A pencil is a line in the projective space of complex homogeneous polynomials of some degree d > 2. The number m of curves whose irreducible components are only lines in some pencils of degree d curves plays an important role for the existence of special line arrangements, which are called (m,d)-nets. It was proved that the number m, independent of d, cannot exceed 4 for an (m,d)-net. When the degree of each irreducible component of a curve is at most 2, this curve is called a conic-line curve and it is a union of lines or irreducible conics in the complex projective plane. Our main goal is to find an upper bound on the number m of such curves in pencils in CP^2 with the number of concurrent lines in these pencils.

In this talk, we study the restrictions on the number m of conic-line curves in special pencils. The most general result we obtain is the relation between upper bounds on m and the number of concurrent lines in these pencils. We construct a one-parameter family of pencils such that each pencil in the family contains exactly 4 conic-line curves.