Let $X$ be an irreducible algebraic variety and suppose that $L$ is a linear space defined by the linear forms $l_1,l_2,\ldots,l_k$. I want to study $L\cap X$. I would like to know whether the following argument is correct and I would appreciate very much a reference:
The linear sections of $X$ are all birational equivalent and the birational map between $X \cap V(l_1)$ and $X \cap V(l')$ will preserve also the degree. So the dimension and the degree of $X\cap L$ will be the same of the dimension and the degree of $X\cap L'$, with $L'$ defined by $l',l_2,\ldots,l_k$.