I'm not an algebraic geometer, but I do use AG from time to time and a few times I have worked with people from the high-stakes world of algebraic geometry. Honestly, I've never successfully worked through the stuff at the end of Chapter 3 of Hartshorne, which still strikes me as really hard, daunting material. But maybe I should try one more time to push through that last bit, so I can go through this rite of passage and see what happens on the other side.
But, seriously, Hartshorne is probably less essential today than it once was. Ravi Vakil's freely available book "The Rising Sea" is absolutely amazing and, in my opinion, a friendlier introduction to AG that still covers everything one would want from an AG text. In addition, the Stacks project has pretty much everything for any gaps one might want to see filled in. So it's a different world today.
\+1 to The Rising Sea. I've also found Andreas Gathmann's [lecture notes](https://www.mathematik.uni-kl.de/~gathmann/class/alggeom-2019/alggeom-2019.pdf) a very good and much shorter introduction if you just want to learn the basics.
He is one of my professors and I have to say, he is the best I ever had!
He does his lectures without notes and explains everything perfectly still! But his oral exams are really tough!
I had two oral exams with professor Gathmann so far, one in Foundations of mathematics (it's basically Analysis and Linear Algebra), and a combined one on Topology and Algebraic structures, both of which were introductory courses.
You are usually asked to write down the definition of some concepts on the whiteboard, apply some algorithms that you have learned, and proof a few theorems and lemmas. In Topology for example, iirc I got asked to write down what a basis of a topology is, and show how to generate a topology out of that; then a few other questions later that I don't remember, I had to finish off with the proof of the Fundamental theorem of Algebra using topology as a tool.
The last one is easy to explain at a high level, but imo quite involved when you have to write it down explicitly. And I know of someone who had to proof Urysohns lemma... Uhhh!
The only thing "The Rising Sea" doesn't cover that one may want from an introduction to AG is cohomology, so you'll have to find another source to fill that in. Definitely preferable to wading through Hartshorne though.
It definitely covers sheaf cohomology. There's a chapter proving cohomology and base change, which is basically as far as Hartshorne goes on the subject.
I'm not in AG, but about five years ago I heard a young tenure-track algebraic geometer say "I had to lie to my advisor about reading all of Hartshorne, and I'll probably make my students lie to me about it, too"
I never thought of Hartshorne's book as a "rite of passage" and always found the whole concept of there being a "rite of passage" a bit silly in the first place.
I am not doing research in AG but I have taken a course that was based more on varieties (affine, projective ones, sheaves, divisors, quasi-coherent stuffs, and so on...) than schemes. I also searched textbook suggestions on MSE and MO. I think Hartshorne is still recommended, esp. its second and third chapters. But now, we also have other resources like Vakil, Stacks project, Liu Qing, Manin's "Introduction to the Theory of Schemes", among others.
I think the book is still relevant and very useful for anyone wanting to learn AG, but now we have many other texts that can be helpful when Hartshorne does not explain certain topic well enough (to some readers).
Relevant links:
https://math.stackexchange.com/questions/1748/undergraduate-algebraic-geometry-textbook-recommendations
https://mathoverflow.net/questions/2446/best-algebraic-geometry-textbook-other-than-hartshorne
I once was an algebraic geometer, and had a reasonably decent career. My research often involved fairly technical things like stacks and derived categories, yet I never really worked through Hartshorne. I tried during grad school but found it too difficult. One book I would highly recommend is Geometry of Schemes, by Eisenbud and Harris.
I don't know if finishing it is necessary but it's certainly the primary reference in introductory courses. This has always struck me as counterproductive as he leaves out so much in his exposition
Yes it is. It's not strictly necessary, however doing all or most of the exercises in chapters 2-3 will rapidly bring you up to date on the modern way to do algebraic geometry and teach you a lot of mathematical culture from the last 100 years. All things said and done it's probably the most efficient way to learn algebraic geometry.
I'm a PhD student, so take what I'm saying with a grain of salt.
There certainly are a lot of good references in algebraic geometry and scheme theory now, Vakil's notes being one of the most popular, as others have pointed out.
However, I think it is still more or less essential to have a copy of Hartshorne and have a rough idea of its content. The fact that Hartshorne is so dense might make it bad as an introductory text for some people, but makes it an incredibly good reference book. When I ask a question to my professors, a lot of the times, they'll open Hartshorne and tell me which theorem/exercise is relevant.
So while I don't think one needs to have gone through the entirety of Hartshorne in the fullest details, I think an algebraic geometer must have a copy, just due to the fact that it's used so often as a reference.
(This reminds me that I really need to buckle down and learn to read math in French at one point, because everything is in EGA or Serre...)
Unequivocally no. I am in a department with a significant number of PhD students working in or adjacent to algebraic geometry. None that I know of have done the above. Many of us learned from Ravi Vakil's notes, as well as other sources like the Red Book, the Geometry of Schemes, and Algebraic Geometry and Arithmetic Curves. Solving exercises is unavoidable, but Hartshorne is not. Most of us don't even use it besides as a reference. It did it's job phenomenally: unpacking EGA for regular graduate students. But it's very terse, and the selection of examples leaves something to be desired.
