# Projection of a hypersurface from a point

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?

• Note that this question was also asked at MO: mathoverflow.net/questions/177526/… (It is considered good form to link to the other question in cases of cross-posting, so that efforts to answer are not duplicated.) Aug 1, 2014 at 14:12

If $Q\in X$, then the projection is a rational map $X\dashrightarrow H$ which is a morphism $X\setminus Q\to H$. Sometimes, this rational map extends to a morphism $X\to H$, so $Q$ has in fact an image, which depends on $X$.