Severi–Brauer variety

Key takeaways

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

Reference excerpt

In mathematics, a Severi–Brauer variety over a field K is an algebraic variety V which becomes isomorphic to a projective space over an algebraic closure of K. The varieties are associated to central simple algebras in such a way that the algebra splits over K if and only if the variety has a rational point over K. Francesco Severi (1932) studied these varieties, and they are also named after Richard Brauer because of their close relation to the Brauer group. In dimension one, the Severi–Brauer varieties are conics. The corresponding central simple algebras are the quaternion algebras. The quaternion algebra (a, b)K, with generators i , j {\displaystyle i,j} and relations i 2 = a , j 2 = b {\displaystyle i^{2}=a,j^{2}=b} and i j = − j i {\displaystyle ij=-ji} , corresponds to the conic C(a, b) with equation

z 2 = a x 2 + b y 2 {\displaystyle z^{2}=ax^{2}+by^{2}}

and the algebra (a, b)K splits, that is, (a, b)K is isomorphic to a matrix algebra over K, if and only if C(a, b) has a point defined over K: this is in turn equivalent to C(a, b) being isomorphic to the projective line over K. Such varieties are of interest not only in diophantine geometry, but also in Galois cohomology. They represent (at least if K is a perfect field) Galois cohomology classes in H1(G(Ks/K),PGLn), where PGLn is the projective linear group, and n is one more than the dimension of the variety V. As usual in Galois cohomology, we often leave the G ( K s / K ) {\displaystyle G(K^{s}/K)} implied. There is a short exact sequence

1 → GL1 → GLn → PGLn → 1 of algebraic groups. This implies a connecting homomorphism

H1(PGLn) → H2(GL1) at the level of cohomology. Here H2(GL1) is identified with the Brauer group of K, while the kernel is trivial because H1(GLn) = {1} by an extension of Hilbert's Theorem 90. Therefore, Severi–Brauer varieties can be faithfully represented by Brauer group elements, i.e. classes of central simple algebras. Lichtenbaum showed that if X is a Severi–Brauer variety over K then there is an exact sequence

0 → P i c ( X ) → Z → δ B r ( K ) → B r ( K ) / ( X ) → 0 . {\displaystyle 0\rightarrow \mathrm {Pic} (X)\rightarrow \mathbb {Z} ~{\stackrel {\delta }{\rightarrow }}~\mathrm {Br} (K)\rightarrow \mathrm {Br} (K)/(X)\rightarrow 0\ .}

Here the map δ sends 1 to the Brauer class corresponding to X. As a consequence, we see that if the class of X has order d in the Brauer group then there is a divisor class of degree d on X. The associated linear system defines the d-dimensional embedding of X over a splitting field L.

Worked examples

Example 1 — a first encounter with Severi–Brauer variety

Start with the simplest possible case. Write down what Severi–Brauer variety 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 Severi–Brauer variety 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 Severi–Brauer variety 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 Severi–Brauer variety

In research: Severi–Brauer variety 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 Severi–Brauer variety 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: Severi–Brauer variety is common in secondary-school and first-year university syllabi. It links to neighbouring topics Algebraic groups, Algebraic varieties, Diophantine geometry, so understanding it makes those chapters shorter.
In everyday life: Look for Severi–Brauer variety 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 Severi–Brauer variety in 20 minutes

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

Frequently asked questions

What is Severi–Brauer variety in simple terms?

In mathematics, a Severi–Brauer variety over a field K is an algebraic variety V which becomes isomorphic to a projective space over an algebraic closure of K. The varieties are associated to central simple algebras in such a way that the algebra splits over K if and only if the variety has a ratio…

Why does Severi–Brauer variety 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 Severi–Brauer variety?

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 Severi–Brauer variety.