Akademik Veri Yönetim Sistemi
Topology of Arrangements with an Eye to Applications, Pisa, İtalya, 1 - 05 Eylül 2025, (Yayınlanmadı)
A pencil of degree d > 2 curves is a line in the projective space of all homogeneous polynomials in ℂ[x0,x1,x2] of degree d. The k > 2 curves whose irreducible components are only lines in some pencil of degree d curves 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 number k, independent of d, 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 a conic-line curve and it is a union of lines or irreducible conics in the complex projective plane. The number m of such curves in pencils cannot exceed 6.
We study the restrictions on the number m of 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 with k = 4. Moreover, we show the combinatorics of the irreducible components of conic-line curves in odd degree pencils with m = 6.