{"title":"Non-reduced Components of the Hilbert Scheme of Curves Using Triple Covers","authors":"Youngook Choi, Hristo Iliev, Seonja Kim","doi":"10.1007/s00009-024-02668-3","DOIUrl":null,"url":null,"abstract":"<p>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 <span>\\(\\gamma \\)</span> and degree <i>e</i> in <span>\\({\\mathbb {P}}^{e-\\gamma }\\)</span>. 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 <span>\\(\\gamma \\ge 3\\)</span> and <span>\\(e \\ge 4\\gamma + 5\\)</span>, there exists a non-reduced component <span>\\({\\mathcal {H}}\\)</span> of the Hilbert scheme of smooth curves of genus <span>\\(3e + 3\\gamma \\)</span> and degree <span>\\(3e+1\\)</span> in <span>\\({\\mathbb {P}}^{e-\\gamma +1}\\)</span>. We show that <span>\\(\\dim T_{[X]} {\\mathcal {H}} = \\dim {\\mathcal {H}} + 1 = (e - \\gamma + 1)^2 + 7e + 5\\)</span> for a general point <span>\\([X] \\in {\\mathcal {H}}\\)</span>.</p>","PeriodicalId":1,"journal":{"name":"Accounts of Chemical Research","volume":null,"pages":null},"PeriodicalIF":16.4000,"publicationDate":"2024-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Accounts of Chemical Research","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00009-024-02668-3","RegionNum":1,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"CHEMISTRY, MULTIDISCIPLINARY","Score":null,"Total":0}
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 \([X] \in {\mathcal {H}}\).
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.