Best Books to Learn Algebraic Geometry, in Order
Algebraic geometry has the worst reputation-to-accessibility ratio in mathematics, mostly because people are pointed straight at Hartshorne and discover that Chapter II assumes a commutative algebra course they have not taken. The subject is actually reachable, provided you approach it the way the literature was written: computationally first, where a variety is a set of polynomial solutions you can manipulate with a computer algebra system; then classically, through curves and surfaces you can picture; then, only once the geography is familiar, through the scheme-theoretic machinery that makes the general theory work. This path is built on that ramp, and it puts the commutative algebra where it belongs — before the schemes, not after them.
Computational Entry
BeginnerLearn what a variety and an ideal are by computing with them — Grobner bases, elimination, the ideal-variety correspondence — before meeting any abstraction.
▸ Study plan for this stage
Pace: Four to six months for Cox, Little and O'Shea's Ideals, Varieties, and Algorithms, working through chapters one to four with the exercises. This is a textbook, not a popular account, and the honest unit of measurement is not pages but exercises: two to three sections a week, with every computational
- The ideal-variety correspondence: V(I) and I(V), and the fact that the two operations are not inverse to each other without conditions
- Monomial orderings, and why the choice of ordering changes the answer a computation gives you
- Gröbner bases and Buchberger's algorithm — the whole reason the subject can be computed with at all
- The Nullstellensatz in its weak and strong forms, made concrete: radical ideals correspond to varieties over an algebraically closed field
- Elimination theory, which is what turns 'solve this system' into a mechanical procedure
- Implicitization: converting a parametrised curve or surface into the polynomial equations that cut it out
- Resultants, local rings and free resolutions from the sequel, which are the first genuinely algebraic tools on the path
- The habit of checking an abstract claim against a computed example, which is the single most useful thing this stage installs
- State the strong Nullstellensatz precisely, including the hypothesis on the field, and give an example over the reals showing why the hypothesis is needed.
- Why does a Gröbner basis solve the ideal membership problem, and what would you do without one?
- How does lexicographic ordering produce an elimination ideal, and why does that constitute solving a system?
- Given a parametrisation of a surface, describe the procedure that recovers its implicit equation, and say where it can fail.
- What does a free resolution tell you about an ideal that a generating set does not?
- Run Buchberger's algorithm by hand, on paper, on at least three of the small two- and three-variable examples in chapter two of Cox, Little and O'Shea. Doing it once mechanically is what makes every later assertion about Gröbner bases mean something.
- Take the same ideal and compute its Gröbner basis under lex, grlex and grevlex orderings, then compare the sizes of the answers. The blow-up under lex is the practical fact the theory does not convey.
- Solve one of the book's polynomial systems twice: once by hand using elimination, and once in Macaulay2 or Singular. Reconcile the two answers, including any solutions at infinity the elimination step lost.
- Implicitize the parametrised surfaces given in chapter three and verify each answer by substituting the parametrisation back into the implicit equation and confirming it vanishes identically.
- From Using Algebraic Geometry, work one full resultant computation and one free resolution of a small monomial ideal. These are the two objects the later abstract chapters will assume you have physically produced at least once.
Next up: You can now compute with varieties and ideals, which means the geometric language of the next stage — projective space, smoothness, dimension — will attach to objects you have already manipulated rather than to definitions alone.
Cox, Little and O'Shea's book is the right first text for almost everyone: it needs only linear algebra, it makes the Nullstellensatz concrete, and Grobner bases give you something to actually calculate. Everything abstract later on has a computational shadow here that you will keep referring back to.
The same authors' sequel, pushing the computational machinery into resultants, local rings, free resolutions and applications from robotics to statistics. Read it selectively rather than cover to cover — its function on this path is to show that the abstractions ahead are answers to real problems.
The Gentle Bridges
IntermediateMeet the classical geometric objects — projective space, smoothness, dimension, divisors, Riemann surfaces — in short books written specifically to be readable.
