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The following snapshot comes from the paper Latent Variable Graphical Model Selection Via Convex Optimization:Latent Variable Graphical Model Selection Via Convex Optimization

I know little about algebraic geometry so I have several basic questions:

  1. How is the dimension of $\mathcal{S}(k)$ and $\mathcal{L}(r)$ calculated? I've found an explanation for the latter one, but it's quite complicated. Can you give hints for me?
  2. Are some points nonsmooth because they have a smaller dimension?
  3. How is the tangent space calculated?
  4. Finally, how is the curvature calculated?

Thank you very much for answering. Reference materials are also welcomed.

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