Algebraic geometry is precise enough to prove deep theorems, and subtle enough that a small mismatch of hypotheses can invalidate an argument without any obvious sign. Most recurring errors are not about missing a clever idea. They are about silently switching languages: treating a geometric statement as if it were a ring-theoretic statement, or treating a scheme-theoretic phenomenon as if it were set-theoretic, or moving between affine and projective worlds without carrying the correct baggage.
This article collects common mistakes that appear again and again in reading, writing, and solving problems in algebraic geometry. Each mistake comes with a concrete diagnostic and a reliable fix.
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Mistake: Forgetting which category you are in
Many confusions begin with a phrase like “let X be a variety.” Different authors use the same word for different categories. Later, you use a theorem whose statement assumes a specific category, and the proof seems to go through anyway, until it does not.
Fix: write the ambient category on the first line of your notes. A good habit is to include a header like:
- base field k, possibly not algebraically closed
- X is a scheme of finite type over k, Noetherian
- reduced and irreducible assumptions: stated explicitly
- separated or proper: stated explicitly
Then, whenever you invoke a theorem, you check it against that header.
Mistake: Confusing points with geometric points
On an affine scheme X = Spec(A), a point is a prime ideal p ⊂ A. A closed point corresponds \to a maximal ideal m, but even “closed point” depends on the base field. Over a non algebraically closed field, a closed point can have residue field k(p) that is a nontrivial extension of k. Many geometric intuitions are really about geometric points, meaning morphisms Spec(k̄) → X for an algebraic closure k̄ of k.
A typical error: you prove something “for all points” and later use it “for all k-rational points.” Over a non algebraically closed field, those are different statements.
Fix: distinguish three notions, and label them in your argument.
| Notion | What it is | What it controls |
|—|—|—|
| scheme point x ∈ X | prime ideal p | local ring O_{X,x} and specialization |
| closed point | maximal ideal m | classical set of “visible” points in Zariski topology |
| k-rational point | morphism Spec(k) → X | solutions with coordinates in k |
Mistake: Treating Spec as if it were a set
A map of rings φ: A → B induces a map of affine schemes f: Spec(B) → Spec(A), but the direction reverses, and the topology and structure sheaf matter. A frequent error is to reason as if f were a function on underlying sets that you can analyze by element chasing alone.
Fix: translate carefully using the algebraic dictionary.
- f is a closed immersion if and only if φ is surjective.
- f is dominant if and only if ker(φ) is contained in the nilradical of A.
- f is an open immersion onto a principal open D(g) when B is isomorphic \to a localization A_g.
If you do not remember a translation, derive it from the definition of pullback on primes.
Mistake: Forgetting nilpotents exist and they matter
Schemes carry nilpotents, and nilpotents can change deformation theory, tangent spaces, and intersection multiplicities. A common error is to treat every scheme as if it were reduced.
A minimal example is the dual numbers k[ε]/(ε^2). Its spectrum has a single point, so set-theoretically it looks like Spec(k). But its structure sheaf remembers ε, which encodes an infinitesimal thickening.
Fix: check reducedness at the start when you intend set-theoretic geometry. If your argument only uses the underlying set of points, either assume reducedness or explicitly pass to the reduced subscheme X_red. When you need tangent space or intersection information, do not pass \to X_red unless you mean to discard multiplicities.
Mistake: Mixing affine and projective habits
Affine geometry is controlled by rings and localization. Projective geometry is controlled by graded rings, homogeneous ideals, and line bundles. A classic error is to attempt to define a map of projective varieties by writing coordinate functions that are not homogeneous of the same degree.
On an integral projective variety over an algebraically closed field, global regular functions are often just constants. If you forget this, you may attempt to use global sections of O_X \to build a morphism that cannot exist.
Fix: choose the correct object of functions. In the projective world, the \right “functions” are sections of line bundles, not global regular functions. When defining a map X → P^n, use a base-point-free linear system, meaning a tuple of global sections of a line bundle L that do not vanish simultaneously.
