I want to understand how smoothness is checked for algebraic varieties given as zero sets of functions.
Let me consider a "textbook" example to help fix ideas. Let the algebraic variety be given by these three polynomial equations,
Firstly I observe that if I assume $x_3$ and $x_2$ to be nonzero then the third equation follows from the first two.
Does that allow me to totally forget about the third equation?
Now I can write down the Jacobian matrix for the above set. I guess then I am supposed to check if the rank of that Jacobian evaluated on the common zeroset = or not to the number of functions.
But I can't figure out how this calculation has to be done. How do I determine the rank and further so on the common zero set?
I would be grateful if someone can walk me through the steps in this specific example.