I'm finishing up an undergrad degree in mathematics and am beginning to think about areas of research. I know that the work of Grothendieck is considered the cornerstone of modern algebraic geometry, and I'd like to get a deeper overview of the subject (the questions it asks, the methods it uses, etc.) and particularly of Grothendieck's contribution. All I can tell from reading several sources online is (1) Grothendieck is very general and abstract (defining points abstractly, etc.) (2) Grothendieck output is very large (~10,000 pages in total). (3) His work is scattered in several books and series, and much remains untranslated. Given these points, I can't find an angle to approach the subject (all sources seem to repeat the letters "EGA, SGA" as a mantra) that isn't either in French or difficult to find or both. Is there a good place to begin that would be suitable for an advanced undergrad? Or is there a standard "path" of subjects to study before learning French and tackling EGA?

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    $\begingroup$ Not sure this is appropriate for MathOverflow, probably more appropriate on MathStackExchange, so don't be surprised if it's closed. But I'd suggest (1) learn some commutative algebra at the level of, say, Atiyah-Macdonald's or Eisenbud's books and then (2) read Hartshorne's Algebraic Geometry, which is a standard introduction to Grothendieck-style algebraic geometry. $\endgroup$ – Joe Silverman Feb 8 '16 at 1:29
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    $\begingroup$ I'm making this CW. It might not stay open as Joe Silverman says, but if it does: there is some divergence of opinion on the level of abstraction which is appropriate in the teaching and learning of modern algebraic geometry at the level of graduate introductory courses. Just yesterday I was reading a fascinating thread on this: sbseminar.wordpress.com/2009/08/06/… $\endgroup$ – Todd Trimble Feb 8 '16 at 1:41
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    $\begingroup$ I mostly agree with Joe Silverman. You certainly need some commutative algebra first. But I'd start with Mumford's red book rather than Hartshorne. The first part is a very clear presentation of a (somewhat) more classical approach so you understand what schemes are intended to generalize, and the second part is a very clear presentation of scheme theory with emphasis on why it's the right generalization. There's no cohomology, but first things should come first. $\endgroup$ – Steven Landsburg Feb 8 '16 at 2:28
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    $\begingroup$ @StevenLandsburg Good point. I learned the non-scheme version from Shafarevich's book, which is (I thought) quite well written. Mumford, of course, is also a good choice. Best of all, I'd say, is to take a graduate algebraic geometry course a couple of times from different teachers who present it from different viewpoints. My own experience was first with Gieseker (quite geometric), second with Hironaka (very algebraic), with the summer between spent reading Hartshorne and doing lots of the problems with a group of other grad students. $\endgroup$ – Joe Silverman Feb 8 '16 at 3:49
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    $\begingroup$ Possible duplicate of A learning roadmap for algebraic geometry $\endgroup$ – user9072 Feb 8 '16 at 9:50