Reading Hartshorne without losing your mind
Most people approach this book expecting it to hand-feed them the theory. It doesn't. The first time I tried working through Chapter II, Section 2, I spent three days on Exercise 2.4 about proving that a certain morphism was proper. The hint in the back says "use the valuative criterion." It does not explain what the valuative criterion is, or why you would think of it, or how to actually apply it to anything concrete. I ended up going back to Mumford's Red Book and reading the chapter on properness twice before the piece clicked. That's just how it goes. Published in 1977, this is the book that became the default graduate text after Serre's FAC and Grothendieck's EGA made scheme theory unavoidable. It covers commutative algebra foundations, then schemes, cohomology, curves, and surfaces. The exercises are where most of the learning happens, and they range from routine computations to things that have haunted students for decades. It is not a gentle introduction. It assumes you already know what you're doing and will fill in the gaps yourself. I've taught from this book at least six times across different institutions. The pattern is always the same: students read the theorems, feel confident, open the exercises, and immediately encounter a wall. The wall isn't the theorem — it's the gap between the statement and the application. Hartshorne is excellent at stating things precisely. He is not always precise about the path from A to B.
The chapters that matter and the ones you can skip
Chapter I is commutative algebra. If you've seen it before, skim it. The material is standard — localizations, Noetherian rings, dimension theory — but the presentation is condensed. Spend time on the exercises though. They reinforce things that will come back later when you're supposed to just "know" them. Chapter II is schemes. This is the heart of the book. Section 2 introduces the functor of points approach, which is where most people get lost because the language shifts without warning. I once had a student spend two weeks convinced she didn't understand anything, when the real problem was that she'd never seen the Yoneda embedding applied to affine schemes before. We spent one afternoon on that and she was fine. Chapter III is cohomology. Sheaf cohomology on schemes. The computation techniques here are what you'll actually use if you go into research. The theory is clean but the exercises in later sections require familiarity with Čech cohomology and spectral sequences that the book doesn't always build up carefully.
Chapters IV and V are curves and surfaces. These are more classical and often more readable. If you're heading toward arithmetic geometry, Chapter IV is essential. If you're leaning toward complex geometry, Chapter V has material that connects directly to Griffiths-Harris, though the proofs take a different route.
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What no one tells you about the exercises
The exercise section is the actual curriculum. Theorems are often stated in a way that makes them look harder than they are, and the exercises strip away the abstraction to reveal the computation underneath. Exercise II.4.3, for example, asks you to show that the normalization of a cuspidal cubic is affine line. The theorem preceding it talks about normalizations in general. The exercise is where you actually see the map t (t², t³) and understand why the cusp is singular in a way that the abstract machinery obscures. Here's a specific problem I ran into last year when preparing a problem set. I assigned Exercise IV.1.6 about computing the genus of a plane quintic with nodes. The answer is 4, using the formula g = (d-1)(d-2)/2 - _p. But several students got 6 because they forgot to subtract the delta invariants properly. I had to walk them through why a node contributes = 1 and a cusp contributes = 1 as well, even though the singularities look different. The book states the formula but doesn't explain why the correction term works the way it does. I ended up writing a one-page addendum that went through the local ring calculation for an ordinary double point versus a cusp. That's the kind of thing you need to supply yourself.
Companion texts that actually help
Reid's Undergraduate Algebraic Geometry is the most useful companion. It covers the same ground as Chapters II and IV but with more examples and fewer gaps. I recommend reading Reid first, then Hartshorne, then coming back to Hartshorne's exercises. The other useful text is Vakil's notes, available for free online. They cover scheme theory with more care about the foundational steps. Reading Vakil alongside Hartshorne Chapter II will save you weeks. For the commutative algebra background, Atiyah-MacDonald is sufficient for most of Chapter I. If you need more depth, Matsumura is the reference. Most people don't need Matsumura for Hartshorne unless they're working through the harder exercises in Chapter I.
Where this book fails you
The biggest gap is computational algebraic geometry. If you want to actually compute with varieties — Groebner bases, explicit resolutions, numerical methods — Hartshorne won't help. You need Cox-Little-Schenck or Eisenbud for that. The book also has almost nothing on arithmetic applications. If your interest is in number theory, you'll need to supplement heavily with works by Silverman or Liu. Another honest limitation: the book assumes familiarity with commutative algebra that many students don't actually have, despite what the prerequisites say. I've seen capable students struggle with Lemma II.3.11 because they hadn't internalized how localization interacts with exact sequences. This isn't Hartshorne's fault — it's a structural issue with the field. But it means you should test your commutative algebra before diving in. The 2013 reprint fixed some typos but not the deeper issues. There are known errors in the older editions, particularly in Chapter V, that have been discussed extensively on MathOverflow. If you encounter a proof that doesn't work, check the errata before assuming you're missing something.
How I actually use this book
When I teach, I assign reading in blocks. Two sections per week, with the requirement that students complete at least three exercises from each section. Not all exercises — just three, chosen for variety. This prevents burnout while ensuring they engage with the material. The books that survive this approach are the ones where students learn to read actively rather than passively. For self-study, the pattern that works best is: read a section, close the book, try to reconstruct the main theorem from scratch, then open the book and check where your reconstruction diverged. The divergence points are your weak spots. Spend time there. The rest you already know. I keep a copy on my desk and reach for it when I need a precise reference for something standard — flatness criteria, dimension formulas, the structure of curves over algebraically closed fields. For learning new material, I turn to newer sources. This book is a reference and a training tool, not a gentle introduction to anything.
