I would like to understand if the following statement is actually proven in Shafarevich's book "Basic algebraic geometry" (or just learn its proof in the spirit of Shafarevich's book).


**Statement**. Let $K$ be an algebraically closed field and let $X, Y$ be irreducible projective subvarieties in $\mathbb P\_K^n$. Suppose that $\dim X+\dim Y\ge n$. Then $X\cap Y$ is non-empty.


It is proven in section 1.6, Theorem 5, that provided a form $F$ is non-vanishing on $X$, all irreducible components of $X\cap F=0$ have dimension $\dim X-1$. So if $Y$ were a complete intersection the statement $X\cap Y\ne \emptyset$ would immediately follow from this theorem. At the same time $Y$ need not be a complete intersection…. A few lines after Theorem 6 in the same section Shafarevich mentions that the **statement** holds. But I can not find the proof in the book. So, is the proof really missing or did I just miss something simple?



Intersection of two projective submanifolds in $P^n$ treatment in Shafarevich book
I don't know how Shafarevich proves it or intended to prove it, but here is a way to do it.


**Claim:** Let $X,Y\subseteq \mathbb A^n$ irreducible of dimension $a,b$. Then every (non-empty!) irreducible component of $X\cap Y$ has dimension at least $a+b-n$.


**Sketch of Proof:** Step 1: using the Krull principle ideal theorem prove this in the case $Y$ is a hypersurface and then iterating it implies it if $Y$ is a complete intersection.


Step 2: use the diagonal trick: let $\Delta\subset \mathbb A^n\times \mathbb A^n$ be the diagonal and $$\delta:\mathbb A^n\to \Delta$$ the map given by $x\mapsto (x,x)$, then $\delta$ induces an isomorphism between $X\cap Y$ and $(X\times Y)\cap \Delta$. Since $\Delta\subset \mathbb A^n\times \mathbb A^n$ is a complete intersection, you can use your argument for this intersection (and you can check that the dimensions work out, too).


Now to prove the projective case, notice that if you take the cones over the projective subvarieties, then they necessarily intersect and hence the above claim can be applied to their intersection.
The claim in this case implies that (each irreducible component of) the intersection of the affine cones has to have at least dimension one, which implies that then there is a point other than the origin in it, which gives you an actual point on the intersection of the projective varieties.


