I read in Harris' book at page 148, proposition 11.37 and got slightly confused regarding the argument.
Harris mentions a projective space $\mathbb{P}^{2n+1}$ and a linear subspace $\mathbb{P}^{n}$ denoted $L$, disjoint from a projective variety $J$, which is embedded in $\mathbb{P}^{2n+1}$. He then denotes by $\pi_L:\mathbb{P}^{2n+1}\longrightarrow L$, what he refers to as "the projection from $L$", which is a term I both never heard of, nor ever encountered in my reading. He then proceeds to project $J$, claiming that as $J\cap L=\emptyset$, the projection is regular map. He finally adds that it is a general fact that the projection of a projective variety from a linear subspace disjoint from it is finite.
My questions are:
1) How is the projection from a linear subspace defined? I am familiar with the notion of projection from a point.
2) Is this projection a regular morphism?
3) Why does a regular morphism in this context has finite fibers?
4) Why is the projection of a projective variety from a linear subspace disjoint from it is finite in general?
I believe all my questions stem from not understanding the answer to 1). Any help is appreciated.
Ciao!
Any linear subspace $L$ of dimension $n$, after a change of variables in $\mathbb{P}^{2n+1}$ can be assumed to be given by $x_0=x_1=\ldots=x_n=0$. Then the projection from $L$, $\mathbb{P}^{2n+1}\to \mathbb{P}^n$ is given by the map $(x_0:\ldots:x_{2n+1})\mapsto (x_0:\ldots: x_n)$. This is a morphism outside $L$ and thus gives a morphism from any closed subset of $\mathbb{P}^{2n+1}$ which is disjoint from $L$. The rest (3), and (4) should be clear.