I'm a rookie in algebraic geometry, trying to learn. Recently I noticed that I really don't understand the following topics very well. I am asking you if there is comprehensive references (one maybe with examples) to help me get comfortable and grasp these ideas:

  • Quasi-Coherent Sheaves, Coherent Sheaves, etc.
  • Construction of Quasi-Coherent Sheaves on a Projective Variety
  • Invertible Sheaves I: Specifically the Sheaf $\mathscr{O}_{\mathbb{P}^n}(m)$.
  • Weil and Cartier Divisors
  • Invertible Sheaves II: Picard Group $\mathrm{Pic}(X)$.
  • Proof of Bezout's Theorem using Divisors

I've read more or less all of the above from Kempf's Algebraic Varieties multiple times. Truth is although I can read through it, I get lost in all the abstraction, losing sight of what is the core of the discussion. Every time I read those chapters, I think this time I got it. But then I encounter a problem related to one of these, and realize I don't exactly know what I'm doing. I have no insight...

To get you a feeling of how bad the situation is and how great is my confusion, the other day I was trying to solve a problem dealing with sheaves $\mathscr{O}_{\mathbb{P}^1}(-n)$. At one point I argued (shamefully) that "Since $\mathscr{O}_{\mathbb{P}^1}(-n)$ is quasi-coherent but not coherent, blah blah blah"... Not only $\mathscr{O}_{\mathbb{P}^1}(-n)$ is coherent it is actually rank one by definition! It was right there and then that I told myself "OK, if you are making such a trivial mistake, then you have a serious issue."

I looked up Hartshorne, Shafarevich (II) and Ravi Vakil's notes other than Kempf. One thing these sources all have in common is that they describe these ideas on a scheme. I don't really know about schemes (I know the definition, and I can read through them if I have to. But that is again an extra layer of abstraction...)

  • $\begingroup$ Schemes are good; anyway, have you seen Gathmann's notes? Those might help. There's also Mumford's Algebraic Geometry I: Complex Projective Varieties (not the Red Book). $\endgroup$ – Hoot Nov 5 '15 at 21:55
  • $\begingroup$ No I haven't thank you I will take a look. $\endgroup$ – Hamed Nov 6 '15 at 0:16
  • $\begingroup$ Btw I have nothing against schemes, I'd love to learn about them and that's the plan. All in good time. $\endgroup$ – Hamed Nov 6 '15 at 2:43

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