{"title":"Non-K3 Weierstrass numerical semigroups","authors":"Jiryo Komeda, Makiko Mase","doi":"10.1007/s00233-024-10406-0","DOIUrl":null,"url":null,"abstract":"<p>We generalize the result of Reid (J Lond Math Soc 13:454–458, 1976), namely, we prove that a curve of genus <span>\\(\\geqq g^2+4g+6\\)</span> having a double cover of a hyperelliptic curve of genus <span>\\(g\\geqq 2\\)</span> does not lie as a non-singular curve on any K3 surface. Applying this result we construct non-K3 Weierstrass numerical semigroups. A numerical semigroup <i>H</i> is said to be <i>Weierstrass</i> if there exists a pointed non-singular curve (<i>C</i>, <i>P</i>) such that <i>H</i> consists of non-negative integers which are the pole orders at <i>P</i> of a rational function on <i>C</i> having a pole only at <i>P</i>. We call the numerical semigroup <i>K3</i> if we can take the curve <i>C</i> as a curve on some K3 surface. A <i>non-K3 numerical semigroup</i> means that it cannot be attained by a pointed non-singular curve on any K3 surface. We also give infinite sequences of non-K3 Weierstrass numerical semigroups.</p>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00233-024-10406-0","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
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Abstract
We generalize the result of Reid (J Lond Math Soc 13:454–458, 1976), namely, we prove that a curve of genus \(\geqq g^2+4g+6\) having a double cover of a hyperelliptic curve of genus \(g\geqq 2\) does not lie as a non-singular curve on any K3 surface. Applying this result we construct non-K3 Weierstrass numerical semigroups. A numerical semigroup H is said to be Weierstrass if there exists a pointed non-singular curve (C, P) such that H consists of non-negative integers which are the pole orders at P of a rational function on C having a pole only at P. We call the numerical semigroup K3 if we can take the curve C as a curve on some K3 surface. A non-K3 numerical semigroup means that it cannot be attained by a pointed non-singular curve on any K3 surface. We also give infinite sequences of non-K3 Weierstrass numerical semigroups.
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