I'm slowly working my way through 'Ideals, Varieties and Algorithms' by Cox, Little and O'Shea and the bit about subvarieties has confused me. If those in the know have access to the book it's page 236 I'm talking about.(adsbygoogle = window.adsbygoogle || []).push({});

Definition: Given an affine variety [tex]V[/tex] in [tex]k^{n}[/tex], let [tex]k[V][/tex] represent the collection of polynomial functions [tex]\phi : V \to k[/tex]

No problem there, you define a variety somehow (often as the common zeros of a set of functions) and then you have a set of functions defined on the points of V such that they take a point and map it to the field k, so points in n dimensional space to some number (typically k is the complex numbers).

Definition: Given [tex]V[/tex] in [tex]k^{n}[/tex],

1) For any ideal [tex]J = \langle \phi_{1},\ldots,\phi_{m}\rangle \subset k^{n}[/tex] we can define [tex]\mathbf{V}_{V}(J) = \{ (a_{1},\ldots,a_{n}) \in V \, : \, \phi(a_{1},\ldots,a_{n}) = 0 \, \forall \phi \in J\}[/tex]. This is a subvariety of V.

2) For each subset [tex]W \subset V[/tex], we define [tex]\mathbf{I}_{V}(W) = \{ \phi \in k[V] \, : \, \phi(a_{1},\ldots,a_{n}) =0 \, \forall (a_{1},\ldots,a_{n}) \in W\}[/tex]

Definition one is picking out the zeros of the ideal J, fine. Definition two pick out the "functions everywhere zero" on a subset of the variety. Up to this point I feel I've got a decent grip of things. However the book has an example which kinda demolishes that :

"Let [tex]V = \mathbf{V}(z-x^{2}-y^{2}) \subset \mathbb{R}^{3}[/tex]. Let [tex]J = \langle [x]\rangle \in \mathbb{R}[V][/tex]. Then we have [tex]W = \mathbf{V}_{V}(J) = \{(0,y,y^{2}) : y \in \mathbb{R}\} \subset V[/tex]"

Fine, no problem. J basically says "Let x be zero, what's the part of V with this true? It's W, parameterised by y. Happens to be, as pointed out by the book, the same as [tex]\mathbf{V}(z-x^{2}-y^{2},x) \subset \mathbb{R}^{3}[/tex]. The book then continues :

"Similarly, if we let [tex]W = \{(1,1,2)\} \subset V[/tex], then we leave it as an exercise to show that [tex]\mathbf{I}_{V}(W) = \langle [x-1],[y-1]\rangle[/tex]."

This I don't get. If [tex]\mathbf{I}_{V}(W)[/tex] is picking out the "functions everywhere zero on W in V", shouldn't it be [tex]\mathbf{I}_{V}(W) = \langle [x-1],[y-1],[z-2]\rangle[/tex] ? Otherwise you've not specificed in your new ideal that z=2 leads to the function being zero. Or are you supposed to 'glean' this information from the fact the ideal is defined from Win V[/tex] and since V is defined by [tex]\mathbf{V}(z-x^{2}-y^{2})[/tex], you have this additional bit of information?

I'm just generally confused entirely by that second part of the example. I'd be really grateful if someone could spell out what exactly is going on. Thanks very much.

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# Subvarieties in algebraic geometry

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