Varieties with quadratic entry locus. II

Abstract

For a complex projective manifold Xn⊂PN, the secant variety SX is defined to be the locus of chords of X, viz. SX=⋃x,y∈X,x≠y⟨x,y⟩−−−−−−−−−−−−, where ⟨x,y⟩ denotes the line passing through points x and y. For a general point p∈SX, the entry locus Σp is the locus of endpoints of the chords of X passing through p ; the nonnegative integer δ=dimΣp=2n+1−dimSX is called the secant defect of X. The authors are particularly interested in the class of varieties for which Σp is a quadric ("manifolds with quadratic entry locus'', or "QELMs'') and a larger class for which Σp contains a quadric component ("manifolds with local quadratic entry locus'', or "LQELMs''). For δ=0, QELMs are just the classical varieties with one apparent double point. In the paper under review the authors establish various properties of QELMs and LQELMs, such as rationality and special structure of the varieties of lines passing through a general point x∈X, and consider some applications