Hilbert curves of conic fibrations over smooth surfaces

Abstract

Let (X, L) be a complex polarized threefold which is a conic fibration over a smooth surface. The complex affine cubic Γ representing the Hilbert curve of (X, L) is studied, paying special attention to its reducibility. In particular, Γ contains a specific line ℓ₀ if and only if X has no singular fibers. This leads to characterize the existence of a triple point simply in terms of numerical invariants of X. Other lines may cause the reducibility of Γ, which in this case depends also on the polarization. This situation is analyzed for a special class of conic fibrations.

Keywords:

• conic fibration
• scroll
• Hilbert curve

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 14C20
• 14N30
• Secondary: 14J35
• 14M99

Acknowledgments

We thank anonymous referee for the careful reading of the manuscript.

Additional information

Funding

The first author is supported by PRIN 2017SSNZAW. Both authors are members of INdAM-GNSAGA.