[Math] Basics on anabelian geometry and Grothendieck’s section conjecture

Question

Even I can find similar questions and some answers on that questions, most of them are not quite unsatisfactory to me. Maybe this is a very stupid question, but there is no other place that I can ask this. I want to undersatnd Grothendieck's section conjecture and its recent results and also I want to study anabelian geometry. I just finished to read Hartshorne's algebraic geometry book. Is there any good start point for those things which contains some basics? If my background is not enough, please let me konw a kind of 'road-map' for anabelian geometry and section conjecture.

Answer 1

if you just read the Hartshorne, your background is not enough for anabelian geometry. You need

A) To learn about etale morphismes: Hartshorne just mentions that notion in exercises.

B) to learn about the étale fundamental group Grothendieck,

before C) learning anabelian geometry proper.

Let me begin by B), because for this I think there is one best reference, which is the original one: SGA I. It is certainly a non-trivial read, but not nearly as difficult as it is often imagined by beginners. After all, it is the first SGA, made for an audience who consisted only of beginners. And it is a good initiation to the grothendieck's style of thought, of which you will need plenty if you are to learn anabelian geometry.

Before learning the étale fundamental groups, you have to learn étale morphisms, as I said in A). Here you have several choices. You can stick with SGA I, whose first chapters contain everything you need on the subject. But those chapters may be a little terse for someone who just knows the Hartshorne. Alternatively, you can find a good introduction to étale morphisms in the texts about étale cohomology, such as Milne's or Freitag-Kiehl's. And of course, you can also read the relevant chapters in EGA, which will probably lead you to read other chapters first, but if you want to work in algebraic geometry, that's no wasted time.

For C), there is much to read and I am not an expert, but for an introduction you can begin with "The Grothendieck Conjecture on the Fundamental Groups of Algebraic Curves" by Hiroaki Nakamura, Akio Tamagawa, Shinichi Mochizuki.