How do we check if the variety defined by two homogeneous irreducible polynomials of degree $2$ in ${\mathbb C}[X_1, X_2, \ldots, X_n]$ is irreducible ?
Is there a nice condition for such things ? I guess this might be hard to do for general varieties, but what if the number of generators is 2 ? Are things simpler then ?
An intersection of two quadrics is not integral (i.e., reducible or non-reduced) of codimension 2 if and only if there is a linear combination of these two quadrics which has rank at most 2.
One direction is easy, because a quadric of rank 2 is a union of two hyperplanes. For the other direction, note that the intersection has degree 4 (in the projective space), so, if it is not integral, it has a component of degree at most 2. This component has to be a quadric of dimension n-2, and the linear space of quadrics passing through it contains a space of rank 2 quadrics as a hyperplane.