▸ Study plan for this stage
Pace: Four to six months for three short books that are short in very different ways. Karen Smith's An Invitation to Algebraic Geometry is about 160 pages, needs almost no prerequisites, and is genuinely readable in a weekend if you do not attempt the exercises — read it that way first, as orientation. Mi
- Projective space, and why compactifying is not a technicality — Bézout's theorem is false without it
- The projective plane curve as the basic object: degree, singular points, smoothness via the Jacobian criterion
- Dimension of a variety, defined several inequivalent-looking ways that turn out to agree
- The 27 lines on a smooth cubic surface, which is Reid's set-piece and the classical result most worth having seen early
- Riemann surfaces: a smooth complex projective curve is a compact Riemann surface, and the same object has an algebraic, an analytic and a topological description
- Genus, and the degree-genus formula for a smooth plane curve
- The Riemann-Roch theorem in the form Kirwan proves it, which is the first genuinely deep statement on this path
- Divisors and linear systems as the bookkeeping that makes Riemann-Roch statable
- Why is Bézout's theorem stated in projective space over an algebraically closed field, and what goes wrong if you drop either condition?
- Give the Jacobian criterion for smoothness and apply it to decide whether a specific plane cubic of your choosing is smooth.
- State the degree-genus formula and use it to give the genus of a smooth plane quartic.
- State Riemann-Roch as Kirwan does. What does each term mean geometrically, and what does the theorem let you compute that you could not before?
- In what sense are the algebraic, analytic and topological descriptions of a complex curve the same object, and where in Kirwan is that equivalence actually established?
- Read Smith's Invitation straight through in one or two sittings without stopping at anything you do not follow, then write a one-page summary of what the subject is about. Reread the same page after Kirwan and note what has changed.
- Work Reid's construction of the 27 lines on the cubic surface in full, including the incidence count. This is the classical computation the entire nineteenth-century subject is organised around and it is done once and remembered.
- Compute the genus of at least five smooth plane curves of different degrees by the degree-genus formula, then confirm two of them independently by triangulating or by an Euler characteristic argument as Kirwan sets up.
- Work every Riemann-Roch example Kirwan gives for curves of genus 0, 1 and 2, and produce the dimension of the space of functions with poles bounded by a specific divisor in each case. Riemann-Roch is a calculating tool and this is how it becomes one.
- Take a singular plane cubic — a node and a cusp — and resolve each by the procedure Kirwan describes, tracking what happens to the genus.
Next up: You now have curves, projective space and one hard theorem in hand, which is exactly the stock of concrete objects the classical treatments will assume when they start generalising.
About 150 pages taking a reader from affine varieties to sheaves and moduli with almost no prerequisites, deliberately sacrificing completeness for narrative. The best possible orientation: read it in a weekend and you will know what the subject is about before committing to a real course.
Short, opinionated, and centred on plane curves and cubic surfaces — the concrete objects the classical theory was built to explain. Reid's asides on what the subject is for are worth as much as the mathematics.
An undergraduate route into curves over the complex numbers, ending at Riemann-Roch with genuinely minimal prerequisites. Read it third here: it is the first book on the path where a serious theorem gets proved in full.
The Classical Theory
IntermediateBuild a working geographical knowledge of varieties, curves and surfaces, and see the classification results that motivated everything abstract that followed.
▸ Study plan for this stage
Pace: A year, realistically, for three substantial graduate texts. Joe Harris's Algebraic geometry — the 1992 first course, catalogued under a bare title, so check the author — is 328 pages of almost pure examples and assumes a solid abstract algebra course plus some complex analysis; three to four months
- The example-first method: Veronese, Segre, rational normal curves, Grassmannians, determinantal varieties — the standard stock every algebraic geometer carries
- Degree and dimension computed for those specific families, which is what makes a general theorem checkable
- The Grassmannian and its Plücker embedding, the first moduli space most people meet
- Divisors, linear systems and maps to projective space — the mechanism by which an abstract curve acquires an embedding
- Riemann-Roch and the Hurwitz formula worked in full in Miranda, over branched coverings
- Sheaves appearing first as bookkeeping for local data, before any scheme-theoretic weight is put on them
- Shafarevich's systematic classical treatment of varieties in projective space: singularities, normality, blow-ups, and the birational point of view
- The classification impulse — what it would even mean to classify curves or surfaces, which is the question that motivated everything abstract that came after
- Compute the dimension and degree of the Veronese embedding of the plane in P^5, and explain what makes the computation typical of Harris's method.
