mathematics

Porteous formula

Porteous formula 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 Porteous formula rather than just read about it. In short: In mathematics, the Porteous formula, or Thom–Porteous formula, or Giambelli–Thom–Porteous formula, is the expression for the fundamental class of a degeneracy locus (or determinantal variety) of a morphism of vector bundles in terms of Chern classes. Giambelli's formula is roughly the special case when the vector bundles are sums of line bundles over projective space.

Key takeaways

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

Reference excerpt

In mathematics, the Porteous formula, or Thom–Porteous formula, or Giambelli–Thom–Porteous formula, is the expression for the fundamental class of a degeneracy locus (or determinantal variety) of a morphism of vector bundles in terms of Chern classes. Giambelli's formula is roughly the special case when the vector bundles are sums of line bundles over projective space. Thom (1957) pointed out that the fundamental class must be a polynomial in the Chern classes and found this polynomial in a few special cases, and Porteous (1971) found the polynomial in general. Kempf & Laksov (1974) proved a more general version, and Fulton (1992) generalized it further.

Statement Given a morphism of vector bundles E, F of ranks m and n over a smooth variety, its k-th degeneracy locus (k ≤ min(m,n)) is the variety of points where it has rank at most k. If all components of the degeneracy locus have the expected codimension (m – k)(n – k) then Porteous's formula states that its fundamental class is the determinant of the matrix of size m – k whose (i, j) entry is the Chern class cn–k+j–i(F – E).

References Fulton, William (1992), "Flags, Schubert polynomials, degeneracy loci, and determinantal formulas", Duke Mathematical Journal, 65 (3): 381–420, doi:10.1215/S0012-7094-92-06516-1, ISSN 0012-7094, MR 1154177 Kempf, G.; Laksov, D. (1974), "The determinantal formula of Schubert calculus", Acta Mathematica, 132: 153–162, doi:10.1007/BF02392111, ISSN 0001-5962, MR 0338006 Porteous, Ian R. (1971) [1962], "Simple singularities of maps", Proceedings of Liverpool Singularities Symposium, I (1969/70), Lecture Notes in Mathematics, vol. 192, Berlin, New York: Springer-Verlag, pp. 286–307, doi:10.1007/BFb0066829, ISBN 978-3-540-05402-3, MR 0293646 Thom, René (1957), Les ensembles singuliers d'une application différentiable et leurs propriétés homologiques, Séminaire de Topologie de Strasbourg

Worked examples

Example 1 — a first encounter with Porteous formula

Start with the simplest possible case. Write down what Porteous formula 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 Porteous formula 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 Porteous formula 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 Porteous formula

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

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

Frequently asked questions

What is Porteous formula in simple terms?

Why does Porteous formula 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 Porteous formula?

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 Porteous formula.