Algebraic Geometry: Varieties and Schemes

Algebraic geometry is the study of solutions to systems of polynomial equations. The objects that arise from such solution sets are called algebraic varieties. Although varieties capture much of the classical intuition, modern research often prefers the more flexible language of schemes. This page gives a concise overview of both concepts, explains how they differ, and shows why the scheme-theoretic point of view is indispensable in contemporary mathematics.

1. Classical Algebraic Varieties

A variety (in the sense of classical algebraic geometry) is a subset \(V\subseteq \mathbb{A}^n_k\) defined as the common zero set of a collection of polynomials \[V = \{\,x\in k^n \mid f_1(x)=\dots =f_r(x)=0\,\}.\]Here \(k\) is an algebraically closed field (most often the complex numbers). Varieties are endowed with the Zariski topology: the closed sets are exactly the algebraic sets defined by polynomial equations. A variety is called affine if it can be presented as such a zero set inside an affine space \(\mathbb{A}^n_k\); it is projective if it lives in projective space \(\mathbb{P}^n_k\) and is cut out by homogeneous equations.

The algebra of polynomial functions on an affine variety \(V\) is the coordinate ring\[k[V] = k[x_1,\dots,x_n]/I(V),\]where \(I(V)\) is the ideal of all polynomials vanishing on \(V\). Many geometric properties can be read directly from this ring: for example, \(V\) is irreducible precisely when \(k[V]\) is an integral domain. Morphisms of varieties correspond to \(k\)-algebra homomorphisms between their coordinate rings.

2. Limitations of the Classical Theory

Despite its elegance, the classical notion of variety has several drawbacks. First, it forces the ground field to be algebraically closed; over fields like \(\mathbb{Q}\) or \(\mathbb{R}\) one cannot describe many natural objects as varieties without passing to their algebraic closures. Second, varieties cannot encode infinitesimal information: the Zariski topology is so coarse that it cannot distinguish a double point from a smooth point. Finally, many operations in algebraic geometry (e.g., taking fibers, completing along a subvariety, or forming intersections with multiplicities) naturally lead to objects that are not varieties in the classical sense.

3. Schemes: A Modern Framework

A scheme is a space that locally looks like the spectrum of a commutative ring. The basic building block is\[\operatorname{Spec} A = \{\text{prime ideals of }A\}\]equipped with the Zariski topology and a structure sheaf \(\mathcal{O}_{\operatorname{Spec} A}\) that records, for each open set, the ring of functions on that set. A scheme is obtained by gluing together such affine pieces along open subsets, exactly as manifolds are built from charts.

Schemes retain the algebrageometric dictionary: points correspond to prime ideals, and the stalk of the structure sheaf at a point \(p\) is the localized ring \(A_{p}\). In particular, a point can have a nonreduced local ring, which captures nilpotent infinitesimal directions that are invisible in the classical picture. When every local ring of a scheme is reduced (has no nilpotents) and each irreducible component is of finite type over a field, the scheme is called a variety. Hence varieties are simply a special class of schemes.

4. From Varieties to Schemes

Given an affine variety \(V\subseteq\mathbb{A}^n_k\), its coordinate ring \(k[V]\) is reduced, so \(\operatorname{Spec} k[V]\) has no nilpotent elements. The underlying topological space of \(\operatorname{Spec} k[V]\) coincides with the Zariski set of \(V\). Consequently the variety can be recovered as the reduced scheme\[V_{\text{red}} = \operatorname{Spec}(k[V]).\]Conversely, any reduced scheme of finite type over \(k\) determines a classical variety. The passage from varieties to schemes therefore amounts to forgetting the reduced condition, allowing us to work with rings that may contain nilpotent elements.

One particularly important example is the double line in the plane. As a variety it would be the line \(\{y=0\}\) in \(\mathbb{A}^2_\mathbf{C}\). As a scheme we can define\[X = \operatorname{Spec}\bigl(\mathbf{C}[x,y]/(y^2)\bigr).\]The coordinate ring contains the nilpotent class of \(y\), and the scheme \(X\) remembers that the line has multiplicity two. This extra data is essential for intersection theory, where counting intersections with multiplicity is a fundamental requirement.

5. Why Schemes Matter

Schemes provide a language that works uniformly over arbitrary base rings, not only algebraically closed fields. This flexibility is crucial in number theory: the spectrum \(\operatorname{Spec}\mathbb{Z}\) is a scheme whose points correspond to prime numbers, and many deep results (e.g., the Weil conjectures) are phrased in terms of schemes over \(\mathbb{Z}\) or over finite fields.

Moreover, the scheme-theoretic approach makes many constructions canonical. For instance, the fiber of a morphism \(f : X \to Y\) over a point \(y\in Y\) is defined as the fiber product\[X_y = X \times_Y \operatorname{Spec} \kappa(y),\]where \(\kappa(y)\) is the residue field at \(y\). This definition works even when \(y\) is a nonclosed point, something that has no analogue for varieties.

Finally, modern developments such as derived algebraic geometry, motivic homotopy theory, and the theory of stacks all rely on the scheme framework. Schemes are the first step toward these more sophisticated objects, and understanding them is a prerequisite for exploring the frontier of contemporary algebraic geometry.

6. Illustrative Examples

• Affine line: \(\mathbb{A}^1_k = \operatorname{Spec} k[t]\). Its points correspond to prime ideals \((0)\) (the generic point) and \((t-a)\) for each \(a\in k\) (closed points).
• Projective line: constructed by gluing two copies of \(\operatorname{Spec} k[t]\) along the open subset where \(t\neq 0\). The resulting scheme is \(\mathbb{P}^1_k\), a compactification of \(\mathbb{A}^1_k\).
• Spec of a nonreduced ring: \(\operatorname{Spec}(k[\epsilon]/(\epsilon^2))\) is a single point whose local ring contains a nilpotent \(\epsilon\). This fat point encodes an infinitesimal thickening of a usual point.
• Intersection of curves: The schemetheoretic intersection of \(V(f)\) and \(V(g)\) in \(\mathbb{A}^2_k\) is \(\operatorname{Spec}(k[x,y]/(f,g))\). Its length (dimension of the coordinate ring as a \(k\)-vector space) gives the intersection multiplicity.

7. Further Reading

For a gentle introduction, see Hartshornes Algebraic Geometry (Chapter II) or the notes by Vakil The Rising Sea. For a deeper dive into schemes, Eisenbud and Harriss Geometry of Schemes provides many concrete examples. Online resources such as the Stacks Project give a comprehensive reference for all technical aspects.

Understanding varieties and schemes is a stepping stone toward many advanced topics. Whether you are interested in pure geometry, arithmetic applications, or modern homotopical methods, the language of schemes offers a unifying framework that can accommodate all of them.