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: $$ (x_2-x_1)(y_{2i}-y_{1j})-(x_3-x_1)(y_{3k}-y_{1j})=0, $$ 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 $m:n:r\neq 1:1:1$.