Let $k$ be an algebraically closed field. We consider the projective space $\mathbb P\_n$ over defined over $k$, the point $Q=(0:\dots:1)$, the hyperplane $H=\{X\_n=0\}$ and a hypersurface $X$. We want to study the image of $X$ under the projection from $Q$ to $H$. If $Q\notin X$ everything is clear and every point $(x\_0:\dots:x\_n)\in X$ is mapped to $(x\_0:\dots:x\_{n-1}:0)$.


What happens if $Q\in X$? I've read that the projection must be thought using the blow up but I am still confused... what does the projection look like? I need this to (try to) extend a result but I would like to walk on "solid ground". To be precise, I am interested in the actual shape of the map and, in particular, in the ramification divisor.


I found hundreds of examples about blowing up the affine plane at the origin, dozens about an affine curve but it looks like nobody wants to lose time giving details in in this case...


A reference or a few hints would be very appreciated!



Projection of a hypersurface from a point
I agree that this question is perhaps not suitable for MO. Anyway, as you received no satisfactory answers on MSE, let me give you some examples that (I hope) can improve a bit your understanding of the situation. You are strongly encouraged to fill the details by yourself.


**Example 1.** Take a smooth quadric surface $X \subset \mathbf{P}^2$ and consider the projection $\pi\_P \colon X \dashrightarrow \mathbf{P}^2$, where $p \in X$. The birational map $\pi\_P$ is *not* defined at the point $p$, and it induces a birational morphism $\hat{\pi} \colon \widehat{X} \to \mathbf{P}^2$, where $\widehat{X}$ is the blow-up of $X$ at $p$. The morphism $\pi$ contracts the two lines $L\_1$ and $L\_2$ of $X$ passing through $p$ to two points $q\_1, \, q\_2 \in \mathbf{P}^2$. The image of the exceptional divisor $E \subset \widehat{X}$ is the line $\overline{q\_1 q\_2}$. 


**Example 2.** Take a smooth cubic surface $X \subset \mathbf{P}^2$ and consider the projection $\pi\_P \colon X \dashrightarrow \mathbf{P}^2$, where $p \in X$. The birational map $\pi\_P$ is *not* defined at the point $p$ and has generically degree $2$. It induces a flat double cover $\hat{\pi} \colon \widehat{X} \to \mathbf{P}^2$, where $\widehat{X}$ is the blow-up of $X$ at $p$, whose branch locus is a smooth quartic curve $B \subset \mathbf{P}^2$. It is well known (Plücker formulas) that $B$ has precisely $28$ bitangents. They arise as follows: $27$ of them are the images of the $27$ lines on the cubic surface, and the remaining one is the image of the exceptional divisor $E \subset \widehat{X}$. 


**Example 3.** In this example the center of the projection is a singular point of $X$. Take a Kummer quartic surface $X \subset \mathbf{P}^2$, i.e. a quartic with $16$ nodes, and choose a node $p$. The birational map $\pi\_P \colon X \dashrightarrow \mathbf{P}^2$ is *not* defined at the point $p$ and has generically degree $2$. It induces a flat double cover $\hat{\pi} \colon \widehat{X} \to \mathbf{P}^2$, where $\widehat{X}$ is the blow-up of $X$ at $p$, whose branch locus is a sextic curve $B \subset \mathbf{P}^2$, that splits as the union of six lines $L\_1, \ldots, L\_6$. These lines intersects pairwise in $6(6-1)/2=15$ points, that correspond to the $15$ nodes of $\widehat{X}$. The exceptional divisor $E \subset \widehat{X}$ is mapped to a conic $C$ which is tangent to all the $L\_i$: in particular, the $L\_i$ are *not* in general position.


