Two Lines and One Quadric 12

    The set of lines that meet 2 general fixed lines l1 and l2 is isomorphic to the product l1xl2, and is
a quadric surface in a P3, as it is defined by the quadratic Plücker equation and 2 linear equations.
Restricting the tangent equation

  =   0
to the quadric surface l1xl2 gives a bihomogeneous form, FQ, of bidegree (2,2) on l1xl2.
[Defining a genus-1 or (2,2)-curve.]
    This gives a map
f   :   General Quadric Q (2,2)-form FQ.
P9 P8
whose fibre over FQ are those other quadrics having the same set of common transversal tangents with l1 and l2 as does Q.
The fibres of  f solve our problem of determining `generic' configurations having
    infinitely many transversal tangents.