# Hartshorne IV.3.9 To prove a Lemma of Bertini regarding degree $d$ curves in $\mathbb{P}^3$.

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)?

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?