My algebraic geometry is more of a disconnected set of ideas; however I need to understand well the Zariski tangent space. My commutative algebra though is decent. Towards this end, I don't find Hartshorne helpful, neither Eisenbud or Shokurov-Danilov as their presentation is dense. What I am looking for is a comprehensive treatment in terms of classical varieties, not in terms of schemes. Any recommendations?
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Have you looked at Shafarevich, Varieties in Projective Space (chapter 2 I think)? He starts out by calculating with polynomials what a tangent space should look like, and then derives from there the abstract definition.