- What does the Plücker embedding do, and what is the dimension of the Grassmannian of lines in P^3?
- State the Hurwitz formula and use it to compute the genus of a specific branched cover of the Riemann sphere.
- In what sense do the algebraic, analytic and topological accounts of a curve in Miranda give the same Riemann-Roch theorem?
- What is a birational map, why is it the right notion of equivalence for a great deal of classical geometry, and where does Shafarevich show that isomorphism is too strong?
- Work at least ten of Harris's example lectures completely, computing the dimension and degree of each variety he introduces. Keep the results on one sheet; this becomes the reference stock you test every later general theorem against.
- Compute the Plücker relations for G(1,3) explicitly and verify that the image is a quadric hypersurface in P^5.
- Work the Hurwitz formula on at least four explicit branched coverings from Miranda, including one with a branch point of order greater than two, and confirm the genus each time.
- Blow up a nodal plane curve at its singular point following Shafarevich's construction, writing down the charts explicitly and checking that the strict transform is smooth.
- Take one variety from Harris's stock and describe it three ways: by equations, as an image of a parametrisation, and by the linear system that embeds it. Being able to move between those three descriptions is the actual skill of this stage.
Next up: You have the classical geography; the next stage supplies the algebra that scheme theory is built out of, deliberately taken before the schemes rather than looked up in a panic afterwards.
Harris teaches by examples — Veronese, Segre, Grassmannians, determinantal varieties — on the principle that you cannot understand a general theory without a stock of objects to test it on. Catalogued simply as Algebraic geometry, so check the author. It is the single best cure for abstraction with nothing underneath it.
The book that shows the same object — a curve — through algebraic, analytic and topological eyes at once, and works Riemann-Roch and the Hurwitz formula properly. The deepest thing on this path that still stays entirely concrete.
Shafarevich's first volume covers varieties in projective space classically and at length, with a geometer's instincts throughout. Read it after Harris: it is the systematic treatment that his example-driven course deliberately is not.
The Commutative Algebra You Actually Need
BeginnerAcquire the algebra that scheme theory is built out of — localisation, primary decomposition, integral extensions, dimension theory — before it is needed rather than after.
▸ Study plan for this stage
Pace: Six months to a year, and this is the stage people skip and then regret. Atiyah and Macdonald's Introduction to Commutative Algebra is 128 pages, which is deceptive: essentially all the real content is in the exercises, and the standard way to use it is a chapter a fortnight with every exercise atte
- Localisation, which is the algebraic operation corresponding to restricting attention to a neighbourhood
- Prime and maximal ideals as the points of the eventual space, and why prime rather than maximal is the right choice
- Primary decomposition and the associated primes of a module — the algebraic content of decomposing a variety into components with multiplicities
- Integral extensions, the going-up and going-down theorems, and Noether normalisation
- Dimension theory: Krull dimension, systems of parameters, and the agreement of several definitions
- Completion and the local rings that describe a singularity's analytic type
- Flatness, which is the algebraic condition that makes families of varieties behave, and which is where Eisenbud's geometric commentary earns its length
- Exact sequences and the elementary homological algebra — Tor and Ext — that Hartshorne's chapter three will assume outright
- What does localising at a prime do to the ideal lattice, and why is the result a local ring?
- State the going-up theorem and explain what it says geometrically about a finite morphism.
- Give the primary decomposition of a specific non-radical ideal of your choosing and say what each primary component means geometrically.
- State Noether normalisation and explain why it gives a proof of the Nullstellensatz.
- What does flatness mean for a family of varieties, and what is the standard example of a non-flat family that misbehaves?
- Work every exercise in chapters one to three of Atiyah and Macdonald. Not most — every one. The book's main text is a summary; the mathematics is in the exercises and reading it without them teaches almost nothing.
- Compute the primary decomposition of at least three explicit ideals by hand, then verify each with Macaulay2 or Singular. The computer agrees or you have made an error, and both outcomes are informative.
- Carry out an explicit Noether normalisation for a specific finitely generated algebra over a field, writing down the change of variables.
- For a chosen singular point on a plane curve, compute the local ring, its completion, and identify the singularity type. This is the calculation that makes 'local' a technique rather than a word.
