Non-reduced Components of the Hilbert Scheme of Curves Using Triple Covers

IF 16.4 1区 化学 Q1 CHEMISTRY, MULTIDISCIPLINARY
Youngook Choi, Hristo Iliev, Seonja Kim
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引用次数: 0

Abstract

In this paper, we consider curves on a cone that pass through the vertex and are also triple covers of the base of the cone, which is a general smooth curve of genus \(\gamma \) and degree e in \({\mathbb {P}}^{e-\gamma }\). Using the free resolution of the ideal of such a curve found by Catalisano and Gimigliano, and a technique concerning deformations of curves introduced by Ciliberto, we show that the deformations of such curves remain on cones over a deformation of the base curve. This allows us to prove that for \(\gamma \ge 3\) and \(e \ge 4\gamma + 5\), there exists a non-reduced component \({\mathcal {H}}\) of the Hilbert scheme of smooth curves of genus \(3e + 3\gamma \) and degree \(3e+1\) in \({\mathbb {P}}^{e-\gamma +1}\). We show that \(\dim T_{[X]} {\mathcal {H}} = \dim {\mathcal {H}} + 1 = (e - \gamma + 1)^2 + 7e + 5\) for a general point \([X] \in {\mathcal {H}}\).

Abstract Image

使用三重盖的曲线希尔伯特方案的非还原成分
在本文中,我们考虑了圆锥上的曲线,这些曲线通过顶点,同时也是圆锥底面的三重盖,而圆锥底面是 \(\gamma \) 属性和 \({\mathbb {P}}^{e-\gamma }\) 度为 e 的一般平滑曲线。利用卡塔利萨诺(Catalisano)和吉米利亚诺(Gimigliano)发现的这种曲线理想的自由解析,以及西里贝托(Ciliberto)引入的关于曲线变形的技术,我们证明了这种曲线的变形保持在基曲线变形的圆锥上。这让我们能够证明,对于 \(\gamma\ge 3\) 和 \(e\ge 4\gamma + 5\), 在 \({\mathbb {P}}^{e-\gamma +1}\) 中存在一个非还原的平滑曲线的希尔伯特方案的组成部分 \({mathcal {H}}\) genus \(3e + 3\gamma\) and degree \(3e+1\).我们证明(dim T_{[X]}{\mathcal {H}} = \dim {\mathcal {H}}+ 1 = (e - \gamma + 1)^2 + 7e + 5\) for a general point \([X] \ in {\mathcal {H}}\).
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来源期刊
Accounts of Chemical Research
Accounts of Chemical Research 化学-化学综合
CiteScore
31.40
自引率
1.10%
发文量
312
审稿时长
2 months
期刊介绍: Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.
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