Scorza variety

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 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 P^N 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 Segre 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 R. 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 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

• Hartshorne, Robin (1974), "Varieties of small codimension in projective space", Bulletin of the American Mathematical Society 80: 1017–1032, doi:10.1090/S0002-9904-1974-13612-8, ISSN 0002-9904, MR 0384816 
• Zak, F. L. (1981), "Projections of algebraic varieties", Matematicheski\u\i Sbornik. Novaya Seriya, 116(158) (4): 593–602, 608, ISSN 0368-8666, MR 665860 
• Lazarsfeld, Robert; Van de Ven, Antonius (1984), Topics in the geometry of projective space, DMV Seminar 4, Birkhäuser Verlag, ISBN 978-3-7643-1660-0, MR 808175 
• Zak, F. L. (1985), "Severi varieties", Matematicheskii Sbornik. Novaya Seriya, 126(168) (1): 115–132, 144, ISSN 0368-8666, MR 773432 
• Zak, F. L. (1993), Tangents and secants of algebraic varieties, Translations of Mathematical Monographs 127, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-4585-1, MR 1234494