1994/12/13 by J. M. Landsberg, Landsberg, J. M.
Mathematics · #Advanced Algebra and Geometry #Advanced Differential Equations and Dynamical Systems #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #Differential Geometry (math.DG) #FOS: Mathematics #alg-geom #dg-ga #math.AG #math.DG
paper · pdf · doi:10.48550/arxiv.alg-geom/9412012
Exposition altered according to the helpful recommendations of the referee. AMSTeX
openalex publication_date 1994/12/13 · arxiv created 1995/10/20 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the local differential geometry of varieties Xn⊂ \Bbb C\Bbb Pn+a with degenerate secant and tangential varieties. We show that the second fundamental form of a smooth variety with degenerate tangential variety is subject to certain rank restrictions. The rank restrictions imply a slightly refined version of Zak's theorem on linear normality and a short proof of the Zak-Fantecchi theorem on the superadditivity of multisecant defects. We show there is a vector bundle defined over general points of TX whose fibers carry the structure of a Clifford algebra. This structure implies additional restrictions of the size of the secant defect. The Clifford algebra structure, combined with further local computations, yields a new proof of Zak's theorem on Severi varieties that is substantially shorter than the original. We also prove local and global results on the dimension of the Gauss image of degenerate tangential varieties, refining the results in [GH].