Theorem of Bertini

Key takeaways

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

Reference excerpt

In algebraic geometry, the theorem of Bertini is an existence and genericity theorem for smooth connected hyperplane sections for smooth projective varieties over algebraically closed fields, introduced by Eugenio Bertini. This is the simplest and broadest of the "Bertini theorems" applying to a linear system of divisors; simplest because there is no restriction on the characteristic of the underlying field, while the extensions require characteristic 0.

Statement for hyperplane sections of smooth varieties

Let X be a smooth quasi-projective variety over an algebraically closed field, embedded in a projective space P n {\displaystyle \mathbf {P} ^{n}} . Let | H | {\displaystyle |H|} denote the complete system of hyperplane divisors in P n {\displaystyle \mathbf {P} ^{n}} . Recall that it is the dual space ( P n ) ⋆ {\displaystyle (\mathbf {P} ^{n})^{\star }} of P n {\displaystyle \mathbf {P} ^{n}} and is isomorphic to P n {\displaystyle \mathbf {P} ^{n}} . The theorem of Bertini states that the set of hyperplanes not containing X and with smooth intersection with X contains an open dense subset of the total system of divisors | H | {\displaystyle |H|} . The set itself is open if X is projective. If dim ⁡ ( X ) ≥ 2 {\displaystyle \dim(X)\geq 2} , then these intersections (called hyperplane sections of X) are connected, hence irreducible. The theorem hence asserts that a general hyperplane section not equal to X is smooth, that is: the property of smoothness is generic. Over an arbitrary field k, there is a dense open subset of the dual space ( P n ) ⋆ {\displaystyle (\mathbf {P} ^{n})^{\star }} whose rational points define smooth hyperplane sections of X. When k is infinite, this open subset then has infinitely many rational points and there are infinitely many smooth hyperplane sections in X. Over a finite field, the above open subset may not contain rational points and in general there is no hyperplanes with smooth intersection with X. However, if we take hypersurfaces of sufficiently big degrees, then the theorem of Bertini holds.

Outline of a proof We consider the subfibration of the product variety X × | H | {\displaystyle X\times |H|} with fiber above x ∈ X {\displaystyle x\in X} the linear system of hyperplanes that intersect X non-transversally at x. The rank of the fibration in the product is one less than the codimension of X ⊂ P n {\displaystyle X\subset \mathbf {P} ^{n}} , so that the total space has lesser dimension than n {\displaystyle n} and so its projection is contained in a divisor of the complete system | H | {\displaystyle |H|} .

General statement Over any infinite field k {\displaystyle k} of characteristic 0, if X is a smooth quasi-projective k {\displaystyle k} -variety, a general member of a linear system of divisors on X is smooth away from the base locus of the system. For clarification, this means that given a linear system f : X → P n {\displaystyle f:X\rightarrow \mathbf {P} ^{n}} , the preimage f − 1 ( H ) {\displaystyle f^{-1}(H)} of a hyperplane H is smooth -- outside the base locus of f -- for all hyperplanes H in some dense open subset of the dual projective space ( P n ) ⋆ {\displaystyle (\mathbf {P} ^{n})^{\star }} . This theorem also holds in characteristic p>0 when the linear system f is unramified.

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Theorem of Bertini

Start with the simplest possible case. Write down what Theorem of Bertini 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 Theorem of Bertini 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 Theorem of Bertini 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 Theorem of Bertini

In research: Theorem of Bertini 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 Theorem of Bertini 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: Theorem of Bertini is common in secondary-school and first-year university syllabi. It links to neighbouring topics Geometry of divisors, Theorems in algebraic geometry, so understanding it makes those chapters shorter.
In everyday life: Look for Theorem of Bertini 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.

How to study Theorem of Bertini in 20 minutes

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

Frequently asked questions

What is Theorem of Bertini in simple terms?

In algebraic geometry, the theorem of Bertini is an existence and genericity theorem for smooth connected hyperplane sections for smooth projective varieties over algebraically closed fields, introduced by Eugenio Bertini. This is the simplest and broadest of the "Bertini theorems" applying to a li…

Why does Theorem of Bertini 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 Theorem of Bertini?

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 Theorem of Bertini.