Local Quadratic Entry Locus Manifolds and Conic Connected Manifolds

  • Francesco Russo
Chapter
First Online:
Part of the Lecture Notes of the Unione Matematica Italiana book series (UMILN, volume 18)

Abstract

We recall the definition of QEL, respectively LQEL, manifolds as those \(X \subset \mathbb{P}^{N}\) whose general entry locus is a quadric hypersurface of dimension equal to the secant defect of X, respectively the union of quadric hypersurfaces of dimension equal to the secant defect of X. We present the main results of the theory of LQEL-manifolds introduced in Russo (Math. Ann. 344:597–617, 2009), leading to the Divisibility Theorem for the secant defect of LQEL-manifolds. The main applications concern the classification of LQEL-manifolds of type \(\delta \geq \frac{n} {2}\), which also include Severi varieties. The rest of the chapter surveys the classification of conic-connected manifolds obtained in Ionescu and Russo (J. Reine Angew. Math. 644:145–158, 2010) and some of the results of Ionescu and Russo (Compos. Math. 144:949–962, 2008) and of Ionescu and Russo (Math. Res. Lett. 21:1137–1154, 2014) on dual defective manifolds. In particular, an astonishing simple proof of the famous Landman Parity Theorem for dual defective manifolds appears as an application of the tools developed in Chap.  2

Keywords

Complete Intersection Fano Manifold Composition Algebra Secant Defect Quadric Hypersurface 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Francesco Russo
    • 1
  1. 1.Dipartimento di Matematica e InformaticaUniversità degli Studi di CataniaCataniaItaly

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