It is known (see the MO question " Varieties cut by quadrics") that every projective variety can be realized as a scheme-theoretic intersection of quadrics. Is there a way to determine if the intersection of irreducible quadric hypersurfaces is irreducible? For example, consider equations of the following form: $$ (x_2-x_1)(y_{2i}-y_{1j})-(x_3-x_1)(y_{3k}-y_{1j})=0. $$ Is there a way to determine if the intersection is irreducible?