
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 ndimensional 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 f1,...,fr be a set of generators for I(V) (where by fi I mean f sub index i). L is tangent to V at p iff for each j, fj(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 (dfj/dxi) 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 (Worried)

I believe a real example will help you understand the motivation.
Consider a surface M defined by in our Euclid space . 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
. If b=0 f(L(t)) is identically zero.

Thank you for the example. That has helped me get my head round it somewhat.
With regards to the other questions, any takers? (Doh)