- Take one construction from Eisenbud — flatness, or the Koszul complex — and write out both the algebraic definition and the geometric statement it corresponds to. Eisenbud gives both; producing the pair yourself is how the book is meant to be used.
Next up: With localisation, primes, dimension and flatness in hand, the definition of a scheme will read as a natural construction rather than an act of gratuitous abstraction.
Atiyah and Macdonald: 128 pages, almost all of the real content in the exercises, and the standard prerequisite for every graduate algebraic geometry course. Work the exercises; there is no other way to use this book.
The much larger companion, written explicitly to supply what a geometer needs, with the geometric meaning of each algebraic construction stated. Use it as the reference alongside Hartshorne rather than reading it linearly.
Schemes and the Graduate Texts
BeginnerMove to the modern language — schemes, sheaves, cohomology — and understand why the generality is worth its cost.
▸ Study plan for this stage
Pace: One to two years, and possibly more. This is the stage where an honest estimate matters most. Shafarevich's Basic Algebraic Geometry 2 is 277 pages and the gentlest possible first look at schemes; two to three months. Eisenbud and Harris's The Geometry of Schemes is 294 pages and should be read imme
- Spec of a ring as a topological space with a structure sheaf, and why nilpotents are kept rather than discarded
- Generic points, and the fact that a non-closed point is a feature rather than a pathology
- Sheaves, sheafification, and the fact that a presheaf of functions is usually not a sheaf
- The functor of points, which is the perspective that makes moduli problems statable
- Fibre products and base change, which is how a family and its fibres are formalised
- Sheaf cohomology via derived functors, Čech cohomology as the computable model, and Serre duality
- Why the generality is worth its cost: schemes over Z let a single argument cover geometry and number theory at once
- Arithmetic surfaces, models of curves and reduction modulo a prime, which is what Liu's book is for
- Describe Spec Z[x] as a topological space — what are its points, and which are closed?
- Give an example of a scheme with nilpotents and explain what geometric information the nilpotents record that a variety would lose.
- What is a generic point, and how does its existence simplify a statement about a family of varieties?
- Compute the cohomology of the twisting sheaves O(n) on projective space, and say why that single computation carries so much of chapter III of Hartshorne.
- What does it mean to reduce a curve modulo a prime, and what does good or bad reduction correspond to geometrically?
- Work the Spec computations at the start of Hartshorne chapter II by hand: Spec of a field, of a discrete valuation ring, of Z, of k[x], of k[x,y], and of Z[x]. Draw each one. Everything abstract in the chapter is a generalisation of a picture you should be able to sketch.
- Work through the nilpotent and non-reduced examples in Eisenbud and Harris — the double point, the ribbon, the degenerating family — and in each case state what the scheme records that the underlying set does not.
- Compute the cohomology of O(n) on P^1 and P^2 from the Čech complex explicitly, with the actual covers and the actual coboundary maps written out, and check your answer against Hartshorne III.5.
- Do the exercises in Hartshorne chapter II, sections 2 and 3, in full. They are the course, they are where the definitions become usable, and skipping them is the single most common reason people stall at chapter III.
- Take one smooth projective curve over Q and follow Liu's procedure to write down a model over Z, then identify at least one prime of bad reduction and describe the fibre there.
Next up: This is the end of the path and the beginning of the subject: with schemes, cohomology and a model of a curve over Z in hand you are equipped to read current papers and to choose between the geometric and the arithmetic directions the field runs in.
The second volume, covering schemes, varieties over the complex numbers and the topological picture. The gentlest possible first encounter with schemes, because you already have the first volume's classical objects to translate.
Eisenbud and Harris on what a scheme actually is and why anyone wanted one — nilpotents, generic points, functors of points — worked through examples rather than axioms. Read it immediately before Hartshorne; it is the book that makes Chapter II survivable.
The standard graduate text and the common language of the field: varieties, then schemes, then cohomology, then curves and surfaces. It is famously terse and the exercises are the course. Catalogued as Algebraic geometry — Hartshorne is the 1977 one.
The arithmetic direction: schemes over Dedekind rings, models of curves, reduction. Read it after Hartshorne if you are heading toward number theory, which is where a large share of modern algebraic geometry actually goes.
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