On fundamental forms and osculating bundles

Abstract: We define higher order fundamental forms and osculating spaces of projective algebraic varieties, using sheaves of principal parts. We show that the $m$th fundamental form can be viewed as the differential of the $(m-1)$th Gauss map, and explain why the vanishing of the $m$th fundamental form implies that the variety is contained in a general $(m-1)$th osculating space. Pointwise, the fundamental forms give linear systems on the projectivized tangent spaces. We show that, at each point, the Jacobian of the $m$th fundamental form is contained in the $(m-1)$th fundamental form. In the case of ruled varieties, we describe these linear systems. We discuss conditions for a surface to be ruled, in terms of the second fundamental form and the Fubini cubic.