Algebraic Geometry

Prerequisite path Preparation & skip guidance · Commutative Algebra · Sheaves & gluing

Studying Algebraic Geometry

Algebraic geometry becomes easier to navigate when equations, coordinate rings and local functions remain visible together. Milne starts with varieties over an algebraically closed field; Vakil develops schemes and makes sheaf language part of the working machinery. Choose according to the gap you need to close: geometric examples and regular maps, or the passage from rings to spaces that can retain nilpotents. Reading both from the beginning is unnecessary.

What to work on

Translate between affine algebraic sets and radical ideals over an algebraically closed field.
Compute coordinate rings and describe regular maps through ring homomorphisms.
Compute a Zariski tangent space and detect a singular point.
Explain what additional information a scheme's structure sheaf records.

What linearization sees

Smooth at the origin

Singular at the origin

These drawings show real points, not the whole complex curve. For

F=y-x^2

, the linear part at the origin is

y

, giving a tangent line. For

F=y^2-x^3

, the linear part vanishes: its Zariski tangent space is the entire plane (shaded), even though the visible cusp has a horizontal limiting tangent direction.

Enlarge: What linearization sees

A starting text

Algebraic Geometry — James S. Milne. Read chapter 2, sections a-e and h-j on algebraic sets, Nullstellensatz, irreducibility and regular maps. Then use chapter 4, sections a and d on tangent spaces. Read chapter 3, sections a-f only when ready for the sheaf description of affine varieties.

Choosing what to read

Start by translating the parabola and a singular plane curve between equations and coordinate rings. Compute a tangent space before accepting its abstract definition. In the scheme route, verify the sheaf axioms on continuous functions, then inspect distinguished opens in an affine spectrum. Keep localization and gluing calculations beside the definitions; return to commutative algebra when those calculations become formal gestures.

James S. Milne

Build the equation-to-variety dictionary before general schemes.

How it reads: Varieties first, with coordinate rings and regular maps.
What it expects: Rings, ideals, fields and localization; some module language.
What to keep in mind: The initial selection works over an algebraically closed field; it is not a general scheme course.
Practice: Chapter 2 exercises, printed p. 58; chapter 4 exercises, p. 98. Public solution coverage has not been audited.

Complete author-hosted PDF.

Edition, reading range and source

Version 6.10: chapter 2, sections a-e and h-j; chapter 4, sections a and d. Optional sheaf bridge: chapter 3, sections a-f.

Source · Checked 2026-10-17

The Rising Sea: Foundations of Algebraic Geometry

Ravi Vakil

Move from gluing functions to affine spectra and the structure sheaf.

How it reads: Schemes through sheaves, examples and embedded exercises.
What it expects: Commutative algebra, topology and category language.
What to keep in mind: A long development; the selected opening does not cover cohomology or intersection theory.
Practice: Work Exercises 2.2.A and 2.2.D-E on presheaves and sheaves. Public worked solutions not verified.

Complete October 21, 2025 author edition on Stanford; distinct from the catalogue's July 2024 archive.

Edition, reading range and source

October 21, 2025 edition: sections 2.1-2.4, 3.1-3.5 and 4.1-4.3. Review sections 1.1-1.3 if category language is unfamiliar.

Source · Checked 2026-10-17

Other ways into the subject

The Rising Sea: Foundations of Algebraic Geometry, October 21, 2025 draft — Ravi Vakil. Sections 2.1-2.4 on sheaves and stalks; 3.2-3.5 on Spec and its topology; 4.1-4.3 on the structure sheaf, nilpotents and schemes. Take the scheme continuation after localization and sheaves; this is a focused selection from a long draft.

Practice

Affine varieties and tangent-space exercises. End-of-chapter exercises for chapter 2, printed page 58; chapter 3, page 80; and chapter 4, page 98.. Exercises provided; public solutions not verified.

Try these before moving on

Try a proof or calculation before opening the hint. These solutions address the checks below, not the exercises in the linked textbooks.

Over C, compute the coordinate ring and irreducible components of V(xy) in affine two-space and identify its minimal prime ideals.
A hint
A product vanishes when at least one factor vanishes. For the algebra, use that every prime ideal containing $xy$ contains $x$ or $y$ .

