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I have the following problem:

I have an algebraic variety given as a zero locus of a set of polynomials. I know that there are points on this variety, which are singular (i.e. the dimension of the tangent space at these points is bigger than that of the variety). Now fixing one of these points, I would like to know, what is the number of irreducible components of the variety containing this point and what is their respective dimension. How can one do this?