimes \mathbb C^2\otimes \mathbb C^2\otimes \mathbb C^2$.

So a curve on this space is given by $\gamma:(a,b)\rightarrow \mathbb C^2\otimes \mathbb C^2\otimes \mathbb C^2\otimes \mathbb C^2$, $\gamma(t)=a(t)\otimes b(t)\otimes c(t)\otimes d(t)$. Computing the tangent space to $x=\gamma(0)=a\otimes b\otimes c\otimes d$, we have that $$ T_x(C(\sigma(\mathbb P^1\times\mathbb P^1\times\mathbb P^1\times\mathbb P^1)))=\mathbb C^2\otimes b\otimes c\otimes d+\dots +a\otimes b \otimes c \otimes \mathbb C^2, $$ So via Terracini I would have that $$T_p\sigma_2(\mathbb P^1\times\mathbb P^1\times\mathbb P^1\times\mathbb P^1)=\langle T_x\sigma(\mathbb P^1\times\mathbb P^1\times\mathbb P^1\times\mathbb P^1), T_y\sigma(\mathbb P^1\times\mathbb P^1\times\mathbb P^1\times\mathbb P^1)~\rangle $$ I cannot see what I am missing. Any suggestion or tips?

Thanks in advance!

Answer 1

The four 2-dimensional summands of $T_x$ in your formula all contain the 1-dimensional space $a \otimes b \otimes c \otimes d$. Consequently, their sum is 5-dimensional.