Let $V\subseteq \mathbb{P}^n$ be a smooth projective variety. Let $X, Y$ be two irreducible subvarieties of $V$. I would like to show that $\dim_V(X)+\dim_V(Y)\leq \dim_V(X\cap Y)+\dim(V)$. In particular, if $\dim_V(X)+\dim_V(Y)\geq \dim(V)$, then $\dim_V(X\cap Y)$ is nonempty.
If $V=\mathbb{P}^n$, then a proof can be found in Shafarevich's book, Basic Algebraic Geometry, 3rd edition, pp.75, Theorem 1.24. That proof reduces to the affine case by looking at the affine cones, and uses the diagonal embedding of $\mathbb{A}^{n+1}$ to $\mathbb{A}^{n+1}\times \mathbb{A}^{n+1}$. The affine case follows by seeing that $\mathbb{A}^{n+1}$ is a complete intersection. In particular, the argument for the affine case holds for any smooth affine variety.
To adapt that proof to the case when $V$ is a proper smooth subvariety of $\mathbb{P}^n$ meets a difficulty (?). That is, even though $V$ is smooth as a projective variety, its affine cone is not smooth at the origin point.
Could you please let me know how to overcome this difficulty? Or is there any inherent problem with the above statement? Thanks.
