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How to like schemes in the right way

Some people like schemes too much. Others don’t like them at all. Here is my view on how to like schemes for the right reasons.

There are several good things about schemes. I usually study varieties over a field, but there are various ways one can encounter more general schemes in the course of a given problem.

One good thing is the way schemes automatically include information about “multiplicities”. For example, given a polynomial f in one variable over a field k, we would like to say that the number of zeros of f is equal to the degree of f. But there are obvious obstacles to that statement. For example, the polynomial x2+1 over the real numbers actually has no roots in R, so we are really talking about roots of f in some algebraic closure of k. But even then, you have to keep track of “multiplicities” for the statement about deg(f) to be true; e.g., the polynomial (x-1)2 has a root at x=1 “with multiplicity 2”.

Both these complications are avoided by the language of schemes. In that language, the zero set of a polynomial f(x) of degree d over a field k is always a 0-dimensional closed subscheme of the affine line A1 over k, and that subscheme always has degree d. This shows two different good aspects of the notion of a scheme. First, a subscheme Y of a variety X determines not only a subset Y(k) of X(k) (where X(k) means the set of solutions in k of the equations defining a variety X), it also determines a subset Y(E) of X(E) for every extension field E of k. (For example, the subscheme Y of the affine line over the real numbers defined by the equation x2+1 = 0 is not the empty scheme, even though Y(R) is the empty set, because we can see that Y(C) is not empty.) Second, a subscheme Y of a variety X contains more subtle “nilpotent” information than just the subsets Y(E) of X(E) for field extensions E of k. For example, the subscheme Y of the affine line over the real numbers defined by the equation (x-1)2=0 is different from the point Z defined by x-1=0, even though Y(E)=Z(E) for all field extensions E of the real numbers. We picture Y as a “fat point”, a point together with an “infinitesimal neighborhood of length 1”.

In this simple situation of polynomials in one variable, one could live without the geometric language of schemes. But for the same phenomena of “multiplicities” and so on in higher dimensions, the language of schemes is unavoidable (and useful). For example, consider the intersection of two smooth curves at a point in a smooth surface. If they intersect transversely, then the intersection (as a scheme) is “reduced”, i.e., it’s just the point. But if the intersection is not transverse, then the intersection is a 0-dimensional subscheme of degree greater than 1, i.e. it’s a “fattened” or “non-reduced” version of the point. (This is relevant even in the familiar situation of varieties over the complex numbers.)

The standard textbook on schemes is Hartshorne’s Algebraic Geometry. Unfortunately the actual definition of schemes is done in a very dry way there. Still, there is a huge amount of illuminating stuff in that book: the informal essay introducing schemes on pp. 55-59 is good, many exercises are interesting, many good examples and so on. By the end of chapter II on schemes, he gets to crucial geometric ideas like divisors (section 6) and differentials (i.e., the tangent bundle, or more precisely its dual). Chapters IV and V on curves and surfaces are appealing and (I think) can be read before going through all the earlier parts of the book. Appendices A, B, C introducing more advanced topics in an informal way are even better: don’t miss these.

There are by now lots of other books introducing schemes, although none has displaced Hartshorne. I have heard good things about Mumford’s Red Book of Varieties and Schemes, but I haven’t read it myself.

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Books for beginning research

I don’t remember exactly when or why I wrote the list below. I suspect it was a couple of years ago for Part III algebraic geometry students.

If you want to be a mathematician, there’s no substitute for knowing some math. You might as well learn it from great writers.

“To learn to write well, one should read Serre, Bott, Milnor,…” (I think I’m quoting Steve Hurder here, but I believe this too.)

The following list includes both short, readable books that everyone should read and longer reference books.

Undergraduate-level books

The standard topics in pure mathematics are: real analysis including Lebesgue integration (I recommend Royden, Real Analysis); complex analysis; topology (I recommend Armstrong, Basic Topology); and algebra including Galois theory. Fourier series are also fundamental; I recommend Dym and McKean, Fourier Series and Integrals, with a variety of applications in a short space.

Part III-level books

Representation theory: Serre, Linear Representations of Finite Groups. Fulton-Harris, Representation Theory (of semisimple Lie algebras, or equivalently of compact Lie groups). By concentrating on examples, Fulton-Harris’s book is wonderfully readable although somewhat long.

Commutative algebra: Atiyah and Macdonald, Introduction to Commutative Algebra. Very clear in a short space.

Number theory: Serre, A Course in Arithmetic. Cassels, Local Fields.

Topology: Bott and Tu’s Differential Forms in Algebraic Topology is a very readable introduction to smooth manifolds and goes far; everyone should read it. Hatcher, Algebraic Topology.

Riemannian geometry: Gallot-Hulin-Lafontaine’s Riemannian Geometry is one of several gentle introductions. Warner’s Foundations of Differentiable Manifolds and Lie Groups is heavier, but is indispensable for giving the only understandable proof of the Hodge theorem for a Riemannian manifold.

Analysis: Royden, Real Analysis. Lieb and Loss, Analysis.

Graduate-level books

Algebraic geometry: Hartshorne, Algebraic Geometry. Griffiths and Harris, Principles of Algebraic Geometry. These are long references, indispensable for the working algebraic geometer (emphasizing algebraic and analytic approaches, respectively). Huybrechts’s Complex Geometry is a good simplification of Griffiths-Harris.

On more specific topics in algebraic geometry, some outstanding books are Mukai, An Introduction to Invariants and Moduli, and Mumford, Abelian Varieties. There are several other great books (both easier and harder) by Mumford. Borel, Linear Algebraic Groups.

Topology: Milnor’s Characteristic Classes and Morse Theory are magnificent books: short, readable, with a tremendous range of applications. Everyone should read them. There are several other great books by Milnor.

McCleary’s A User’s Guide to Spectral Sequences covers a lot of algebraic topology beyond the basics. Thurston, Three-Dimensional Geometry and Topology.

Symplectic geometry: Arnold, Mathematical Methods in Classical Mechanics.

Homological algebra: Brown, Cohomology of Groups, is an excellent book applying topological ideas to algebra. Weibel, An Introduction to Homological Algebra. Benson, Representations and Cohomology (2 vols.) S. MacLane, Categories for the Working Mathematician.

Number theory: Serre, Local Fields, among several other great books. Lang, Algebraic Number Theory. Miyake, Modular Forms. Silverman, The Arithmetic of Elliptic Curves.

Geometric group theory: Serre, Trees. De la Harpe, Topics in Geometric Group Theory, gives quick treatments of a rich variety of topics.

Analysis: Zimmer, Basic Results in Functional Analysis, treats the fundamental topics and applications in a very short space. Krylov, Lectures on Elliptic and Parabolic Equations in H¨older Spaces, is one of the few graduate-level introductions to serious PDE theory. The big reference books are Gilbarg-Trudinger, Elliptic Partial Differential Equations of Second Order, and Evans, Partial Differential Equations.

Dynamical systems: Walters, An Introduction to Ergodic Theory, is a standard short introduction. Hasselblatt and Katok, An Introduction to the Modern Theory of Dynamical Systems, is the standard big reference book.

Collected papers

Everyone interested in algebraic geometry, number theory, and many aspects of topology and group theory should look at Serre’s Oeuvres: Collected Papers. Atiyah’s Collected Papers are fundamental for topology, with links to analysis and differential geometry.

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Lazarsfeld’s good book

Why not think about reading Rob Lazarsfeld’s Positivity in Algebraic Geometry I. Classical Setting: Line Bundles and Linear Series? Two or three people in the past couple of weeks have asked me questions that this book sheds light upon, and I’ve decided that more people need to know about this excellent book.

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