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Ruled join

Ruled join 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 Ruled join rather than just read about it. In short: In algebraic geometry, given irreducible subvarieties V, W of a projective space Pn, the ruled join of V and W is the union of all lines from V to W in P2n+1, where V, W are embedded into P2n+1 so that the last (resp. first) n + 1 coordinates on V (resp. W) vanish.

Key takeaways

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

Reference excerpt

In algebraic geometry, given irreducible subvarieties V, W of a projective space Pn, the ruled join of V and W is the union of all lines from V to W in P2n+1, where V, W are embedded into P2n+1 so that the last (resp. first) n + 1 coordinates on V (resp. W) vanish. It is denoted by J(V, W). For example, if V and W are linear subspaces, then their join is the linear span of them, the smallest linear subcontaining them. The join of several subvarieties is defined in a similar way.

See also Secant variety

References

Dickenstein, Alicia; Schreyer, Frank-Olaf; Sommese, Andrew J. (2010-07-10). Algorithms in Algebraic Geometry. Springer Science & Business Media. ISBN 9780387751559. Fulton, William (1998), Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., vol. 2 (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-3-540-62046-4, MR 1644323 Flenner, H.; O'Carroll, L.; Vogel, W. (29 June 2013). Joins and Intersections. ISBN 9783662038178. Russo, Francesco. "Geometry of Special Varieties" (PDF). University of Catania. Retrieved 7 March 2018.

Worked examples

Example 1 — a first encounter with Ruled join

Start with the simplest possible case. Write down what Ruled join 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 Ruled join 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 Ruled join 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 Ruled join

In research
Ruled join 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 Ruled join 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
Ruled join is common in secondary-school and first-year university syllabi. It links to neighbouring topics Algebraic geometry, Algebraic geometry stubs, so understanding it makes those chapters shorter.
In everyday life
Look for Ruled join 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 Ruled join in 20 minutes

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

Frequently asked questions

What is Ruled join in simple terms?

In algebraic geometry, given irreducible subvarieties V, W of a projective space Pn, the ruled join of V and W is the union of all lines from V to W in P2n+1, where V, W are embedded into P2n+1 so that the last (resp. first) n + 1 coordinates on V (resp. W) vanish.

Why does Ruled join 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 Ruled join?

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 Ruled join.

Tags

  • Algebraic geometry
  • Algebraic geometry stubs

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