# On dimension of Segre embedding of lattice translations

Consider three lattices $L_1$, $L_2$ in $\Bbb Z^{n+1}$ and $L$ in $\Bbb Z^{2n+1}$.

Let $L_1+v_1$, $L_2+v_2$ and $L+v$ be their respective translationsfor some $v_1,v_2\in\Bbb Z^{n+1}\backslash\{(0,\dots,0)\}$ and $v\in\Bbb Z^{2n+1}\backslash\{(0,\dots,0)\}$.

Consider the Segre map $f:\Bbb Z^{n+1}\times\Bbb Z^{n+1}\rightarrow\Bbb Z^{(n+1)^2}$ given by $$f(x_0,x_1,\dots,x_n,y_0,y_1,\dots,y_n)=(x_0y_0,x_0y_1,x_1y_0,\dots,x_{n-1}y_n,x_ny_{n-1},x_ny_n)$$

$T\cap (L+v)$ is not linear where $T=f(L_1+v_1,L_2+v_2)\subseteq\Bbb Z^{(n+1)^2}$.

What is the dimension of $T\cap (L+v)$ as a variety in $\Bbb R^{(n+1)^2}$ whenever $|T\cap (L+v)|>1$ holds?