Hilbert scheme

Key takeaways

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

Reference excerpt

In algebraic geometry, a branch of mathematics, a Hilbert scheme is a scheme that is the parameter space for the closed subschemes of some projective space (or a more general projective scheme), refining the Chow variety. The Hilbert scheme is a disjoint union of projective subschemes corresponding to Hilbert polynomials. The basic theory of Hilbert schemes was developed by Alexander Grothendieck (1961). Hironaka's example shows that non-projective varieties need not have Hilbert schemes.

Hilbert scheme of projective space The Hilbert scheme H i l b ( n ) {\displaystyle \mathbf {Hilb} (n)} of P n {\displaystyle \mathbb {P} ^{n}} classifies closed subschemes of projective space in the following sense: For any locally Noetherian scheme S, the set of S-valued points

Hom ⁡ ( S , H i l b ( n ) ) {\displaystyle \operatorname {Hom} (S,\mathbf {Hilb} (n))}

of the Hilbert scheme is naturally isomorphic to the set of closed subschemes of P n × S {\displaystyle \mathbb {P} ^{n}\times S} that are flat over S. The closed subschemes of P n × S {\displaystyle \mathbb {P} ^{n}\times S} that are flat over S can informally be thought of as the families of subschemes of projective space parameterized by S. The Hilbert scheme H i l b ( n ) {\displaystyle \mathbf {Hilb} (n)} breaks up as a disjoint union of pieces H i l b ( n , P ) {\displaystyle \mathbf {Hilb} (n,P)} corresponding to the Hilbert scheme of the subschemes of projective space with Hilbert polynomial P. Each of these pieces is projective over Spec ⁡ ( Z ) {\displaystyle \operatorname {Spec} (\mathbb {Z} )} .

Construction as a determinantal variety Grothendieck constructed the Hilbert scheme H i l b ( n ) {\displaystyle \mathbf {Hilb} (n)} of n {\displaystyle n} -dimensional projective P n {\displaystyle \mathbb {P} ^{n}} space as a subscheme of a Grassmannian defined by the vanishing of various determinants. Its fundamental property is that for a scheme T {\displaystyle T} , it represents the functor whose T {\displaystyle T} -valued points are the closed subschemes of P n × T {\displaystyle \mathbb {P} ^{n}\times T} that are flat over T {\displaystyle T} .

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Hilbert scheme

Start with the simplest possible case. Write down what Hilbert scheme 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 Hilbert scheme 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 Hilbert scheme 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 Hilbert scheme

In research: Hilbert scheme 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 Hilbert scheme 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: Hilbert scheme is common in secondary-school and first-year university syllabi. It links to neighbouring topics Algebraic geometry, Differential geometry, Moduli theory, so understanding it makes those chapters shorter.
In everyday life: Look for Hilbert scheme 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 Hilbert scheme in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Hilbert scheme 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 Hilbert scheme out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Hilbert scheme in simple terms?

In algebraic geometry, a branch of mathematics, a Hilbert scheme is a scheme that is the parameter space for the closed subschemes of some projective space (or a more general projective scheme), refining the Chow variety. The Hilbert scheme is a disjoint union of projective subschemes corresponding…

Why does Hilbert scheme 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 Hilbert scheme?

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 Hilbert scheme.