Obstructions to deforming space curves lying on a smooth cubic surface

TL;DR

This paper investigates the deformation theory of space curves on smooth cubic surfaces, proving a conjecture about maximal families and providing conditions for obstructions based on lines on the surface.

Contribution

It proves a conjecture regarding maximal families of space curves on cubic surfaces and offers a new criterion for when such curves are obstructed in deforming within projective space.

Findings

Proved a conjecture by Kleppe and Ellia on maximal families of space curves.

Established a sufficient condition for obstructions based on lines on the surface.

Utilized Hilbert-flag schemes and recent results on primary obstructions in the proofs.

Abstract

In this paper, we study the deformations of curves in the projective 3-space $\mathbb P^3$ (space curves), one of the most classically studied objects in algebraic geometry. We prove a conjecture due to J. O. Kleppe (in fact, a version modified by Ph. Ellia) concerning maximal families of space curves lying on a smooth cubic surface, assuming the quadratic normality of its general members. We also give a sufficient condition for curves lying on a cubic surface to be obstructed in $\mathbb P^3$ in terms of lines on the surface. For the proofs, we use the Hilbert-flag scheme of $\mathbb P^3$ as a main tool and apply a recent result on primary obstructions to deforming curves on a threefold developed by S. Mukai and the author.