In fact the answer _is_ in some sense tautological: every projective variety can be realized as a scheme-theoretic intersection of quadrics!  See e.g.

D. Mumford,   "Varieties defined by quadratic equations", Questions on algebraic varieties, C.I.M.E. Varenna, 1969 , Cremonese  (1970)  pp. 29–100,

for quantitative results in this direction.