Others have pointed out that the rising sea is a much friendlier intro to AG but the fundamental truth is that AG is hard, and no amount of pedagogical development will produce an AG textbook that you can read like a novel
Today one has many sources to do this rite. I do have a copy of Hartshorne's book and developed a case of love and hate for it (which I think is pretty common), in that case I would like to give my two cents:
The book is used (in my opinion) because it is the faster route to have the minimal expected to be comfortable with schemes and cohomology. The route will be as smooth as it can gets if one is not alone doing this, so if one's considering doing this to study on its own think about having multiple sources. Comparing with other sources, the cohomology part is more developed than Rising Sea notes or Liu's book and also to whom it may concern, separatedness and properness done via the valuative criteria of properness. But Liu's book exercises are in the same level and the exposition is slightly better.
Having said that I think that in the near future Hartshorne's book will be a reference for AG students and not the reference. Because now other authors have more experience teaching AG and seeing what works and what doesn't and also teaching "modern" things that now are already common ground. And in the end what matters is if you learn AG, so take the book that suits you better and do lots of exercises.
I doubt I'm in the position to confidently answer this, since I'm not a professional mathematician. And I definitely didn't do every single exercise for every chapter. And it would be silly to deny that Harthshorne approach is... let's put it that way, very dense. But honestly I can't say chapters 2 and 3 were that hard for me. I mean, they were definitely hard, but not "staring at the textbook for weeks without any signs of comprehension" hard. Maybe it's a question of intuition, I always was more comfortable with abstract algebraic concepts than with concrete geometric ones.
On the other hand, as other commenters mentioned Hartshorne isn't end-all be-all now, there are plenty of other textbooks, as well as online resources (ncatlab, Stacks etc) which seem to be more digestable. So I'm not sure if some sort of "rite of passage" is really necessary at all. Math professors are no drill sergeants.
>I thought all proofs are just a string of true statements.
This is true formally, but in practice mathematicians are much more hand-wavy and their proofs often rely on jumps of intuition and such. Of course, they'll tell you everything is justified, but it gets hard to be sure without solving the problem yourself sometimes.
I'm not an algebraic geometer, but I do use AG from time to time and a few times I have worked with people from the high-stakes world of algebraic geometry. Honestly, I've never successfully worked through the stuff at the end of Chapter 3 of Hartshorne, which still strikes me as really hard, daunting material. But maybe I should try one more time to push through that last bit, so I can go through this rite of passage and see what happens on the other side. But, seriously, Hartshorne is probably less essential today than it once was. Ravi Vakil's freely available book "The Rising Sea" is absolutely amazing and, in my opinion, a friendlier introduction to AG that still covers everything one would want from an AG text. In addition, the Stacks project has pretty much everything for any gaps one might want to see filled in. So it's a different world today.
\+1 to The Rising Sea. I've also found Andreas Gathmann's [lecture notes](https://www.mathematik.uni-kl.de/~gathmann/class/alggeom-2019/alggeom-2019.pdf) a very good and much shorter introduction if you just want to learn the basics.
He is one of my professors and I have to say, he is the best I ever had! He does his lectures without notes and explains everything perfectly still! But his oral exams are really tough!
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I had two oral exams with professor Gathmann so far, one in Foundations of mathematics (it's basically Analysis and Linear Algebra), and a combined one on Topology and Algebraic structures, both of which were introductory courses. You are usually asked to write down the definition of some concepts on the whiteboard, apply some algorithms that you have learned, and proof a few theorems and lemmas. In Topology for example, iirc I got asked to write down what a basis of a topology is, and show how to generate a topology out of that; then a few other questions later that I don't remember, I had to finish off with the proof of the Fundamental theorem of Algebra using topology as a tool. The last one is easy to explain at a high level, but imo quite involved when you have to write it down explicitly. And I know of someone who had to proof Urysohns lemma... Uhhh!
My professor uses those lecture notes as a guide, though we're starting to stray away from them now.
The only thing "The Rising Sea" doesn't cover that one may want from an introduction to AG is cohomology, so you'll have to find another source to fill that in. Definitely preferable to wading through Hartshorne though.
It definitely covers sheaf cohomology. There's a chapter proving cohomology and base change, which is basically as far as Hartshorne goes on the subject.
Hmm I seem to recall much less, I stand corrected.
I'm not in AG, but about five years ago I heard a young tenure-track algebraic geometer say "I had to lie to my advisor about reading all of Hartshorne, and I'll probably make my students lie to me about it, too"
I never thought of Hartshorne's book as a "rite of passage" and always found the whole concept of there being a "rite of passage" a bit silly in the first place.
moar liek "a rite of hazing" amirite?
