codimension one more than the dimension of a variety

If I have a variety $V$ with dimension $k$ and degree $d$, then a subspace of codimension $k$ in general position w.t.r. to $V$ intersects the variety in at most $d$ points. What can one say about a subspace of codimension $k+1$ tha contains a finite numbers of points? I am actually interested in the Segre variety $\Sigma$ that is the product of $n$ lines. I know that the degree is $n!$ and the dimension is $n$. I want to know what happens if I intersect $\Sigma$ with a subspace of codimension $n+1$ that contains a finite number of points.

-

It depends on the position of the points. If you take $n-k$ general points $x_1,...,x_{n-k}$ on $V\subset\mathbb{P}^n$ irreducible and non-degenerate variety. Then $\left\langle x_1,...,x_{n-k}\right\rangle$ is a codimension $k+1$ linear subspace intersecting $V$ in $x_1,...,x_{n-k}$.
On the other hand consider the Segre $V = (\mathbb{P}^1)^3\subset\mathbb{P}^7$. Take $v_1,...,v_4\in \Delta\cong\mathbb{P}^1$. Here $\Delta$ is the diagonal $\{p=q=s\}\subset (\mathbb{P}^1)^3$. The image of $\Delta$ in $\mathbb{P}^4$ is a twisted cubic $C\subset\mathbb{P}^7$. Let $x_1,...,x_4$ be the images of $v_1,...,v_4$. Then $H = \left\langle x_1,...,x_4\right\rangle$ is a $3$-plane intersecting $C$ in four points. Now, $deg(C) = 3$ implies $C\subset H$. Therefore $H\cap V = C$.