Work through the solution
The zero set is the union $V(x)\cup V(y)$ of the two coordinate lines. Its coordinate ring is $A=\mathbb C[x,y]/(xy)$ : the ideal $(xy)=(x)\cap(y)$ is radical. Each line is irreducible, since its coordinate ring is a polynomial ring in one variable over a field.

The minimal prime ideals of $A$ are $(\bar x)$ and $(\bar y)$ . Their quotients are respectively $\mathbb C[y]$ and $\mathbb C[x]$ , so they are prime. Any prime containing $xy$ before passing to the quotient contains one of $x,y$ , which proves minimality and exhausts the possibilities.

This ring has zero divisors, because $\bar x\bar y=0$ with both factors nonzero, but it has no nonzero nilpotents. Reducibility and nonreducedness are different issues.
Compute the Zariski tangent spaces of y squared = x cubed at (0,0) and (1,1), compare their dimensions with the curve dimension, and identify which point is singular.
A hint
For a plane curve $F(x,y)=0$ , the tangent vectors at $p$ lie in the kernel of the linear map $dF_p$ . What happens when both partial derivatives vanish?

Work through the solution
Let $F(x,y)=y^2-x^3$ . At $p=(a,b)$ , a tangent vector $(u,v)$ satisfies

$-3a^2u+2bv=0.$

At the origin, this imposes no linear condition, so $T_{(0,0)}X=\mathbb C^2$ has dimension two. The curve itself has dimension one; the origin is singular.

At $(1,1)$ , the condition is $-3u+2v=0$ , hence $T_{(1,1)}X=\operatorname{span}((2,3))$ has dimension one, and the point is smooth. The corresponding affine tangent line in the drawing plane is $y-1=\tfrac32(x-1)$ .

At the cusp, the visible limiting tangent direction is horizontal, but the Zariski tangent space is the whole ambient plane. The latter is defined by the linear terms of the equations, not by choosing the line that best fits a sketch.
Let k be a field and X=Spec(k[t]). On U=D(t) take s_U=t squared/t; on V=D(1-t) take s_V=(t-t squared)/(1-t). Prove U and V cover X, verify agreement on the overlap, and identify the unique global regular function they define.
A hint
A prime ideal cannot contain both t and 1-t. Simplify each section in its localization, then use the sheaf gluing axiom.
Work through the solution
A point of $X$ is a prime ideal of $k[t]$ . If it were outside both $D(t)$ and $D(1-t)$ , it would contain both $t$ and $1-t$ , hence their sum $1$ , which is impossible. Thus $U\cup V=X$ , and $U\cap V=D(t(1-t))$ .

In $k[t]_t$ , $s_U=t^2/t=t$ . In $k[t]_{1-t}$ , $s_V=t(1-t)/(1-t)=t$ . Their restrictions to $k[t]_{t(1-t)}$ are therefore equal. The structure sheaf gluing axiom gives a unique global section, explicitly the polynomial $t\in k[t]=\Gamma(X,\mathcal O_X)$ . Existence can be seen directly by restricting this polynomial. For uniqueness, any other polynomial with the same restriction to $U$ differs from $t$ by an element becoming zero in $k[t]_t$ . Some power of $t$ annihilates that difference; since $k[t]$ is a domain, the difference is zero. Local expressions need agreement on overlaps; different written denominators alone are no obstruction.

If this was the sticking point:
- Commutative Algebra — Review principal localizations and equality of fractions.
- Point-Set Topology — Review open covers and restrictions before the sheaf gluing axiom.

Already comfortable with this? Skip affine-variety review if you can do both checks and explain the reversed direction of maps on coordinate rings. Enter Vakil's scheme selection directly only if localization, sheaf gluing and stalks are already familiar; nilpotents still need attention even when the underlying spaces look familiar.

Local fractions, one global function

For any field

k

, the principal opens

D(t)

and

D(1-t)

cover

\operatorname{Spec}(k[t])

: no prime contains both

t

and

1-t

. The local expressions equal

t

in their respective localizations and agree on

D(t(1-t))

. They glue to the unique global regular function

t

. The overlapping ellipses depict cover membership schematically, not the geometry of the spectrum.

