# Hartshorne Exercise I.7.7

I'm trying to solve the following exercise from Hartshorne's Algebraic Geometry, namely Exercise I.7.7

Exercise I.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 $$P Q$$, where $$Q \in Y, Q \neq P$$.

(a) Show that $$X$$ is a variety of dimension $$r+1$$.

(b) Show that $$\operatorname{deg} X. [Hint: Use induction on dim $$Y$$.]

I think I've solved part (a) by showing that $$X$$ is birational to a cone, but my main concern is about part (b). For part (b), what I want to do is to pick a hyperplane, $$H$$, not containing $$Y$$, through $$P$$ such that $$Y\cap H$$ has just one irreducible component, and the intersection multiplicity along this irreducible component is $$1$$. Thereafter, I would replace $$Y$$ by $$Y\cap H$$ and thereby reduce $$\text{dim } Y$$ by $$1$$. Now, $$X\cap H$$ is the closure of the union of all lines through $$P$$ to points of $$Y\cap H$$. This would enable me to get an inductive argument running, as suggested in the hint. Degrees are behave exactly as we want them to, i.e. $$\text{deg } Y = \text{deg } Y\cap H$$, and likewise for $$X$$, as can be easily seen from Theorem I.7.7 (page 53). However, the trouble is that I can't see a good way to actually pick $$H$$, if at all this approach can be made to work. I would be very grateful is someone could provide a hint as to what I should try to do now.

Thank you.

• I took exactly this approach when I had to deal with this problem on my first time through the book. The tool you need to make it work is a slightly stronger version of Bertini's theorem, which is given as theorem II.8.18. From what I remember of my time with this section of the book, the stronger version is true and not so bad to prove over $\Bbb C$ by analytic methods, but I could not quite find the appropriate version in characteristic $p$. +1 and good luck in getting an answer, I'm curious to know the resolution. Commented Aug 30, 2021 at 6:56