Start with Ideals, varieties, and algorithms by David Cox, John Little and Donal O'Shea. It assumes linear algebra and a little abstract algebra, introduces the rest as it goes, and it is computational — Groebner bases, elimination, solving actual polynomial systems — so you finish with objects you can manipulate rather than definitions you can recite. If you want the geometric picture faster and with fewer computations, An Invitation To Algebraic Geometry by Karen Smith and co-authors is a short, deliberately informal overview that sketches where the subject goes without proving everything, and Undergraduate algebraic geometry by Miles Reid is a slim, cheap, classical treatment of varieties that is one of the best-value mathematics books in print. The same three authors wrote a sequel, Using algebraic geometry, which pushes the computational side further into resultants, modules and applications; it is a graduate-level book and optional unless you want the algorithmic direction.
Be realistic about the scale. This is graduate mathematics, and the standard texts here are multi-year books, not multi-week ones. Nobody reads Hartshorne over a summer. The usual failure is jumping to the graduate text without the commutative algebra it silently assumes, then concluding you are not able to do the subject. You need the prerequisites, and they are listed below for each book.
Curves, and the classical picture
Before schemes, spend time on curves, where the geometry is still visible. Complex Algebraic Curves by Frances Kirwan assumes some complex analysis and basic topology and works over the complex numbers, where curves are Riemann surfaces. Algebraic curves and Riemann surfaces by Rick Miranda covers the same territory from both sides at once — the complex-analytic and the algebraic — and it is the standard bridge between them; expect a full graduate course of work.
Algebraic geometry by Joe Harris is the example-driven first course: it is packed with concrete varieties, Grassmannians, determinantal loci and constructions, and it is deliberately light on foundational proofs. It is the best book here for developing intuition about what the objects look like, and a poor one for learning to prove things rigorously. Note that Harris's title and Hartshorne's are identical, so always check the author.
Basic Algebraic Geometry 1 by Igor Shafarevich is the classical treatment of varieties in projective space, written with unusual care about motivation, and it is the gentler alternative to Harris if you want proofs. Its companion, Basic Algebraic Geometry 2, moves on to schemes and complex manifolds.
The commutative algebra you cannot skip
Introduction to commutative algebra by Michael Atiyah and Ian Macdonald is about 130 pages and is the standard prerequisite. It is famously terse and a large fraction of the content lives in the exercises, which are not optional. Work through it properly and Hartshorne's Chapter II becomes possible; skip it and it does not.
Commutative algebra with a view toward algebraic geometry by David Eisenbud is the opposite kind of book — roughly eight hundred pages, discursive, full of geometric motivation for every algebraic result. It is the reference you consult rather than read cover to cover, and it is the place to go when Atiyah and Macdonald prove something in four lines that you do not believe.
Schemes, and then Hartshorne
The geometry of schemes by David Eisenbud and Joe Harris exists precisely because schemes are usually introduced too abruptly. It is short, motivating, and full of worked examples of what a scheme actually is before the machinery arrives. Read it before, or alongside, the main text.
Algebraic geometry by Robin Hartshorne is the standard graduate text and has been since 1977. Chapter I is classical varieties, Chapter II is schemes, Chapter III is cohomology, and Chapters IV and V apply the machinery to curves and surfaces. It assumes a solid commutative algebra course, it is compressed, and its exercises are legendary both for their difficulty and for containing material the main text needs later. Most people work through it in a seminar or a course rather than alone.
Algebraic Geometry and Arithmetic Curves by Qing Liu is the alternative for anyone heading toward number theory: it develops schemes over Dedekind domains and arithmetic surfaces, covering material Hartshorne leaves out, and many readers find its exposition more patient.
Two widely used texts are not in our catalogue and worth knowing about: Ravi Vakil's notes, The Rising Sea, which are freely available and now a common substitute for Hartshorne, and William Fulton's Algebraic Curves, also free, which makes a good short first course.
Follow the full reading path in order, and expect it to take years rather than months.
Follow the full ordered path here: How to Learn Algebraic Geometry from Books, in Order.
