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Math Help - Tangent Spaces

  1. #1
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    Tangent Spaces

    Hi,

    HUGE gratitude to anyone that can help me on the following "problem". I have tried looking it up in algebraic geometry text books though they are either too advanced or contextulised in a different manner. I don't think it is particularly hard, in fact I am sure it is standard text book stuff, though while I don't understand it properly, it limits my progress on the topic.

    Anway,the "problem" can be broken down into a number of steps:

    1) My lecture notes first consider an affine algebraic variety V in an n-dimensional affine space over a field K and a point p belonging to it. They say a line L given in parametric form {p + tv: t belongs to K, v a non zero vector} is tangent to V iff for every f belonging to I(V), f(p + tv) has a zero of atleast multiplicty 2 at t = 0 or is identically 0.

    What is the motivation for this definition?

    2)The tangent space TpV is then defined to be the union of the lines tangent to V at p. The following proposition is then given:

    Let f|1,...,f|r be a set of generators for I(V) (where by f|i I mean f sub index i). L is tangent to V at p iff for each j, f|j(p +tv) has a zero of atleast multiplicty 2 at t = 0 or is identically 0.

    What is the proof for this?

    3) As a corollary to this TpV is an affine subspace of dimension n - rankJp where J is the Jacobian matrix (df|j/dx|i) and Jp is J evaluated at p.

    Again what is the proof for this?

    Thanks for any help. Rereading this I realise it is a lot to ask, though I've exhausted all other avenues
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  2. #2
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    I believe a real example will help you understand the motivation.
    Consider a surface M defined by \{f=0|f(x,y,z)=x^2+y^2-1\} in our Euclid space E^3. It is a cylinder. Give a point p=(1, 0, 0) on M, choose any vector v=(a,b,c), the line L defined by L(t)=p+tv = (1+at, bt, ct). When L is tangent to M, v must be a vector orthogonal to the unit vector (1, 0, 0). That is, a = 0. and p+tv=(1, bt, ct). Obviously f(L(0))=0, and
    \frac{df(L(t))}{dt}(0)=\frac{d(b^2t^2)}{dt}(0)=0. If b=0 f(L(t)) is identically zero.
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  3. #3
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    Thank you for the example. That has helped me get my head round it somewhat.

    With regards to the other questions, any takers?
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