The projective second fundamental form at a generic smooth point x of
a subvariety X(n) of projective space CP(n+a) may be considered as a l
inear system of quadratic forms \II\x on the tangent space T(x)X. We p
rove this system is subject to certain restrictions (4.1), including a
bound on the dimension of the singular locus of any quadric in the sy
stem \II\x. (The only previously known restriction was that if X is sm
ooth, the singular locus of the entire system must be empty). One cons
equence of (4.1) is that smooth subvarieties with 2(a-1) < n are such
that their third and all higher fundamental forms are zero (4.14). Thi
s says that the infinitesimal invariants of such varieties are of the
same nature as the invariants of hypersurfaces, giving further evidenc
e towards the principle (e.g. [H]) that smooth subvarieties of small c
odimension should behave like hypersurfaces. Further restrictions on t
he second fundamental form occur when one has more information about t
he variety. In this paper we discuss additional restrictions when the
variety contains a linear space (2.3) and when the variety is a comple
te intersection (6.1). These rank restrictions should prove useful bot
h in enhancing our understanding of smooth subvarieties of small codim
ension, and in bounding from below the dimensions of singularities of
varieties for which local information is more readily available than g
lobal information.