# Obstructions to deforming space curves lying on a smooth cubic surface

@article{Nasu2019ObstructionsTD, title={Obstructions to deforming space curves lying on a smooth cubic surface}, author={Hirokazu Nasu}, journal={arXiv: Algebraic Geometry}, year={2019} }

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…

## References

SHOWING 1-10 OF 25 REFERENCES

Obstructions to deforming curves on a 3-fold, III: Deformations of curves lying on a K3 surface

- Mathematics
- 2016

We study the deformations of a smooth curve $C$ on a smooth projective threefold $V$, assuming the presence of a smooth surface $S$ satisfying $C \subset S \subset V$. Generalizing a result of Mukai…

Obstructions to deforming curves on a prime Fano 3‐fold

- MathematicsMathematische Nachrichten
- 2019

We prove that for every smooth prime Fano 3‐fold V, the Hilbert scheme HilbscV of smooth connected curves on V contains a generically non‐reduced irreducible component of Mumford type. We also study…

Murphy’s law in algebraic geometry: Badly-behaved deformation spaces

- Mathematics
- 2004

We consider the question: “How bad can the deformation space of an object be?” The answer seems to be: “Unless there is some a priori reason otherwise, the deformation space may be as bad as…

Obstructions to deforming curves on a 3-fold, I: A generalization of Mumford’s example and an application to schemes

- Mathematics
- 2006

We give a sufficient condition for a first order infinitesimal deformation of a curve on a 3-fold to be obstructed. As application we construct generically non-reduced components of the Hilbert…

The Hilbert scheme of space curves sitting on a smooth surface containing a line

- Mathematics
- 2017

We continue the study of maximal families W of the Hilbert scheme, $$ {{\mathrm{H}}}(d,g)_{sc}$$H(d,g)sc, of smooth connected space curves whose general curve C lies on a smooth degree-s surface S…

Castelnuovo curves and unobstructed deformations

- Mathematics
- 1980

Now it is a well-known fact from Kodai ra [8], that if H i ( N ) = 0 , then there exists a non-singular family ~ for which p is surjective. In classical language, this is the "completeness of the…

On generically non‐reduced components of Hilbert schemes of smooth curves

- Mathematics
- 2017

A classical example of Mumford gives a generically non‐reduced component of the Hilbert scheme of smooth curves in P3 such that a general element of the component is contained in a smooth cubic…

Le schéma de Hilbert des courbes gauches localement Cohen-Macaulay n'est (presque) jamais réduit

- Mathematics
- 1996

— Let k be an algebraically closed field of characteristic zéro. Let Hd,g dénote thé Hilbert scheme of locally Cohen-Macaulay curves of degree d and genus g contained in thé projective space P|. If d…

Components of the Hilbert scheme of space curves on low-degree smooth surfaces

- Mathematics
- 2012

Electronic version of an article published as Kleppe, J. O., & Ottem, J. C. (2015). Components of the Hilbert scheme of space curves on low-degree smooth surfaces. International Journal of…