Life after Hartshorne (the book, not the person)...

Question

I was wondering what material in algebraic geometry is crucial and is a logical step for a serious graduate student in algebraic geometry once they've finished Hartshorne. Good answers could include a list of areas of algebraic geometry or important topics that an algebraic geometer must learn along with good references (i.e. accessible to someone with the background of Hartshorne), preferably in the order he or she should/could learn them. Papers in algebraic geometry tend to draw from so many areas within the field itself that I was wondering what people thought was the best order and way of acquiring that material.

Answer 1

'Intersection Theory' by Fulton

Every algebraic geometer needs to know at least the basics of intersection theory. Fulton's book is the standard reference and serves both as a textbook and a reference.

'Principles of Algebraic Geometry' by Griffiths and Harris

This is because Hartshorne does not really talk about complex geometry, Hodge theory or more classical algebraic geometry. It might also be good to see the classical approach to the theory developed in chapters 4 and 5 in Hartshorne which of course existed way before sheaf cohomology and schemes.

EGA by Grothendieck and Dieudonné.

This is if you want more of the Hartshorne style algebraic geometry. I would not say it is essential to read the entire EGA, but since it is the standard reference, it is at least worth getting to know it.

Here is a few more suggestions for more specialized subjects:

Birational geometry (Kollar-Mori or Matsuki). Toric varieties (Fulton). Hodge theory (Voisin). Arithmetic geometry (Cornell-Silverman). Abelian varieties (Mumford). Deformation theory (Hartshorne). Moduli spaces (Mukai).

After Hartshorne, you could start specializing. Find something that seems interesting to you - you'll pick up a lot of new algebraic geometry even though you are studying a specific subtopic. After finishing Hartshorne's book you should be able to read these books without too much trouble.

Answer 2

Dear anonymous, Mumford wrote a short book Lectures on Curves on an Algebraic Surface which, according to the preface, was written for the reader you have in mind (although at the time Hartshorne's book didn't exist yet). The book corresponds to oral lectures and the sections ( called Lecture $n$) are essentially the notes that had been distributed in class after the lectures, which makes for easy to digest little units.

The book contains the construction of the Picard scheme of a surface and the Hilbert scheme of curves on that surface. Lectures 3 to 10 (out of 27) are recollections of the general theory of schemes, with very interesting insights on the functor of points aspect. For example Mumford describes $\mathbb P^n(S)$ in terms of invertible sheaves on the scheme $S$ and their sections, he explains how to describe the Zariski tangent space of a functor defined on schemes even if the functor is not representable, etc.

The actual goal of the booklet is to prove a theorem of completeness of a characteristic linear system on a surface. The theorem was proved in characteristic zero analytically by Poincaré in 1910 but algebraically in all characteristics only in the 1960's by Grothendieck through systematic use of nilpotent elements. But as in all good books, the road is at least as interesting as the final destination, and much can be learned even if the book is not read to the end.

Answer 3

Lazarsfeld's book ``Positivity in Algebraic geometry'' contains a wealth of important material and is masterfully written. Anyone doing algebraic geometry today will greatly benefit from being familiar with the contents of this book.

Edit: here is a blog post by Burt Totaro on the importance of this topic/book:

http://burttotaro.wordpress.com/2011/01/11/why-you-should-care-about-positivity/

Answer 4

Perhaps the first advice I could give is to ask your advisor or algebraic geometers at the university at which you are located, if you are a student, they might already have areas/problems in mind for you to work on and so can give you the best advice relative to those problems.

A couple books which have not yet been mentioned (some of which I wish I had gone through more carefully):

Higher-Dimensional Algebraic Geometry, by Olivier Debarre.

This is a nice somewhat more informal introduction that covers many of the topics in Kollar-Mori with I would say more examples. It also covers some of the material in "rational curves" .

Moduli of Curves, by Harris and Morrison

A standard introduction / reference on the topic (which is again heavily studied).

Rational curves on algebraic varieties, by János Kollár.

The study of algebraic varieties by studying their rational curves is a major area of investigation in algebraic geometry. This book is fairly technical but contains a lot of information.

Hodge Theory and Complex Algebraic Geometry I: Volume 1 & 2, by Claire Voisin

Hodge Theory is an important tool and field of study as well.

