  In the present paper we compute Alexander polynomials for certain classes of
conic-line arrangements in the complex projective plane which are related to
pencils. We prove two general results for curve arrangements coming from
Halphen pencils of index $k\geq 2$. Then we apply them to the Hesse or the
Chilean arrangement of 12 conics and to some of its degenerations. The results
are completed by computations using computer algebra. In particular, we
construct conic-line arrangements which are non-reduced pencil-type
arrangements and have as roots of their Alexander polynomials roots of unity of
order 7. Such roots are not known and are conjectured not to exist in the class
of line arrangements.
