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The problem is this: let $X$ be a degree $d$ curve in $\mathbb{P}^3$. Show that the set of hyperplanes $H$ in $\mathbb{P}^3$ such that $X\cap H$ has $d$ distinct points and no three of which are coplanar form a Zariski dense subset of $(\mathbb{P}^{3})^{*}$ (the space of hyperplanes in $\mathbb{P}^3$). Essentially the point is that a general hyperplane intersects a degree $d$ curve at $d$ points.

I am not sure how to prove this properly. I only have heuristics and I can totally believe that this result should be true. One of my approaches is as follows.

Fix an embedding $X\hookrightarrow\mathbb{P}^3$ given by a linear system $\mathfrak{d}$. This linear system has dimension $3$ and the divisors in it are degree $d$. I want to show that if I take a general hyperplane divisor $H$ in $\mathbb{P}^3$, then pulling back gives an element of the linear system. If I do this, then I have shown that $X\cap H$ consists of $d$ points (counted with multiplicity).

So essentially, I need to do three things (which I am stuck on):

(1) I can count without multiplicity.

(2) I can make sure that the $d$ distinct points are not coplanar (which follows from the linear system being $3$-dimensional)?

(3) These form a Zariski dense subset of $(\mathbb{P}^3)^*$?

Can I just get hints for these three things (complete solutions would not be appreciated since I feel very close)?

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  1. Hint: find a condition that guarantees the intersection of a hyperplane $H$ and your curve $C$ at a point $c\in C$ has multiplicity 1. Then show that the locus of hyperplanes which fail to have this property is a proper closed subset of $(\Bbb P^3)^*$. (You should be reminded of the proof of Bertini's theorem here.)

  2. Hint: you've seen the condition "three collinear points" before in the text of section IV.3. What's the name of the line through these three collinear points? How many such lines can a curve have? What subset of hyperplanes in $(\Bbb P^3)^*$ contain a given line?

  3. By answering 1 and 2 correctly, you'll see that both conditions are generic, hence their intersection is also generic. (Recall two nonempty open subsets of an irreducible space meet and any nonempty subset of an irreducible space is dense.)

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