{"title":"Regular and rigid curves on some Calabi–Yau and general-type complete intersections","authors":"Ziv Ran","doi":"10.1142/s0129167x24420011","DOIUrl":null,"url":null,"abstract":"<p>Let <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><mi>X</mi></math></span><span></span> be either a general hypersurface of degree <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><mi>n</mi><mo stretchy=\"false\">+</mo><mn>1</mn></math></span><span></span> in <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><msup><mrow><mi>ℙ</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span><span></span> or a general <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><mo stretchy=\"false\">(</mo><mn>2</mn><mo>,</mo><mi>n</mi><mo stretchy=\"false\">)</mo></math></span><span></span> complete intersection in <span><math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"><msup><mrow><mi>ℙ</mi></mrow><mrow><mi>n</mi><mo stretchy=\"false\">+</mo><mn>1</mn></mrow></msup><mo>,</mo><mi>n</mi><mo>≥</mo><mn>4</mn></math></span><span></span>. We construct balanced rational curves on <span><math altimg=\"eq-00006.gif\" display=\"inline\" overflow=\"scroll\"><mi>X</mi></math></span><span></span> of all high enough degrees. If <span><math altimg=\"eq-00007.gif\" display=\"inline\" overflow=\"scroll\"><mi>n</mi><mo>=</mo><mn>4</mn></math></span><span></span> or <span><math altimg=\"eq-00008.gif\" display=\"inline\" overflow=\"scroll\"><mi>g</mi><mo>=</mo><mn>1</mn></math></span><span></span>, we construct rigid curves of genus <span><math altimg=\"eq-00009.gif\" display=\"inline\" overflow=\"scroll\"><mi>g</mi></math></span><span></span> on <span><math altimg=\"eq-00010.gif\" display=\"inline\" overflow=\"scroll\"><mi>X</mi></math></span><span></span> of all high enough degrees. As an application we construct some rigid bundles on Calabi–Yau threefolds. In addition, we construct some low-degree balanced rational curves on hypersurfaces of degree <span><math altimg=\"eq-00011.gif\" display=\"inline\" overflow=\"scroll\"><mi>n</mi><mo stretchy=\"false\">+</mo><mn>2</mn></math></span><span></span> in <span><math altimg=\"eq-00012.gif\" display=\"inline\" overflow=\"scroll\"><msup><mrow><mi>ℙ</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span><span></span>.</p>","PeriodicalId":54951,"journal":{"name":"International Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6000,"publicationDate":"2024-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"International Journal of Mathematics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1142/s0129167x24420011","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Let be either a general hypersurface of degree in or a general complete intersection in . We construct balanced rational curves on of all high enough degrees. If or , we construct rigid curves of genus on of all high enough degrees. As an application we construct some rigid bundles on Calabi–Yau threefolds. In addition, we construct some low-degree balanced rational curves on hypersurfaces of degree in .
期刊介绍:
The International Journal of Mathematics publishes original papers in mathematics in general, but giving a preference to those in the areas of mathematics represented by the editorial board. The journal has been published monthly except in June and December to bring out new results without delay. Occasionally, expository papers of exceptional value may also be published. The first issue appeared in March 1990.