mathematics

Local uniformization

Local uniformization 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 Local uniformization rather than just read about it. In short: In algebraic geometry, local uniformization is a weak form of resolution of singularities, stating that a variety can be desingularized near any valuation, or in other words that the Zariski–Riemann space of the array is in some sense non-singular. Local uniformization was introduced by Zariski (1939, 1940), who separated the problem of resolving the singularities of a variety into the problem of local uniformizatio…

Key takeaways

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

Reference excerpt

In algebraic geometry, local uniformization is a weak form of resolution of singularities, stating that a variety can be desingularized near any valuation, or in other words that the Zariski–Riemann space of the array is in some sense non-singular. Local uniformization was introduced by Zariski (1939, 1940), who separated the problem of resolving the singularities of a variety into the problem of local uniformization and the problem of combining the local uniformizations into a global desingularization. Local uniformization of a variety at a valuation of its function field means finding a projective model of the variety such that the center of the valuation is non-singular. It is weaker than resolution of singularities: if there is a resolution of singularities then this is a model such that the center of every valuation is non-singular. Zariski (1944b) proved that if one can show local uniformization of a variety then one can find a finite number of models such that every valuation has a non-singular center on at least one of these models. To complete a proof of resolution of singularities, it is then sufficient to show that one can combine these finite models into a single model, but this seems rather hard. (Local uniformization at a valuation does not directly imply resolution at the center of the valuation: roughly speaking; it only implies resolution in a sort of "wedge" near this point, and it seems hard to combine the resolutions of different wedges into a resolution at a point.) Zariski (1940) proved local uniformization of varieties in any dimension over fields of characteristic 0, and used this to prove resolution of singularities for varieties in characteristic 0 of dimension at most 3. Local uniformization in positive characteristic seems to be much harder. Abhyankar (1956, 1966) proved local uniformization in all characteristics for surfaces and in characteristics at least 7 for 3-folds, and was able to deduce global resolution of singularities in these cases from this. Cutkosky (2009) simplified Abhyankar's long proof. Cossart and Piltant (2008, 2009) extended Abhyankar's proof of local uniformization of 3-folds to the remaining characteristics 2, 3, and 5. Temkin (2013) showed that it is possible to find a local uniformization of any valuation after taking a purely inseparable extension of the function field. Local uniformization in positive characteristic for varieties of dimension at least 4 is (as of 2019) an open problem.

References Abhyankar, Shreeram (1956), "Local uniformization on algebraic surfaces over ground fields of characteristic p≠0", Annals of Mathematics, Second Series, 63 (3): 491–526, doi:10.2307/1970014, JSTOR 1970014, MR 0078017 Abhyankar, Shreeram S. (1966), Resolution of singularities of embedded algebraic surfaces, Springer Monographs in Mathematics, Acad. Press, doi:10.1007/978-3-662-03580-1, ISBN 3-540-63719-2 (1998 2nd edition) Cossart, Vincent; Piltant, Olivier (2008), "Resolution of singularities of threefolds in positive characteristic. I. Reduction to local uniformization on Artin–Schreier and purely inseparable coverings", Journal of Algebra, 320 (3): 1051–1082, doi:10.1016/j.jalgebra.2008.03.032, MR 2427629 Cossart, Vincent; Piltant, Olivier (2009), "Resolution of singularities of threefolds in positive characteristic. II" (PDF), Journal of Algebra, 321 (7): 1836–1976, doi:10.1016/j.jalgebra.2008.11.030, MR 2494751 Cutkosky, Steven Dale (2009), "Resolution of singularities for 3-folds in positive characteristic", Amer. J. Math., 131 (1): 59–127, arXiv:math/0606530, doi:10.1353/ajm.0.0036, JSTOR 40068184, MR 2488485, S2CID 2139305 Temkin, Michael (2013), "Inseparable local uniformization", J. Algebra, 373: 65–119, arXiv:0804.1554, doi:10.1016/j.jalgebra.2012.09.023, MR 2995017, S2CID 115167009 Zariski, Oscar (1939), "The reduction of the singularities of an algebraic surface", Ann. of Math., 2, 40 (3): 639–689, doi:10.2307/1968949, JSTOR 1968949 Zariski, Oscar (1940), "Local uniformization on algebraic varieties", Ann. of Math., 2, 41 (4): 852–896, doi:10.2307/1968864, JSTOR 1968864, MR 0002864 Zariski, Oscar (1944a), "The compactness of the Riemann manifold of an abstract field of algebraic functions", Bulletin of the American Mathematical Society, 50 (10): 683–691, doi:10.1090/S0002-9904-1944-08206-2, ISSN 0002-9904, MR 0011573 Zariski, Oscar (1944b), "Reduction of the singularities of algebraic three dimensional varieties", Ann. of Math., 2, 45 (3): 472–542, doi:10.2307/1969189, JSTOR 1969189, MR 0011006

External links "Local uniformization", Encyclopedia of Mathematics, EMS Press, 2001 [1994]

Worked examples

Example 1 — a first encounter with Local uniformization

Start with the simplest possible case. Write down what Local uniformization 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 Local uniformization 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 Local uniformization 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 Local uniformization

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

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

Frequently asked questions

What is Local uniformization in simple terms?

Why does Local uniformization 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 Local uniformization?

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 Local uniformization.