Differential-geometric Characterizations of Complete Intersections

We characterize complete intersections in terms of local differential geometry. Let X ⊂ CPn+a be a variety. We first localize the problem; we give a criterion for X to be a complete intersection that is testable at any smooth point of X. We rephrase the criterion in the language of projective differential geometry and derive a sufficient condition for X to be a complete intersection that is computable at a general point x ∈ X. The sufficient condition has a geometric interpretation in terms of restrictions on the spaces of osculating hypersurfaces at x. When this sufficient condition holds, we are able to define systems of partial differential equations that generalize the classical Monge equation that characterizes conic curves in CP2. Using our sufficent condition, we show that if the ideal of X is generated by quadrics and a < n−(b+1)+3 3 , where b =dimXsing, then X is a complete intersection. §0. Introduction Local and global geometry Projective differential geometry has been used to study the local geometry of subvarieties of projective space by various authors (e.g. [C], [F], [GH], [JM], [T]). However, there are few examples where global conclusions are drawn from the local picture. One (global) fact about projective varieties that has been encoded into the infinitesimal geometry is the following: if there is a line on a variety X ⊂ CP along which the embedded tangent space is constant, then X must be singular. Griffiths and Harris realized this fact had implications for the projective second fundamental form (that its singular locus must be empty at general points) which 1991 Mathematics Subject Classification. primary 14e335, secondary 533a20.