Enlarge: Local fractions, one global function

Review the prerequisites · Compare the full subject collection

Where this can lead

Commutative Algebra — Review prime ideals and localization when distinguished affine opens or local rings are unclear.
Homological Algebra — Study exact sequences and derived functors before a later course in sheaf cohomology.

Reading-source dates

Algebraic Geometry · 2026-10-16
The Rising Sea: Foundations of Algebraic Geometry, October 21, 2025 draft · 2026-10-16

Reading choices revised 2026-10-17.

Suggest an edit · Report a correction

Keep localization beside the geometry

The useful test here is not how many definitions you can repeat, but whether a fraction in a ring tells you where a function is defined. Keep one affine line in view while reading about sheaves.

What to read for this step

Milne remains the starting text for the variety route: chapter 2, sections a–e and h–j, with chapter 4, sections a and d for tangent spaces. Use the existing chapter exercises rather than reading every preliminary chapter again.

For the scheme continuation, use Vakil’s October 21, 2025 draft: §§2.1–2.4 for sheaves and stalks, §§3.2–3.5 for spectra and topology, then §§4.1–4.3 for the structure sheaf and schemes. The route introduction cites the November 18, 2017 early, out-of-date draft. These are different files: the 2025 section suggestions here do not replace or renumber that citation.

Starting text · Compare reading approaches · Exercises and available answers

One example to keep in view

Put $A=k[t]$ , where $k$ is a field. The opens $D(t)$ and $D(1-t)$ cover $\operatorname{Spec}(A)$ because $(t,1-t)=A$ . On their overlap, compare sections in $A_{t(1-t)}$ , not by comparing the written denominators.

The fractions $t^2/t$ on $D(t)$ and $(t-t^2)/(1-t)$ on $D(1-t)$ both restrict to $t$ . The existing drawing and worked check make this calculation explicit. Try the check before opening its hint; then distinguish the existence of the glued section from its uniqueness.

When to review, when to continue

If the fractions and the cover argument are routine, skip this warm-up and work on stalks or the structure sheaf. If you can compute tangent spaces but cannot restrict a section, review localization and sheaf gluing first; solving the cusp check alone does not settle that gap.

Editorial choice and revision

Milne supplies concrete variety calculations; the selected Vakil sections explain the passage to schemes. They serve different gaps, so neither book is assigned cover to cover.

Added 2026-10-19. AI-assisted drafting; independent human mathematical review has not been recorded. Existing reading-source dates are unchanged.

The study guide above is translated; catalogue annotations below retain their original language.

Algebraic Geometry

PDF

J. S. Milne · Course notes · Introductory

A first graduate course centered on abstract algebraic varieties, providing a bridge toward schemes.

Preparation: Fields, Galois theory and commutative algebra.

Open official PDF About / terms

The Rising Sea: Foundations of Algebraic Geometry

PDF

Ravi Vakil · Textbook / author edition · Advanced

A broad, exercise-rich development of schemes and their geometry. The October 21, 2025 author edition closely matches the published text; substantial algebra is essential.

Preparation: Commutative algebra; topology; category language.

Open official PDF About / terms Open older archive Download older archive

July 27, 2024 archive · CC BY-NC-ND 3.0; unchanged noncommercial copy.

The Stacks Project

Online text

The Stacks Project Authors · Reference work · Advanced

An extensive open reference on schemes and related algebra. Stable tags are useful for precise citations; selective consultation is more practical than cover-to-cover reading.

Preparation: Graduate algebraic geometry.

Read full reference

Further print reading (3 retained references)

From the original catalogue. These are bibliographic references, not free full-text links; an authorized complete free edition is not currently linked here.

Algebraic Geometry — Robin Hartshorne
The Red Book of Varieties and Schemes — David Mumford
Algebraic Curves: An Introduction to Algebraic Geometry — William Fulton (the formerly linked university URL is currently unavailable)

Additional reading

Search the complete catalogue.

Graduate reading

Lectures on an Introduction to Grothendieck’s Theory of the Fundamental Group

J. P. Murre · Long lecture notes · Graduate · English · PDF

Develops the algebraic fundamental group via finite étale covers. Historical notation may differ from current treatments. Individual-use access only; do not redistribute.

Preparation: Schemes; Galois theory.

Use: Classical preparation for étale geometry.