Curves in High Degree Plane Pencils with Bounded Degree Components

In this paper, we study pencils of plane curves of sufficiently large degree $d$ with simple base points, and their reducible curves whose irreducible components have degree at most $k\geq 2$.

Combining techniques from algebraic geometry and combinatorics, we establish an explicit upper bound on the number $m_k$ of such curves in the pencils.

We prove that, for sufficiently large $d$, pencils with more than six such curves do not exist.

Consequently, under the stronger assumption $d\geq\frac{7}{2}k(k-1)-2$, we obtain the bound $m_k\leq 6$, improving the previously known bound for $d \ge 2k$.

We also establish restrictions on pencils containing reducible curves consisting of one irreducible component of degree $k$ together with lines, and obtain nonexistence results for certain such pencils.