Answer 5

Mumford's three part series Tata Lectures on Theta is well worth reading. (I wish I had read them already...I could sure use the information they contain.)

Added: Or really, close your eyes and pick a book by Fulton, Hartshorne, Kollar, Mumford, Silverman....If you get the book by Hartshorne that you've already read, pick again. Otherwise, whatever you picked will be a fine choice.

Answer 6

As others have said, after Hartshorne you could branch into too many different things, whose relative importance is subjective. Here is my take on what could constitute a good post-Hartshorne curriculum:

Intersection theory

• Intersection theory, William Fulton.

Classical foundational book, beautifully written.

• 3264 and all that, David Eisenbud and Joe Harris.

This one actually tells you how to compute Chow rings in a multitude of situations. But it also contains a wealth of useful information apart from how to compute Chow rings, such as geometry of Grassmanians, Chern classes, Hilbert schemes and a toolbox for Riemann-Roch.

Complex Algebraic Geometry

• Complex Geometry, An Introduction, Daniel Huybrechts

This concise book covers nicely the foundational material of complex-analytic approach (which people used to learn from chapters 0 and 1 of Griffiths-Harris).

• Hodge theory and complex complex algebraic geometry, I and II, Claire Voisin.

It took me time to develop love for this book. Probably because it strives to reach modern techniques as fast as possible, making it hard for someone just entering the world of Hodge theory.

Derived categories of coherent sheaves

• Fourier-Mukai Transforms in Algebraic Geometry, Daniel Huybrechts

Just as all other of Huybrechts' books, this one is a true gem. The book teaches you how not to be scared of $D^b (Coh (X))$.

Birational geometry

There is wealth of different literature on this topic, but I think the most useful as a first book is this truly wonderful book:

• Positivity in algebraic geometry I & II, Robert Lazarsfeld

Very clearly written and contains a wealth of examples and beautiful geometry.

Geometry of special varieties.

For special varieties (curves, various types of surfaces, special three- and four-folds, etc.) people have accumulated a large amount of more ad-hoc methods. At the basic level, contemplating the geometry of simplest such varieties provides an excellent playground to truly absorb the power of abstract machinery from Hartshorne. Further on, while doing research you will probably need a very good knowledge of the geometry of your particular variety at hand, so you cannot go wrong with investing some time to learn this material. I will mention here only the basics:

Curves:

• Chapter 4 of Hartshorne or
• Chapter 19 of Ravi Vakil's notes

Provide the absolute minimum here.

Surfaces:

• Chapter 5 of Hartshorne,
• Complex algebraic surfaces, Arnaud Beauville
• Algebraic surfaces and holomorphic vector bundles, Robert Friedman

The first two items are standard sources, while R. Friedman's book is an interesting synthesis of classical geometry of surfaces and analysis of coherent sheaves on them, beautifully written.

• Lectures on K3 surfaces, Daniel Huybrechts

K3 surfaces is truly an amazing class of varieties whose geometry is studied through a multitude of different techniques, and this book does an excellent job showing those multiple facets of research in the area.

• The geometry of cubic hypersurfaces, Daniel Huybrechts

These notes are still work in progress, but as the previous item these notes talk about an amazing variety of techniques: moduli spaces, Hodge theoretic methods, etc. There is also a discussion of higher dimensional cases of three- and four-folds.

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The rest is more idiosyncratic since it is the field I am interested in -

Moduli spaces of sheaves.

• The Geometry of Moduli Spaces of Sheaves, D. Huybrechts and M. Lehn

A systematic treatment of foundations of moduli of semistable sheaves. The book discusses an incredible amount of general techniques used in this area.

• Lectures on Vector Bundles, J.Le Potier.

Not the best read with respect to foundations (for that see the previous item in the list), but fully explains the best-studied case of moduli of sheaves on $\mathbb{P}^2$, which still serves as a beacon for research in this area.

• Fundamental Algebraic Geometry. Grothendieck's FGA explained, B. Fantechi et al.

An introduction into some more advanced fundamental techniques useful for moduli problems: descent theory, Hilbert and Quot schemes, elementary deformation theory and Picard scheme.

• Deformation theory, R. Hartshorne

A nice concise account of algebraic approach to deformation theory, with a lot of examples.

Answer 7

If you are interested in complex manifolds I would recommend Complex Geometry: an Introduction by Huybrechts.

