Doubling rational normal curves

TL;DR

This paper investigates double structures on rational normal curves, computes their invariants, and shows that certain families are irreducible and contain arithmetically Gorenstein curves, advancing understanding of their geometric properties.

Contribution

It characterizes the families of double structures on rational normal curves, computes invariants, and proves irreducibility and Gorenstein properties for specific cases.

Findings

Families with fixed triples are irreducible.

General double curves in certain families are arithmetically Gorenstein.

Double conics of genus ≤ -2 form an irreducible Hilbert scheme component.

Abstract

In this paper, we study double structures supported on rational normal curves. After recalling the general construction of double structures supported on a smooth curve described in \cite{fer}, we specialize it to double structures on rational normal curves. To every double structure we associate a triple of integers $ (2r,g,n) $ where $ r $ is the degree of the support, $ n \geq r $ is the dimension of the projective space containing the double curve, and $ g $ is the arithmetic genus of the double curve. We compute also some numerical invariants of the constructed curves, and we show that the family of double structures with a given triple $ (2r,g,n) $ is irreducible. Furthermore, we prove that the general double curve in the families associated to $ (2r,r+1,r) $ and $ (2r,1,2r-1) $ is arithmetically Gorenstein. Finally, we prove that the closure of the locus containing double conics of genus $ g \leq -2 $ form an irreducible component of the corresponding Hilbert scheme, and that the general double conic is a smooth point of that component. Moreover, we prove that the general double conic in $ \mathbb{P}^3 $ of arbitrary genus is a smooth point of the corresponding Hilbert scheme.