It is known (see the MO question "<a href="http://mathoverflow.net/questions/11488/varieties-cut-by-quadrics">
Varieties cut by quadrics</a>") 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?