Factoring the Ideal 16

    We use the computer algebra system Singular for these computations.
At their conclusion, we apply the factorization Gröbner basis algorithm to further study the ideal I defining the fibre in P9Q(s,t). This factors I into six quartic curves, whose intersection we check is the original ideal I.
Inspecting the output shows that each of these six components factors further, over Q(s1/2,t1/2) into ideals of dimension 1 and degree 2.

The computation.
We conclude that I defines 12 degree 2 curves, which proves our main Theorem:

Theorem. Megyesi, S., Theobald
Fixing 2 lines and 1 quadric in general position, the set of quadrics such that infinitely many transversals to the two lines are also tangent to the 2 quadrics is a connected curve in the 9-dimensional space of quadrics consisting of 12 plane conics.