Toward algebraic geometry

What information disappears when you draw the solutions of an equation? Over the complex numbers, $x=0$ and $x^2=0$ give the same visible point. Their quotient rings differ: in $\mathbb{C}[x]/(x^2)$, the class of $x$ is nonzero but has square zero. This tiny example gives algebraic geometry a reason to look beyond pictures. Equations carry information about infinitesimal behavior, and a useful geometry should retain it. Vakil develops precisely this example in The Rising Sea, §4.2.

There are other questions worth carrying into the subject. How does a curve change when a parameter varies? When can a rationally defined map extend across a missing point? Why should two equations meet in a predictable number of points once multiplicities and points at infinity are included? These questions connect computation with classification. Drawing a few curves helps formulate them; local rings, projective space and families provide languages in which their answers become stable under changes of coordinates.

A first route through schemes

Start with Ravi Vakil's The Rising Sea, using the freely available November 18, 2017 draft linked here. It is an earlier version, explicitly marked as out of date, rather than the published book. Begin with the reader guidance in §0.1, then use Chapters 2–4 to connect sheaves, prime ideals and affine schemes. Treat the exercises as the main work. A good early milestone is to explain what localization changes and what a structure sheaf records, using a polynomial ring you can calculate with.

The preparation is mostly usable algebra: ideals, quotient rings, modules, localization and some field theory. You also need open sets, continuity and compactness as proof tools. Test yourself by describing the prime ideals of $\mathbb{C}[x]$ and comparing localization at a prime with quotienting by it. If the algebraic tools listed above are familiar, begin reading and review individual gaps as they arise. If quotients still feel like symbolic manipulation, work through examples before adding sheaf notation. Category theory can initially be learned through the universal properties actually used.

Varieties first, or a reference?

For a different entry, J. S. Milne's Algebraic Geometry develops varieties before schemes. Chapter 2 builds the dictionary between algebraic sets and ideals; Chapters 3–4 introduce sheaves, affine varieties and local geometry. Choose this alternative if you want geometric objects over algebraically closed fields to become familiar before generalizing the ambient language. Keep one curve in view while reading: compare its coordinate ring, regular functions on open subsets and tangent space at a singular point.

If you already know the elementary examples and want a precise reference, use the Stacks Project's definition of schemes, Tag 01II, following its links backward when a definition is missing. Its example of the punctured affine plane shows that being covered by affine pieces does not make a space affine. This is an effective check on understanding local versus global constructions, though the project's compact reference style makes it a less forgiving first course.

What the algebra is buying you

A common misconception is that the subject begins only after every algebraic prerequisite has been completed. Another is that a scheme is just a more elaborate set of solutions. The first delays contact with geometry; the second loses the ringed structure that motivated the construction. Progress means translating between equations, functions and maps, then knowing which translation a problem needs.

After affine and projective examples, choose a question to organize further study. Curves lead toward divisors and Riemann–Roch; varying objects leads toward moduli; equations over arithmetic rings lead toward arithmetic geometry. Cohomology and homological algebra become useful when a concrete global problem calls for them. These branches share foundations without forming one compulsory reading sequence.

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