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 $\operatorname{deg} X<d$. [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.

  • 1
    $\begingroup$ 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. $\endgroup$
    – KReiser
    Commented Aug 30, 2021 at 6:56


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