Is there a characterization of the class of varieties which can be described as an intersection of quadrics, apart from the taulogical one?


Lots of varieties arise in this way (my favorite examples are the Grassmanianns and Schubert varieties and some toric varieties) and I wonder how far can one go.



Varieties cut by quadrics
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. 