Mistake: Using Nullstellensatz without checking the field
Hilbert’s Nullstellensatz has hypotheses. Over an algebraically closed field k, radical ideals correspond to varieties, and maximal ideals correspond to points. Over a non algebraically closed field, maximal ideals correspond to closed points with residue field extensions, and the “vanishing ideal equals radical” needs a correct formulation.
Fix: state the version you are using.
- Over algebraically closed k: I(V(I)) = √I.
- Over general k: work with k̄-points or use schemes and residue fields.
If your proof uses “every maximal ideal is (x_1 – a_1, x_2 – a_2, up \to x_n – a_n),” then you are implicitly assuming k is algebraically closed and you should say so.
Mistake: Confusing irreducible with connected
Irreducible is a property of the topology: X is not the union of two proper closed subsets. Connected is weaker: X is not the union of two disjoint open subsets. In Zariski topology, irreducible implies connected, but the converse fails.
Fix: use the correct tool.
- Use irreducibility when you need a generic point and “property holds on a dense open” reasoning.
- Use connectedness when you need to avoid decomposition into disjoint open pieces.
Mistake: Forgetting the local nature of many statements
Many statements about morphisms are local on the source and target. For example, “f is an isomorphism” can be checked on an affine open cover by verifying the induced maps on rings are isomorphisms. Properties like being separated, finite type, or smooth also have local criteria.
Fix: ask whether the property is local. If it is, reduce to the affine case immediately.
Mistake: Losing track of dimension and codimension conventions
Dimension can mean Krull dimension, transcendence degree, analytic dimension, or local dimension. Codimension can be defined as the height of a prime ideal, or as a dimension difference for irreducible components, and these agree under suitable hypotheses.
Fix: attach dimension statements to primes. Instead of saying “codimension of Y in X,” phrase it as “height of the prime defining Y in the local ring at the generic point of Y.”
Mistake: Treating “generic” as if it meant “random”
Generic means “on a dense open \subset,” not “with probability one.” A statement like “for a generic point of X, property P holds” means there is a dense open U ⊂ X such that P holds for all points of U.
Fix: produce the open set. You should be able to describe it as the complement of a closed set defined by an ideal or discriminant.
Mistake: Assuming base change preserves every property you care about
You extend scalars k → K and form X_K = X ×_{Spec(k)} Spec(K). Many properties are stable under base change, but not all. Reducedness can fail under base change, and geometric irreducibility is stronger than irreducibility.
Fix: separate arithmetic from geometry.
- Use geometric properties when needed: “geometrically reduced,” “geometrically irreducible,” “geometrically connected.”
- When applying a theorem after base change, check whether it requires algebraic closure or separable closure.
Mistake: Using sheaves as if they were sets of functions
A sheaf is not simply an assignment U ↦ functions on U. It is an assignment with restriction maps, gluing, and a local characterization.
Fix: remember two standard tests.
- To show a morphism of sheaves is an isomorphism, check it on stalks.
- To show two sections are equal, show they agree on an open cover.
Mistake: Forgetting to check separatedness when gluing
Gluing affine schemes along open subsets produces a scheme. But to show a scheme is separated, you check that the diagonal is a closed immersion.
Fix: use the diagonal criterion early. For a scheme X over S, X is separated over S if and only if the diagonal Δ: X → X ×_S X is a closed immersion.
A worked micro-example: when “same points” is not “same scheme”
Let k be a field and consider A = k[x] and B = k[x, ε]/(ε^2, εx). The map A → B sends x \to x. On points, Spec(B) has the same underlying set as Spec(A). But scheme-theoretically B has extra nilpotents supported at x = 0, because ε is killed by x.
A statement like “Spec(B) → Spec(A) is an isomorphism because it is bijective on points” is false. The local ring at x = 0 has nilpotents in B, but not in A.
Closing perspective
Algebraic geometry is a discipline of controlled translation. You translate a question into the language where it is easiest, you solve it there, and you translate back without losing hypotheses. The mistakes above are all failures of controlled translation. The corresponding fixes are habits: stating the ambient category, labeling points correctly, reducing to affines when a property is local, and respecting nilpotents and line bundles.
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