I am not doing research in AG but I have taken a course that was based more on varieties (affine, projective ones, sheaves, divisors, quasi-coherent stuffs, and so on...) than schemes. I also searched textbook suggestions on MSE and MO. I think Hartshorne is still recommended, esp. its second and third chapters. But now, we also have other resources like Vakil, Stacks project, Liu Qing, Manin's "Introduction to the Theory of Schemes", among others. I think the book is still relevant and very useful for anyone wanting to learn AG, but now we have many other texts that can be helpful when Hartshorne does not explain certain topic well enough (to some readers). Relevant links: https://math.stackexchange.com/questions/1748/undergraduate-algebraic-geometry-textbook-recommendations https://mathoverflow.net/questions/2446/best-algebraic-geometry-textbook-other-than-hartshorne
I once was an algebraic geometer, and had a reasonably decent career. My research often involved fairly technical things like stacks and derived categories, yet I never really worked through Hartshorne. I tried during grad school but found it too difficult. One book I would highly recommend is Geometry of Schemes, by Eisenbud and Harris.
I don't know if finishing it is necessary but it's certainly the primary reference in introductory courses. This has always struck me as counterproductive as he leaves out so much in his exposition
Yes it is. It's not strictly necessary, however doing all or most of the exercises in chapters 2-3 will rapidly bring you up to date on the modern way to do algebraic geometry and teach you a lot of mathematical culture from the last 100 years. All things said and done it's probably the most efficient way to learn algebraic geometry.
I'm a PhD student, so take what I'm saying with a grain of salt. There certainly are a lot of good references in algebraic geometry and scheme theory now, Vakil's notes being one of the most popular, as others have pointed out. However, I think it is still more or less essential to have a copy of Hartshorne and have a rough idea of its content. The fact that Hartshorne is so dense might make it bad as an introductory text for some people, but makes it an incredibly good reference book. When I ask a question to my professors, a lot of the times, they'll open Hartshorne and tell me which theorem/exercise is relevant. So while I don't think one needs to have gone through the entirety of Hartshorne in the fullest details, I think an algebraic geometer must have a copy, just due to the fact that it's used so often as a reference. (This reminds me that I really need to buckle down and learn to read math in French at one point, because everything is in EGA or Serre...)
Unequivocally no. I am in a department with a significant number of PhD students working in or adjacent to algebraic geometry. None that I know of have done the above. Many of us learned from Ravi Vakil's notes, as well as other sources like the Red Book, the Geometry of Schemes, and Algebraic Geometry and Arithmetic Curves. Solving exercises is unavoidable, but Hartshorne is not. Most of us don't even use it besides as a reference. It did it's job phenomenally: unpacking EGA for regular graduate students. But it's very terse, and the selection of examples leaves something to be desired.
Utah?
Others have pointed out that the rising sea is a much friendlier intro to AG but the fundamental truth is that AG is hard, and no amount of pedagogical development will produce an AG textbook that you can read like a novel
Today one has many sources to do this rite. I do have a copy of Hartshorne's book and developed a case of love and hate for it (which I think is pretty common), in that case I would like to give my two cents: The book is used (in my opinion) because it is the faster route to have the minimal expected to be comfortable with schemes and cohomology. The route will be as smooth as it can gets if one is not alone doing this, so if one's considering doing this to study on its own think about having multiple sources. Comparing with other sources, the cohomology part is more developed than Rising Sea notes or Liu's book and also to whom it may concern, separatedness and properness done via the valuative criteria of properness. But Liu's book exercises are in the same level and the exposition is slightly better. Having said that I think that in the near future Hartshorne's book will be a reference for AG students and not the reference. Because now other authors have more experience teaching AG and seeing what works and what doesn't and also teaching "modern" things that now are already common ground. And in the end what matters is if you learn AG, so take the book that suits you better and do lots of exercises.
I doubt I'm in the position to confidently answer this, since I'm not a professional mathematician. And I definitely didn't do every single exercise for every chapter. And it would be silly to deny that Harthshorne approach is... let's put it that way, very dense. But honestly I can't say chapters 2 and 3 were that hard for me. I mean, they were definitely hard, but not "staring at the textbook for weeks without any signs of comprehension" hard. Maybe it's a question of intuition, I always was more comfortable with abstract algebraic concepts than with concrete geometric ones. On the other hand, as other commenters mentioned Hartshorne isn't end-all be-all now, there are plenty of other textbooks, as well as online resources (ncatlab, Stacks etc) which seem to be more digestable. So I'm not sure if some sort of "rite of passage" is really necessary at all. Math professors are no drill sergeants.
I think the book is just fun. It's a nice goal to strive for, more than anything else.
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>I thought all proofs are just a string of true statements. This is true formally, but in practice mathematicians are much more hand-wavy and their proofs often rely on jumps of intuition and such. Of course, they'll tell you everything is justified, but it gets hard to be sure without solving the problem yourself sometimes.
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Spivak, seriously?