Lines having contact four with a projective hypersurface

George Alan Jennings

Vol. 129 (1987), No. 2, 321–336

DOI: 10.2140/pjm.1987.129.321

Abstract

Let X ⊂ Pⁿ⁺¹(C) be a projective hypersurface and p ∈ X. The third contact cone of X at p, C_p³, is the set of all lines in Pⁿ⁺¹ having contact ≥ 4 with X at p. If dimX ≥ 3 then the map p↦ (projective moduli of C_p³) usually is a local immersion (answering a conjecture of Griffiths and Harris), and one can prove a rigidity theorem: X is determined by the projective moduli of its C_p³’s and certain fourth order invariants. This immersion property may fail e.g. if X is a homogeneous space. We study this case also.

Mathematical Subject Classification 2000

Primary: 14M07

Secondary: 14H45

Milestones

Received: 10 March 1986

Published: 1 October 1987