Pencils of curves with 4 or 6 Conic-Line Curves

Date

2025-09-01

Author

Suluyer, Hasan

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Apencilof degreed >2curves is a line in the projective space of all homogeneous polynomials inℂ[x0,x1,x2]of degreed. Thek >2curves whose irreducible components are only lines in some pencil of degreedcurves play an important role for(k,d)-nets. The line arrangement comprised of all these irreducible components has a net structure. It was proved that the numberk, independent ofd, cannot exceed 4 for an(k,d)-net. When the degree of each irreducible component of a curve is at most 2, this curve is called aconic-line curveand it is a union of lines or irreducible conics in the complex projective plane. The numbermof such curves in pencils cannot exceed 6.We study the restrictions on the numbermof conic-line curves in special pencils. We present a one-parameter family of pencils of cubics with exactly 4 conic-line curves while there exists only one known net withk= 4. Moreover, we show the combinatorics of the irreducible components of conic-line curves in odd degree pencils withm= 6.

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https://hdl.handle.net/11511/116007

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Department of Mathematics, Conference / Seminar