mathematics

Scorza variety

Scorza variety 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 Scorza variety rather than just read about it. In short: In algebraic geometry, a k-Scorza variety is a smooth projective variety, of maximal dimension among those whose k–1 secant varieties are not the whole of projective space. Scorza varieties were introduced and classified by Zak (1993), who named them after Gaetano Scorza.

Key takeaways

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

Reference excerpt

In algebraic geometry, a k-Scorza variety is a smooth projective variety, of maximal dimension among those whose k–1 secant varieties are not the whole of projective space. Scorza varieties were introduced and classified by Zak (1993), who named them after Gaetano Scorza. The special case of 2-Scorza varieties are sometimes called Severi varieties, after Francesco Severi.

Classification Zak showed that k-Scorza varieties are the projective varieties of the rank 1 matrices of rank k simple Jordan algebras.

Severi varieties The Severi varieties are the non-singular varieties of dimension n (even) in PN that can be isomorphically projected to a hyperplane and satisfy N=3n/2+2.

Severi showed in 1901 that the only Severi variety with n=2 is the Veronese surface in P5. The only Severi variety with n=4 is the Segre embedding of P2×P2 into P8, found by Scorza in 1908. The only Severi variety with n=8 is the 8-dimensional Grassmannian G(1,5) of lines in P5 embedded into P14, found by John Greenlees Semple in 1931. The only Severi variety with n=16 is a 16-dimensional variety E6/Spin(10)U(1) in P26 found by Robert Lazarsfeld in 1981. These 4 Severi varieties can be constructed in a uniform way, as orbits of groups acting on the complexifications of the 3 by 3 hermitian matrices over the four real (possibly non-associative) division algebras of dimensions 2k = 1, 2, 4, 8. These representations have complex dimensions 3(2k+1) = 6, 9, 15, and 27, giving varieties of dimension 2k+1 = 2, 4, 8, 16 in projective spaces of dimensions 3(2k)+2 = 5, 8, 14, and 26. Zak proved that the only Severi varieties are the 4 listed above, of dimensions 2, 4, 8, 16.

References Hartshorne, Robin (1974), "Varieties of small codimension in projective space" (PDF), Bulletin of the American Mathematical Society, 80 (6): 1017–1032, doi:10.1090/S0002-9904-1974-13612-8, ISSN 0002-9904, MR 0384816 Zak, F. L. (1981), "Projections of algebraic varieties", Matematicheskii Sbornik, Novaya Seriya, 116(158) (4): 593–602, 608, ISSN 0368-8666, MR 0665860 Lazarsfeld, Robert; Van de Ven, Antonius (1984), Topics in the geometry of projective space, DMV Seminar, vol. 4, Birkhäuser Verlag, doi:10.1007/978-3-0348-9348-0, ISBN 978-3-7643-1660-0, MR 0808175 Zak, F. L. (1985), "Severi varieties", Matematicheskii Sbornik, Novaya Seriya, 126(168) (1): 115–132, 144, ISSN 0368-8666, MR 0773432 Zak, F. L. (1993), Tangents and secants of algebraic varieties, Translations of Mathematical Monographs, vol. 127, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-4585-1, MR 1234494

Worked examples

Example 1 — a first encounter with Scorza variety

Start with the simplest possible case. Write down what Scorza 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 Scorza 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 Scorza 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 Scorza variety

In research: Scorza 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 Scorza 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: Scorza variety is common in secondary-school and first-year university syllabi. It links to neighbouring topics Algebraic varieties, so understanding it makes those chapters shorter.
In everyday life: Look for Scorza 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.

Ask Teacher Smith questions about this articleOpens your AI tutor with a question about “Scorza variety” →

Affiliate

Preply — study more efficiently by working with a personal tutor. 50% off.

How to study Scorza variety in 20 minutes

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

Frequently asked questions

What is Scorza variety in simple terms?

Why does Scorza 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 Scorza 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 Scorza variety.