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Matsusaka's big theorem

Matsusaka's big theorem is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Matsusaka's big theorem rather than just read about it. In short: In algebraic geometry, given an ample line bundle L on a compact complex manifold X, Matsusaka's big theorem gives an integer m, depending only on the Hilbert polynomial of L, such that the tensor power Ln is very ample for n ≥ m. The theorem was proved by Teruhisa Matsusaka in 1972 and named by Lieberman and Mumford in 1975.

Key takeaways

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

Reference excerpt

In algebraic geometry, given an ample line bundle L on a compact complex manifold X, Matsusaka's big theorem gives an integer m, depending only on the Hilbert polynomial of L, such that the tensor power Ln is very ample for n ≥ m. The theorem was proved by Teruhisa Matsusaka in 1972 and named by Lieberman and Mumford in 1975. The theorem has an application to the theory of Hilbert schemes.

Notes

Worked examples

Example 1 — a first encounter with Matsusaka's big theorem

Start with the simplest possible case. Write down what Matsusaka's big theorem 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 Matsusaka's big theorem 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 Matsusaka's big theorem 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 Matsusaka's big theorem

In research
Matsusaka's big theorem 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 Matsusaka's big theorem 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
Matsusaka's big theorem is common in secondary-school and first-year university syllabi. It links to neighbouring topics Algebraic geometry stubs, Theorems in algebraic geometry, so understanding it makes those chapters shorter.
In everyday life
Look for Matsusaka's big theorem 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.
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How to study Matsusaka's big theorem in 20 minutes

  1. Read the reference excerpt below once, without taking notes.
  2. Close the page and write down what Matsusaka's big theorem means in your own words.
  3. Compare your version with the excerpt and mark what you missed.
  4. Work through the three examples above with pen and paper.
  5. Explain Matsusaka's big theorem out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Matsusaka's big theorem in simple terms?

In algebraic geometry, given an ample line bundle L on a compact complex manifold X, Matsusaka's big theorem gives an integer m, depending only on the Hilbert polynomial of L, such that the tensor power Ln is very ample for n ≥ m. The theorem was proved by Teruhisa Matsusaka in 1972 and named by Li…

Why does Matsusaka's big theorem 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 Matsusaka's big theorem?

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 Matsusaka's big theorem.

Tags

  • Algebraic geometry stubs
  • Theorems in algebraic geometry

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