I'm trying to solve an exercise I.7.7 in Hartshorne's Algebraic Geometry:
7.7. Let $Y$ be a variety of dimension $r$ and degree $d>1$ in $\mathbb{P}^n$. Let $P \in Y$ be a nonsingular point. Define $X$ to be the closure of the union of all lines $\overline{PQ}$, where $Q \in Y, > Q\neq P$.
(a) Show that $X$ is a variety of dimension $r+1$.
(b) Show that $deg X < d$.
I've proved (a); it follows from the fact that there is a dominant rational map from the projective cone over $Y$ to $X$.
However, to prove (b), I'm stuck.
I found that it suffices to prove the following:
Let us consider the set $(\mathbb{P}^n)^*$ of all hyperplanes in $\mathbb{P}^n$ as the projective $n$-space $\mathbb{P}^n$ with its Zariski topology.
Let $W$ be the set of all hyperplanes containing $P$. Then W is a hyperplane in $(\mathbb{P}^n)^*$.
Then, for any (closed) variety $Y \subset \mathbb{P}^n$ which contains $P$ and is nonsingular at $P$, the set
$$\{ H\in W: \mbox{For every irreducible component $Z$ of } Y\cap H, i(Y,H;Z) = 1. \} $$
contains a nonempty open subset of $W$.
Now $i(Y,H;Z)$ is the intersection multiplicity of $Y$ and $H$ along $Z$, i.e., the length of the $\mathcal{O}_Z$-module $\mathcal{O}_Z / (I(Y)+I(H))\mathcal{O}_Z$, where $\mathcal{O}_Z$ is the local ring of $\mathbb{P}^n$ at $Z$.
How can I prove this? Thanks.
Edited: I found that if the ideal $I(Y)+I(H)$ is a radical ideal, then $i(Y,H;Z) = 1$ for all $Z$.
