Scheme-theoretic intersection

Key takeaways

Scheme-theoretic intersection belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Scheme-theoretic intersection to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Scheme-theoretic intersection from memory before moving on to harder problems.

Reference excerpt

In algebraic geometry, the scheme-theoretic intersection of closed subschemes X, Y of a scheme W is X × W Y {\displaystyle X\times _{W}Y} , the fiber product of the closed immersions X ↪ W , Y ↪ W {\displaystyle X\hookrightarrow W,Y\hookrightarrow W} . It is denoted by X ∩ Y {\displaystyle X\cap Y} . Locally, W is given as Spec ⁡ R {\displaystyle \operatorname {Spec} R} for some ring R and X, Y as Spec ⁡ ( R / I ) , Spec ⁡ ( R / J ) {\displaystyle \operatorname {Spec} (R/I),\operatorname {Spec} (R/J)} for some ideals I, J. Thus, locally, the intersection X ∩ Y {\displaystyle X\cap Y} is given as

Spec ⁡ ( R / ( I + J ) ) . {\displaystyle \operatorname {Spec} (R/(I+J)).}

Here, we used R / I ⊗ R R / J ≃ R / ( I + J ) {\displaystyle R/I\otimes _{R}R/J\simeq R/(I+J)} (for this identity, see tensor product of modules#Examples.) Example: Let X ⊂ P n {\displaystyle X\subset \mathbb {P} ^{n}} be a projective variety with the homogeneous coordinate ring S/I, where S is a polynomial ring. If H = { f = 0 } ⊂ P n {\displaystyle H=\{f=0\}\subset \mathbb {P} ^{n}} is a hypersurface defined by some homogeneous polynomial f in S, then

X ∩ H = Proj ⁡ ( S / ( I , f ) ) . {\displaystyle X\cap H=\operatorname {Proj} (S/(I,f)).}

If f is linear (deg = 1), it is called a hyperplane section. See also: Bertini's theorem. Now, a scheme-theoretic intersection may not be a correct intersection, say, from the point of view of intersection theory. For example, let W = Spec ⁡ ( k [ x , y , z , w ] ) {\displaystyle W=\operatorname {Spec} (k[x,y,z,w])} be the affine 4-space and X, Y closed subschemes defined by the ideals ( x , y ) ∩ ( z , w ) {\displaystyle (x,y)\cap (z,w)} and ( x − z , y − w ) {\displaystyle (x-z,y-w)} . Since X is the union of two planes, each intersecting with Y at the origin with multiplicity one, by the linearity of intersection multiplicity, we expect X and Y intersect at the origin with multiplicity two. On the other hand, one sees the scheme-theoretic intersection X ∩ Y {\displaystyle X\cap Y} consists of the origin with multiplicity three. That is, a scheme-theoretic multiplicity of an intersection may differ from an intersection-theoretic multiplicity, the latter given by Serre's Tor formula. Solving this disparity is one of the starting points for derived algebraic geometry, which aims to introduce the notion of derived intersection.

Proper intersection Let X be a regular scheme and V, W closed integral subschemes. Then an irreducible component P of V ∩ W := V × X W {\displaystyle V\cap W:=V\times _{X}W} is called proper if the inequality (due to Serre):

codim ⁡ ( P , X ) ≤ codim ⁡ ( V , X ) + codim ⁡ ( W , X ) {\displaystyle \operatorname {codim} (P,X)\leq \operatorname {codim} (V,X)+\operatorname {codim} (W,X)}

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Scheme-theoretic intersection

Start with the simplest possible case. Write down what Scheme-theoretic intersection claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Scheme-theoretic intersection before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Scheme-theoretic intersection ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Scheme-theoretic intersection

In research: Scheme-theoretic intersection appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Scheme-theoretic intersection in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Scheme-theoretic intersection is common in secondary-school and first-year university syllabi. It links to neighbouring topics Algebraic geometry, so understanding it makes those chapters shorter.
In everyday life: Look for Scheme-theoretic intersection outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Scheme-theoretic intersection in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Scheme-theoretic intersection means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Scheme-theoretic intersection out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Scheme-theoretic intersection in simple terms?

In algebraic geometry, the scheme-theoretic intersection of closed subschemes X, Y of a scheme W is X × W Y {\displaystyle X\times _{W}Y} , the fiber product of the closed immersions X ↪ W , Y ↪ W {\displaystyle X\hookrightarrow W,Y\hookrightarrow W} . It is denoted by X ∩ Y {\displaystyle X\cap Y}…

Why does Scheme-theoretic intersection matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Scheme-theoretic intersection?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Scheme-theoretic intersection.