I am reading Harris' Algebraic Geometry - a first course, and I am stuck at trying to figure out how the chords of a rational normal cubic curve form a projective variety. Could anyone point me in the right direction for understanding this?
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2Do you mean, considered as a subset of the Grassmannian? – Jake Levinson Nov 03 '14 at 02:27
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@JakeLevinson I am just looking at a twisted cubic - an image of the map from $\mathbb{P}^1 \rightarrow \mathbb{P}^3$ given by $[X_0,X_1] \rightarrow [X_0^3,X_0^2X_1,X_0X_1^2,X_1^3]$. The chords of this curve in $\mathbb{P}^3$ are supposed to form a variety. – baltazar Nov 03 '14 at 17:31
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It is impossible for a beginner to understand Lecture 1 of Harris's book and even more impossible to solve the exercises. – Georges Elencwajg Nov 12 '14 at 09:29
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@GeorgesElencwajg what would you suggest to go before it? – baltazar Nov 12 '14 at 16:39
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1http://math.stackexchange.com/questions/1748/undergraduate-algebraic-geometry-textbook-recomendations – Georges Elencwajg Nov 12 '14 at 17:00
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That is excellent, I don't know how I missed it. Thank you. – baltazar Nov 12 '14 at 17:02
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1Another thread ...And yet another one – Georges Elencwajg Nov 12 '14 at 17:09
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@GeorgesElencwajg Thank you very much. This is going to come in very handy in the following months. – baltazar Nov 12 '14 at 18:57
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My pleasure, baltazar: I am glad that the links were helpful. – Georges Elencwajg Nov 12 '14 at 19:02
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This isn't necessarily obvious immediately, but every point of $\mathbb{P}^3$ is on some (unique, in fact) chord of the twisted cubic. So it's not interesting to consider "points in $\mathbb{P}^3$ lying on chords to the twisted cubic". Rather, the point is to look in the Grassmannian,
$$G(2,4) = \{\text{lines } \ell \text{ in } \mathbb{P}^3\} = \{\text{2-dimensional vector subspaces of } \mathbb{C}^4\}.$$
This is, itself, a projective variety, with a standard embedding in $\mathbb{P}^5$ -- if you aren't familiar with it, you might be better off learning a bit more about it before doing this problem. You can probably find this info elsewhere in the textbook, or on the internet. I happen to have written a blog post or two about Grassmannians in general here: http://levjake.wordpress.com/2014/07/09/schubert-calculus-i-geometry-of-gkn/. But I can't promise it'll be accessible.
Anyway, you want to consider the subset of $G(2,4)$ consisting of those lines $\ell$ that are chords (or tangents) to the twisted cubic, and show that this is a closed (algebraic) subset of $G(2,4)$, hence also a projective variety.
This more or less boils down to the fact that the condition "$\ell$ is a chord to the twisted cubic" is a Zariski-closed condition, meaning you can express it in terms of certain algebraic equations that must be satisfied in $G(2,4)$.
Jake Levinson
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For some reason this is one of the first examples in Harris' book, so I thought that I should be able to understand it almost immediately. I'll try reading your blog post. I now understand the problem better. Thank you. – baltazar Nov 03 '14 at 20:36
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