Concentrated reading on any given topic---especially topic—especially one in algebraic geometry, where there is so much technique---is technique—is nearly impossible, at least for people with my impatient idiosyncracy. It's much easier to proceed as follows.

1. Ask an expert to explain a topic to you, the main ideas, that is, and the main theorems. Keep diligent notes of the conversations.
2. Try to prove the theorems in your notes or find a toy analogue that exhibits some of the main ideas of the theory and try to prove the main theorems there; you'll fail terribly, most likely.
3. Once you've failed enough, go back to the expert, and ask for a reference.
4. Open the reference at the page of the most important theorem, and start reading.
5. Every time you find a word you don't understand or a theorem you don't know about, look it up and try to understand it, but don't read too much. At this stage, it helps to have a table of contents of FGA explained-EGA-SGA explained-EGA-SGA where you can quickly look up unknown words---mine may be found here, where the links are to versions of EGA and SGA with the chapters glued together in the correct order. The link to FGA explained has been disabled for legal reasonswords. Keep diligent notes of your progress, and talk to your expert as much as possible. Then go back to step 2.

An example of a topic that lends itself to this kind of independent study is abelian schemes, where some of the main topics are (with references in parentheses):

1. the rigidity lemma (Mumford, GITGeometric invariant theory, Chapter 6),
2. the theorem of the cube (Raynaud, Faisceaux amples sur les schemas ...), schémas),
3. construction of the dual abelian scheme (Faltings-Chai, Degeneration of abelian varieties, Chapter 1),
4. questions of projectivity (Raynaud, Faisceaux amples sur les schemas...),
5. Lang-Neron ),
6. Lang-Néron theorem and K/k $K/k$ traces (Brian Conrad's notes).
7. proof that abelian schemes assemble into an algebraic stack (Mumford, GITGeometric invariant theory, Chapter 7),
8. compactifications of the stack of abelian schemes (Faltings-Chai, Degeneration of abelian varieties; Olsson, Canonical compactifications; Kato and Usui, Classidying spaces of degenerating polarized Hodge structures...; Kato's secret writings on log abelian varieties)

You may amuse yourself by working out the first topics above over an arbitrary base. That's enough to keep you at work for a few years!

A brilliant epitome of SGA 3 and Gabriel-Demazure is Sancho de Salas, Grupos algebraicos y teoria de invariantes. It explains the general theory of algebraic groups, and the general representation theory of reductive groups using modern language: schemes, fppf descent, etc., in only 400 quatro-sized pages!

4 changed a word

Concentrated reading on any given topic---especially one in algebraic geometry, where there is so much technique---is nearly impossible, at least for people with my impatient idiosyncracy. It's much easier to proceed as follows.

1. Ask an expert to explain a topic to you, the main ideas, that is, and the main theorems. Keep diligent notes of the conversations.
2. Try to prove the theorems in your notes or find a toy analogue that exhibits some of the main ideas of the theory and try to prove the main theorems there; you'll fail terribly, most likely.
3. Once you've failed enough, go back to the expert, and ask for a reference.
4. Open the reference at the page of the most important theorem, and start reading.
5. Every time you find a word you don't understand or a theorem you don't know about, look it up and try to understand it, but don't read too much. At this stage, it helps to have a table of contents of FGA explained-EGA-SGA where you can quickly look up unknown words---mine may be found here, where the links are to versions of EGA and SGA with the chapters glued together in the correct order. The link to FGA explained has been disabled for legal reasons. Keep diligent notes of your progress, and talk to your expert as much as possible. Then go back to step 2.

An example of a topic that lends itself to this kind of independent study is abelian schemes, where some of the main topics are (with references in parentheses):

1. the rigidity lemma (Mumford, GIT, Chapter 6),
2. the theorem of the cube (Raynaud, Faisceaux amples sur les schemas ...),
3. construction of the dual abelian scheme (Faltings-Chai, Degeneration of abelian varieties, Chapter 1),
4. questions of projectivity (Raynaud, Faisceaux amples sur les schemas ...),
5. Lang-Neron theorem and K/k traces (Brian Conrad's notes).
6. proof that abelian schemes assemble into an algebraic stack (Mumford, GIT, Chapter 7),
7. compactifications of the stack of abelian schemes (Faltings-Chai, Degeneration of abelian varieties; Olsson, Canonical compactifications ...; Kato's secret writings on log abelian varieties)

You may amuse yourself by working out the first topics above over an arbitrary base. That's enough to keep you at work for a few years!

A brilliant epitome of SGA 3 and Gabriel-Demazure is Sancho de Salas, Grupos algebraicos y teoria de invariantes. It explains the general theory of algebraic groups, and the general representation theory of reductive groups using modern language: schemes, fppf descent, etc., in only 400 quatro-sized pages!

3 added the loop in step 5.

Concentrated reading on any given topic---especially one in algebraic geometry, where there is so much technique---is nearly impossible, at least for people with my impatient idiosyncracy. It's much easier to proceed as follows.

1. Ask an expert to explain a topic to you, the main ideas, that is, and the main theorems. Keep diligent notes.
2. Try to prove the theorems in your notes or find a toy analogue that exhibits some of the main ideas of the theory and try to prove the main theorems there; you'll fail terribly, most likely.
3. Once you've failed enough, go back to the expert, and ask for a reference.
4. Open the reference at the page of the most important theorem, and start reading.
5. Every time you find a word you don't understand or a theorem you don't know about, look it up and try to understand it, but don't read too much. At this stage, it helps to have a table of contents of FGA explained-EGA-SGA where you can quickly look up unknown words---mine may be found here, where the links are to versions of EGA and SGA with the chapters glued together in the correct order. The link to FGA explained has been disabled for legal reasons. Keep diligent notes of your progress, and talk to your expert as much as possible. Then go back to step 2.

An example of a topic that lends itself to this kind of independent study is abelian schemes, where some of the main topics are (with references in parentheses):

1. the rigidity lemma (Mumford, GIT, Chapter 6),
2. the theorem of the cube (Raynaud, Faisceaux amples sur les schemas ...),
3. construction of the dual abelian scheme (Faltings-Chai, Degeneration of abelian varieties, Chapter 1),
4. questions of projectivity (Raynaud, Faisceaux amples sur les schemas ...),
5. Lang-Neron theorem and K/k traces (Brian Conrad's notes).
6. proof that abelian schemes assemble into an algebraic stack (Mumford, GIT, Chapter 7),
7. compactifications of the stack of abelian schemes (Faltings-Chai, Degeneration of abelian varieties; Olsson, Canonical compactifications ...; Kato's secret writings on log abelian varieties)

You may amuse yourself by working out the first topics above over an arbitrary base. That's enough to keep you at work for a few years!

A brilliant epitome of SGA 3 and Gabriel-Demazure is Sancho de Salas, Grupos algebraicos y teoria de invariantes. It explains the general theory of algebraic groups, and the general representation theory of reductive groups using modern language: schemes, fppf descent, etc., in only 400 quatro-sized pages!

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