mathematics

Weighted projective space

Weighted projective space 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 Weighted projective space rather than just read about it. In short: In algebraic geometry, a weighted projective space P(a0,...,an) is the projective variety Proj(k[x0,...,xn]) associated to the graded ring k[x0,...,xn] where the variable xk has degree ak. Properties If d is a positive integer then P(a0,a1,...,an) is isomorphic to P(da0,da1,...,dan).

Key takeaways

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

Reference excerpt

In algebraic geometry, a weighted projective space P(a0,...,an) is the projective variety Proj(k[x0,...,xn]) associated to the graded ring k[x0,...,xn] where the variable xk has degree ak.

Properties If d is a positive integer then P(a0,a1,...,an) is isomorphic to P(da0,da1,...,dan). This is a property of the Proj construction; geometrically it corresponds to the d-tuple Veronese embedding. So without loss of generality one may assume that the degrees ai have no common factor. Suppose that a0,a1,...,an have no common factor, and that d is a common factor of all the ai with i≠j, then P(a0,a1,...,an) is isomorphic to P(a0/d,...,aj-1/d,aj,aj+1/d,...,an/d) (note that d is coprime to aj; otherwise the isomorphism does not hold). So one may further assume that any set of n variables ai have no common factor. In this case the weighted projective space is called well-formed. The only singularities of weighted projective space are cyclic quotient singularities. A weighted projective space is a Q-Fano variety and a toric variety. The weighted projective space P(a0,a1,...,an) is isomorphic to the quotient of projective space by the group that is the product of the groups of roots of unity of orders a0,a1,...,an acting diagonally.

References

Dolgachev, Igor (1982), "Weighted projective varieties", Group actions and vector fields (Vancouver, B.C., 1981), Lecture Notes in Math., vol. 956, Berlin: Springer, pp. 34–71, CiteSeerX 10.1.1.169.5185, doi:10.1007/BFb0101508, ISBN 978-3-540-11946-3, MR 0704986 {{citation}}: Cite uses deprecated parameter |citeseerx= (help) Hosgood, Timothy (2016), An introduction to varieties in weighted projective space, arXiv:1604.02441, Bibcode:2016arXiv160402441H Reid, Miles (2002), Graded rings and varieties in weighted projective space (PDF), archived from the original (PDF) on 2023-06-02, retrieved 2020-11-19

Worked examples

Example 1 — a first encounter with Weighted projective space

Start with the simplest possible case. Write down what Weighted projective space 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 Weighted projective space 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 Weighted projective space 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 Weighted projective space

In research: Weighted projective space 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 Weighted projective space 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: Weighted projective space 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 Weighted projective space 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 Weighted projective space in 20 minutes

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

Frequently asked questions

What is Weighted projective space in simple terms?

In algebraic geometry, a weighted projective space P(a0,...,an) is the projective variety Proj(k[x0,...,xn]) associated to the graded ring k[x0,...,xn] where the variable xk has degree ak. Properties If d is a positive integer then P(a0,a1,...,an) is isomorphic to P(da0,da1,...,dan).

Why does Weighted projective space 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 Weighted projective space?

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 Weighted projective space.