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 refinements of this question.