Consider $\mathbb CP^n$ and let $H\subset \mathbb CP^n$ be a hyperplane. Suppose $\varphi: \mathbb CP^n\to H$ is a rational map that fixes $H$ pointwise. I believe that $\varphi$ must be a projection from a point $p\in \mathbb CP^n\setminus H$ to $H$. How to prove this?


**Added.** F\_L gave a nice counterexample to the above question. But I still hope that the answer is positive if one has the following restriction on $\varphi$:


*The intersection of the locus of indeterminacy of $\varphi$ with $H$ has codimension $\ge 2$ in $H$*.


Does the statement hold under this additional condition? (In the counter-example of F\_L, $p\_1\in L$ is a point of indeterminancy of $\varphi$)



Rational maps from $\mathbb CP^n$ to $\mathbb CP^{n-1}$, fixing $\mathbb CP^{n-1}$
With the additional restriction, $\varphi$ is indeed a linear projection.


Applying a linear change of coordinates, $H$ is given by $x\_0=0$, so $\varphi$ is


$[x\_0:\cdots:x\_n]\mapsto [0:P\_1(x\_0,\ldots,x\_n):\cdots: P\_n(x\_0,\ldots,x\_n)]$


for some homogeneous polynomials $P\_1,\ldots,P\_n\in \mathbb{C}[x\_0,\ldots,x\_n]$ of the same degree $d$. You may choose them without common factor. Your assumption is that the restriction to $H$ is the identity. So this means that


$[0:x\_1:\cdots:x\_n]\mapsto [0:P\_1(0,x\_1,\ldots,x\_n):\cdots: P\_n(0,x\_1,\ldots,x\_n)]$
is the identity on $H$. Hence there is a polynomial $A\in \mathbb{C}[x\_1,\ldots,x\_n]$, homogeneous of degree $d-1$, such that $$P\_i(0,x\_1,\ldots,x\_n)=A\cdot x\_i$$ for $i=1,\ldots,n$.


The locus where $A=0$ and where $x\_0=0$ is then contained in the intersection of the base-locus of $\varphi$ with $H$. As you assumed that this latter should be of codimension $2$ in $H$, it implies that $A$ is a non-zero constant. Hence, $d=1$, so $\varphi$ is of degree $1$ and is a linear projection.


