Smoothness of the Hilbert scheme of a few degrees

JOURNAL OF ALGEBRA

卷:277

Article

Guffroy, S

关键词: Hilbert scheme;    space curves;    normal bundle;    double structure;   

DOI  :  10.1016/j.jalgebra.2003.10.018

来源: Elsevier

【 摘 要 】

We check that the Hilbert scheme, H-d,H-g, of smooth and connected curves of degree d and genus g in projective three-dimensional space over C is smooth provided that d less than or equal to 11. The proof uses essentially our good knowledge of curves lying on cubic surfaces and the possibility to endow a curve having a special normal bundle with a double structure of high arithmetic genus. Then we give some partial results in the case of degree 12. Namely, we obtain that H-12,H-g is smooth for g < 15 except cases g = 11, 12, for which we were able to establish only that H-12,H-g is smooth in codimension 1. This shows that (12, 15) is the lexicographically first pair (d, g) such that H-d,H-g is singular in codimension 1. (C) 2003 Elsevier Inc. All rights reserved.

【 授权许可】

Free