I also think the Toric Varieties by Cox, Little, and Schenck is an excellent introduction to many advanced topics in algebraic geometry. Plus you get to learn a bunch of combinatorics in the process!

Answer 8

I've been told that Néron models by Bosch, Lütkebohmert, and Raynaud is a nice book to read if you want to get better acquainted with techniques in arithmetic geometry.

Answer 9

Abelian varieties.

Answer 10

Q.Liu's "Algebraic geometry and arithmetic curves" is a good reference, if you plan to lean on the more arithmetic side.

Again, a good idea would be to discuss the matter with your advisor(s), since there is a rich variety of fields within the topic (puns intended) -- or ask a more precise question.

Answer 11

I'm far from having read all of Hartshorne, but if I did I would study Compact Complex Surfaces, by Barth, Peters, Van de Ven. Also Geometric Invariant Theory would be a nice topic (I know about the book by Mumford, are there other good books on this topic?).

Ah, I forgot! How about derived categories? Someone suggested that for this topic a good reference is the book by Hartshorne Residues and duality. I had a look at some notes by Caldararu on the arxiv, "Derived categories of sheaves: a skimming", they seem well written.

Answer 12

Geometry of Algebraic curves, by Arbarello, Cornalba, Griffiths, Harris, 2 volumes; Complex abelian varieties, by Lange and Birkenhake; M. Artin, Lectures on deformations of singularities.

The basic idea common to all these suggestions is to use the foundational material from Hartshorne to investigate some more specialized topics, like curves, surfaces, abelian varieties, moduli spaces, singularities, or some more general techniques like intersection theory with applications to general Riemann Roch theorems, and vanishing theorems with applications to classification questions, plus arithmetic and analytic questions.

edit: In answer to a request in a comment below, this is an attempt to give some guidance to reading ACGH, Geometry of Algebraic Curves. I apologize if this bumps it back to the top.

A main theme in ACGH is to give, for a curve C, a model for the abel map C^d-->>W(d) in Pic^d(C), which is a resolution of singularities of the Brill Noether variety W(d), in terms of the natural “kernel resolution” of the locus of singular matrices in the space of all matrices of a given size. One is especially interested in the case d = g-1, where g = genus(C).

I.e. in the space say of square nxn matrices, one has the discriminant locus D of singular ones, defined by the determinant. This locus is to be a local model for the theta divisor W(g-1) of the Jacobian of C = Pic^(g-1)(C). It has a natural resolution in the product space P^(n-1) x Mat(nxn), consisting of the set of those pairs R = {([v],M) where v is in the kernel of M}. The second projection map R-->D is to be a local model for the abel map C^(g-1)-->W(g-1) resolving the theta divisor.

To set this up locally, choose a line bundle L0 in Pic^(g-1)(C), and a general divisor E of degree equal to h^0(L0), so that the map H^0(L0)-->H^0(L0(E)) is an isomorphism. Then H^1(L0(E)) = 0 and the map H^0(L0(E))-->H^0(L0(E)|E) in the exact sequence 0-->H^0(L0)-->H^0(L0(E))-->H^0(L0(E)|E)-->H^1(L0)-->0 is the zero map.

For line bundles L near L0, we map L to an nxn matrix, with n = h^0(L(E)) = h^0(L0(E)), for the corresponding restriction map r(L): H^0(L(E))-->H^0(L(E)|E). This maps a neighborhood of L in Pic^(g-1)(C) to matrix space, sending L0 to the zero matrix, and sending a nbhd of L0 in W(g-1) to a nbhd of zero in the discriminant locus D. Via this map, the abel resolution C^(g-1)-->>W(g-1) is locally the pullback of the natural kernel resolution R-->>D of the discriminant, in particular the fibers of the abel map are the linear series |L|, i.e. the projectivized kernel P(H^0(L)) of the map r(L).

To make all this work, one must study rank loci for matrices, as well as methods of representing the family of linear maps H^0(L(E))-->H^0(L(E)|E), for L in an open set of Pic^(g-1)(C). This takes a lot of heavy foundational machinery in acgh, chapters 2 and 4, constructing Poincare’ bundles and so on. But this is where they are going. See pp.83-4, 176-7. Chapter VI has some nice geometry, and chaps V, VII state nice results.