On Conic-Line Curves of Special Pencils

Date

2026-05-22

Author

Suluyer, Hasan

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A pencil is a line in the projective space of homogeneous polynomials in C[x0,x1,x2] 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 is known that m cannot exceed 4. 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. Using purely algebraic–geometric and combinatorial arguments, we establish explicit upper bounds on m corresponding to the number of members of concurrent lines. In this talk, I will present my research on constraints on the number m of conic-line curves that can appear in special pencils. I establish an upper bound, showing that m ≤ 6. I conjecture that for large values of d, the number m of conic-line curves in a pencil must be at most 4.

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https://hdl.handle.net/11511/119952

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Department of Mathematics, Conference / Seminar