In Deligne-Mumford's "The irreducibility of the space of curves of given genus", the authors use the "Schlessinger's theory", and refer his "thesis". Where can I read it? It seems to be different from his paper "Functors of Artins rings".

This theory of deformation (of singular schemes) are used in many papers, for example Deligne-Rapoport's "Les schemas de modules de courbes elliptiques". (And the theory of deformation of "curves + a section", used in Deligne-Rapoport, seems not to be written in his thesis. Please suggest me some references of it.)

  • $\begingroup$ Have you tried sending him an email asking if he still has any copies? I didn't find any posted on google. $\endgroup$
    – roy smith
    Jun 8 '20 at 0:21

I think by 'Schlessinger's theory' D-M mean both the foundations of deformation theory as described in the functor of Artin rings paper (which can nowadays be found in any book on deformation theory, e.g., Hartshorne's "Deformation Theory" or Sernesi's "Deformation of Algebraic Schemes"), and more specifically the deformation theory of reduced curves (see Hartshorne's book). Schlessinger's thesis 'Infinitesimal Deformations of Singularities' is easy to find on Google. I don't think I know a reference for 'curves + section' in the generality you need, but some basic aspects are addressed in Hartshorne's book (chapter 26). Somewhat analogous deformation theories (e.g., 'variety + line bundle') are described in great detail in Sernesi's book; you may find these helpful to set up the relevant deformation theory yourself.


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