It is known (see the MO question "<a href="https://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:
$$
(x_2-x_1)(y_{2i}-y_{1j})-(x_3-x_1)(y_{3k}-y_{1j})=0,
$$
where $i=1, 2, \dots, M$, $j=1,2,\dots, N$ and $k=1, 2, \dots, R$, and $M$, $N$ and $R$ are fixed natural numbers.
Is there a way to determine if the intersection is irreducible? More generally, what about equations of the following form:

$$
(x_n-x_m)(y_{ni}-y_{mj})-(x_r-x_m)(y_{rk}-y_{mj})=0,
$$ where $m:n:r\neq 1:1:1$.