mathematics

Fulton–Hansen connectedness theorem

Fulton–Hansen connectedness 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 Fulton–Hansen connectedness theorem rather than just read about it. In short: In mathematics, the Fulton–Hansen connectedness theorem is a result from intersection theory in algebraic geometry, for the case of subvarieties of projective space with codimension large enough to make the intersection have components of dimension at least 1. It is named after William Fulton and Johan Hansen, who proved it in 1979.

Key takeaways

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

Reference excerpt

In mathematics, the Fulton–Hansen connectedness theorem is a result from intersection theory in algebraic geometry, for the case of subvarieties of projective space with codimension large enough to make the intersection have components of dimension at least 1. It is named after William Fulton and Johan Hansen, who proved it in 1979. The formal statement is that if V and W are irreducible algebraic subvarieties of a projective space P, all over an algebraically closed field, and if

dim ⁡ ( V ) + dim ⁡ ( W ) > dim ⁡ ( P ) {\displaystyle \dim(V)+\dim(W)>\dim(P)}

in terms of the dimension of an algebraic variety, then the intersection U of V and W is connected. More generally, the theorem states that if Z {\displaystyle Z} is a projective variety and f : Z → P n × P n {\displaystyle f\colon Z\to P^{n}\times P^{n}} is any morphism such that dim ⁡ f ( Z ) > n {\displaystyle \dim f(Z)>n} , then f − 1 Δ {\displaystyle f^{-1}\Delta } is connected, where Δ {\displaystyle \Delta } is the diagonal in P n × P n {\displaystyle P^{n}\times P^{n}} . The special case of intersections is recovered by taking Z = V × W {\displaystyle Z=V\times W} , with f {\displaystyle f} the natural inclusion.

See also Zariski's connectedness theorem Grothendieck's connectedness theorem Deligne's connectedness theorem

References Fulton, William; Hansen, Johan (1979). "A connectedness theorem for projective varieties with applications to intersections and singularities of mappings". Annals of Mathematics. 110 (1): 159–166. doi:10.2307/1971249. JSTOR 1971249. Lazarsfeld, Robert (2004). Positivity in algebraic geometry, Vol. I. Berlin: Springer. ISBN 3-540-22533-1. Lazarsfeld, R. K. (2004). Positivity in algebraic geometry, Vol. II. ISBN 3-540-22534-X.

External links PDF lectures with the result as Theorem 15.3 (attributed to Faltings, also)

Worked examples

Example 1 — a first encounter with Fulton–Hansen connectedness theorem

Start with the simplest possible case. Write down what Fulton–Hansen connectedness 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 Fulton–Hansen connectedness 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 Fulton–Hansen connectedness 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 Fulton–Hansen connectedness theorem

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

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

Frequently asked questions

What is Fulton–Hansen connectedness theorem in simple terms?

Why does Fulton–Hansen connectedness 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 Fulton–Hansen connectedness 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 Fulton–Hansen connectedness theorem.