Abstract
We study the relationships between the algebra of symmetric twisted differentials, the algebra generated by tangentially homogeneous polynomials and the quadric algebra of a smooth projective subvariety whose codimension is small relative to its dimension. It is conjectured that these three algebras coincide for such varieties and we prove this for complete intersections and subvarieties of codimension two. The connection between these three algebras leads to questions about the local projective differential geometry of the variety, its trisecant lines and the linear system of quadrics vanishing on it.