Is it true that every projective sub-variety of degree $d$ in $\mathbb CP^n$ is an intersection of some number of hypersurfaces of degree $d$? Is there some simple proof of this fact? (I believe this is so)



Intersections of hypersurfaces of degree $d$ in $\mathbb CP^n$
It is true. The idea is as follows : suppose your variety, say $X$, has dimension $m$. For any $(n-m-2)$-plane $\pi \subset\mathbb{P}^n$, take the join $\langle \pi ,X\rangle$, that is, the union of the lines $\langle p ,x\rangle$ for $p\in\pi $, $x\in X$. It is easy to see that it is a hypersurface of degree $d$ when $\pi $ is general, and that the intersection of all these hypersurfaces when $\pi $ varies is $X$.


