In mathematics, 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 , who named them after

Gaetano Scorza Bernardino Gaetano Scorza (29 September 1876, in Morano Calabro – 6 August 1939, in Rome) was an Italian mathematician working in algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariat ...

. The special case of 2-Scorza varieties are sometimes called Severi varieties, after

Francesco Severi Francesco Severi (13 April 1879 – 8 December 1961) was an Italian mathematician. He was the chair of the committee on Fields Medal on 1936, at the first delivery. Severi was born in Arezzo, Italy. He is famous for his contributions to algeb ...

Classification

Zak showed that ''k''-Scorza varieties are the projective varieties of the rank 1 matrices of rank ''k'' simple

Jordan algebra In abstract algebra, a Jordan algebra is a nonassociative algebra over a field whose multiplication satisfies the following axioms: # xy = yx (commutative law) # (xy)(xx) = x(y(xx)) (). The product of two elements ''x'' and ''y'' in a Jordan al ...

Severi varieties

The Severi varieties are the non-singular varieties of dimension ''n'' (even) in P^''N'' that can be isomorphically projected to a hyperplane and satisfy ''N''=3''n''/2+2. *Severi showed in 1901 that the only Severi variety with ''n''=2 is the

Veronese surface In mathematics, the Veronese surface is an algebraic surface in five-dimensional projective space, and is realized by the Veronese embedding, the embedding of the projective plane given by the complete linear system of conics. It is named after Giu ...

in P⁵. *The only Severi variety with ''n''=4 is the

Segre embedding In mathematics, the Segre embedding is used in projective geometry to consider the cartesian product (of sets) of two projective spaces as a projective variety. It is named after Corrado Segre. Definition The Segre map may be defined as the map ...

of P²×P² into P⁸, found by Scorza in 1908. *The only Segre variety with ''n''=8 is the 8-dimensional Grassmannian ''G''(1,5) of lines in ''P''⁵ embedded into P¹⁴, found by

John Greenlees Semple John Greenlees Semple (10 June 1904 in Belfast, Ireland – 23 October 1985 in London, England) was a British mathematician working in algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multiva ...

in 1931. *The only Severi variety with ''n''=16 is a 16-dimensional variety ''E''₆/''Spin''(10)''U''(1) in P²⁶ found by

Robert Lazarsfeld Robert Kendall Lazarsfeld (born April 15, 1953) is an American mathematician, currently a professor at Stony Brook University. He was previously the Raymond L. Wilder Collegiate Professor of Mathematics at the University of Michigan. He is the ...

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 2^''k'' = 1, 2, 4, 8. These representations have complex dimensions 3(2^''k''+1) = 6, 9, 15, and 27, giving varieties of dimension 2^''k''+1 = 2, 4, 8, 16 in projective spaces of dimensions 3(2^''k'')+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

* * * * *{{Citation , last1=Zak , first1=F. L. , title=Tangents and secants of algebraic varieties , url=https://books.google.com/books?id=j9ZXQ-ARR7EC , publisher=

American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...

, location=Providence, R.I. , series=Translations of Mathematical Monographs , isbn=978-0-8218-4585-1 , mr=1234494 , year=1993 , volume=127 Algebraic geometry