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Can someone point some good reference (books or lecture notes) for these topics:

Let $X$ a smooth projective variety over an algebraically closed field

  • Sheaves of abelian groups over $X$
  • Quasi-coherent sheaves over $X$
  • Coherent sheaves over $X$
  • Correspondence between locally free sheaves and vector bundles
  • Divisors
  • Degree of $X$
  • slope stability

I need something that encompasses these topics and doesn't get over all the motivations too quickly.

I tried so far R. Vakil's FOAG and Hartshone but I keep getting lost, even though they are very good references and I've already studied sheaves / sheaf cohomology and basic algebraic geometry. Also I found R. Borcherds videos on youtube which are amazing but he just scratches the surface of it.

Also what strategy do you recommend for studying these topics. Is there something to keep track (like a mind map) of the "dependencies" and consequences of these concepts so one can divide and conquer?

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    $\begingroup$ It's a bit unclear what you exactly want to know especially if you already studied sheaf cohomology + basic AG. Do you have a final goal? Anyways, other than Vakil and Hartshorne, I'd say these topics (except slope stability) are well covered in Görtz--Wedhorn. Gathmann's notes are also good and much easier. For slope stability, it's a bit more advanced, I'd suggest Le Potier's book or Mukai "introduction to invariant and moduli". But I would say best way to learn it is through examples, e.g surfaces :-) $\endgroup$
    – Nicolas Hemelsoet
    Commented Jul 28, 2022 at 12:42
  • $\begingroup$ Thanks! The idea is to reach Bridgeland stability conditions on an arbitrary category. Feels like there's a gap. The path looks like: {(Quasi-)Coherent Sheaves} --> {Derived Categories of Coherent Sheaves}. But I'm bogged in between. Also Divisors and Moduli spaces are subjects that I often encounter while trying to understand these topics. But still I couldn't find any reference that covers these topics in a "coherent" fashion (no pun intended). $\endgroup$
    – Abel
    Commented Jul 28, 2022 at 13:03
  • $\begingroup$ For derived categories of coherent sheaves, I'd say one of the best reference is Huybrechts book on Fourier-Mukai transform. Not sure if a coherent reference exists. Divisors and moduli spaces are huge topics. Also this master thesis could be useful for you : maths.mic.ul.ie/kreussler/ciaradalyma.pdf $\endgroup$
    – Nicolas Hemelsoet
    Commented Jul 28, 2022 at 14:31

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