mathematics

Sumihiro's theorem

Sumihiro's 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 Sumihiro's theorem rather than just read about it. In short: In algebraic geometry, Sumihiro's theorem, introduced by (Sumihiro 1974), states that a normal algebraic variety with an action of a torus can be covered by torus-invariant affine open subsets. The "normality" in the hypothesis cannot be relaxed.

Key takeaways

Sumihiro's 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 Sumihiro's theorem to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Sumihiro's theorem from memory before moving on to harder problems.

Reference excerpt

In algebraic geometry, Sumihiro's theorem, introduced by (Sumihiro 1974), states that a normal algebraic variety with an action of a torus can be covered by torus-invariant affine open subsets. The "normality" in the hypothesis cannot be relaxed. The hypothesis that the group acting on the variety is a torus can also not be relaxed.

Notes

References Sumihiro, Hideyasu (1974), "Equivariant completion", J. Math. Kyoto Univ., 14: 1–28, doi:10.1215/kjm/1250523277, hdl:2433/220057.

External links Alper, Jarod; Hall, Jack; Rydh, David (2015). "A Luna étale slice theorem for algebraic stacks". Annals of Mathematics. 191 (3). arXiv:1504.06467. doi:10.4007/annals.2020.191.3.1.

Worked examples

Example 1 — a first encounter with Sumihiro's theorem

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

In research: Sumihiro's 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 Sumihiro's 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: Sumihiro's 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 Sumihiro's 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 Sumihiro's theorem in 20 minutes

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

Frequently asked questions

What is Sumihiro's theorem in simple terms?

Why does Sumihiro's 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 Sumihiro's 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 Sumihiro's theorem.