mathematics

Reider's theorem

Reider'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 Reider's theorem rather than just read about it. In short: In algebraic geometry, Reider's theorem gives conditions for a line bundle on a projective surface to be very ample. Statement Let D be a nef divisor on a smooth projective surface X.

Key takeaways

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

Reference excerpt

In algebraic geometry, Reider's theorem gives conditions for a line bundle on a projective surface to be very ample.

Statement Let D be a nef divisor on a smooth projective surface X. Denote by KX the canonical divisor of X.

If D2 > 4, then the linear system |KX+D| has no base points unless there exists a nonzero effective divisor E such that

D E = 0 , E 2 = − 1 {\displaystyle DE=0,E^{2}=-1} , or

D E = 1 , E 2 = 0 {\displaystyle DE=1,E^{2}=0} ; If D2 > 8, then the linear system |KX+D| is very ample unless there exists a nonzero effective divisor E satisfying one of the following:

D E = 0 , E 2 = − 1 {\displaystyle DE=0,E^{2}=-1} or − 2 {\displaystyle -2} ;

D E = 1 , E 2 = 0 {\displaystyle DE=1,E^{2}=0} or − 1 {\displaystyle -1} ;

D E = 2 , E 2 = 0 {\displaystyle DE=2,E^{2}=0} ;

D E = 3 , D = 3 E , E 2 = 1 {\displaystyle DE=3,D=3E,E^{2}=1}

Applications Reider's theorem implies the surface case of the Fujita conjecture. Let L be an ample line bundle on a smooth projective surface X. If m > 2, then for D=mL we have

D2 = m2 L2 ≥ m2 > 4; for any effective divisor E the ampleness of L implies D · E = m(L · E) ≥ m > 2. Thus by the first part of Reider's theorem |KX+mL| is base-point-free. Similarly, for any m > 3 the linear system |KX+mL| is very ample.

References Reider, Igor (1988), "Vector bundles of rank 2 and linear systems on algebraic surfaces", Annals of Mathematics, Second Series, 127 (2), Annals of Mathematics: 309–316, doi:10.2307/2007055, ISSN 0003-486X, JSTOR 2007055, MR 0932299

Worked examples

Example 1 — a first encounter with Reider's theorem

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

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

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

Frequently asked questions

What is Reider's theorem in simple terms?

In algebraic geometry, Reider's theorem gives conditions for a line bundle on a projective surface to be very ample. Statement Let D be a nef divisor on a smooth projective surface X.

Why does Reider'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 Reider'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 Reider's theorem.