I am a Ph.D. student and starting a side project with a fellow student on Moduli spaces. Our plan was to start with the book on [Invariants and Moduli][1] by Mukai (starting from chapter 5) and use the notes [Moduli problems and geometric invariant theory][2] by Hoskins side by side. After learning about the basics, we want to go two ways, (I) study moduli spaces of vector bundles and study stability conditions from [Bayer - A tour to stability conditions on derived categories][3], (II) study the paper [Geometric invariant theory and flips][4] by Thaddeus.

Now my study partner's guide has asked him to start studying the book ‘The geometry of moduli spaces of sheaves’ by Huybrechts. We (roughly) went through the introduction and the first few sections of the book, and although we understood that the book studies the moduli space of semistable sheaves, focusing on surfaces, it is not clear whether this is just a more general theory than the project we were planning to start, or this demands different prerequisites and ideas. Currently, I am trying to somehow combine both of these together, so that we can start our project in a way that will help us get into the (quite dense) book by Huybrechts.

So, the main question is: What is the moduli space of semistable sheaves? And is it going to be very much different than studying GIT and moduli of curves or vector bundles? It will be awesome if someone can suggest a potential game plan.

Any suggestions or comments would be appreciated.


  [1]: https://www.google.com/books/edition/An_Introduction_to_Invariants_and_Moduli/oUcMZbiM7eAC
  [2]: https://userpage.fu-berlin.de/hoskins/M15_Lecture_notes.pdf
  [3]: https://www.maths.ed.ac.uk/~abayer/dc-lecture-notes.pdf
  [4]: https://www.ams.org/journals/jams/1996-9-03/S0894-0347-96-00204-4/S0894-0347-96-00204-4.pdf