Irreducibility of intersections of quadric hypersurfaces 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 $m:n:r\neq 1:1:1$.
 A: It seems that you are taking 2x2 minors of a matrix. This gives you an irreducible determinantal variety. For more on the subject see the book of Harris, Algebraic Geometry
or a paper of Eisenbud, Linear sections of determinantal varieties.
A: This is not quite an answer to your question, but you may find it useful.  Miles Reid's thesis has a chapter on intersections of two quadrics, which is already an interesting case.  Associated to an intersection of two quadrics is a polynomial: express your two quadrics as symmetric matrices $A$ and $B$, and then look at $\textrm{det}(A + \lambda B)$.  One of the results you'll find in Miles' thesis (Proposition 2.1) states that the intersection is non-singular if that polynomial has distinct roots (over an algebraically closed field).
There is also a section on intersections of two quadrics in Colliot-Thélène, Sansuc & Swinnerton-Dyer, "Intersections of two quadrics and Châtelet surfaces. I", Crelle vol. 373.  One result which you might find useful is Lemma 1.11: let $\Phi_1, \Phi_2$ be the two quadric forms; suppose that they have no common factor; that there exists a form of rank at least 5 in the pencil $\lambda \Phi_1 + \mu \Phi_2$; and there is no non-zero form of rank strictly less than 3 in that pencil.  Then the variety is geometrically integral.  (They point out that this criterion is definitely not necessary, since it's impossible to satisfy it for intersections of two quadrics in $\mathbb{P}^3$, yet there certainly are integral intersections of two quadrics in $\mathbb{P}^3$.)
