Families of Smooth Rational Curves of Small Degree on the Fano Variety of Degree 5 of Main Series
In this paper we consider some families of smooth rational curves of degree 2, 3 and 4 on a smooth Fano threefold X which is a linear section of the Grassmanian G(1, 4) under the Pl¨ucker embedding. We prove that these families are irreducible. The proof of the irreducibility of the families of curv...
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| Format: | Article |
| Language: | English |
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Yaroslavl State University
2013-06-01
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| Series: | Моделирование и анализ информационных систем |
| Subjects: | |
| Online Access: | https://www.mais-journal.ru/jour/article/view/198 |
| _version_ | 1797877985421819904 |
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| author | M. S. Omelkova |
| author_facet | M. S. Omelkova |
| author_sort | M. S. Omelkova |
| collection | DOAJ |
| description | In this paper we consider some families of smooth rational curves of degree 2, 3 and 4 on a smooth Fano threefold X which is a linear section of the Grassmanian G(1, 4) under the Pl¨ucker embedding. We prove that these families are irreducible. The proof of the irreducibility of the families of curves of degree d is based on the study of degeneration of a rational curve of degree d into a curve which decomposes into an irreducible rational curve of degree d−1 and a projective line intersecting transversally at a point. We prove that the Hilbert scheme of curves of degree d on X is smooth at the point corresponding to such a reducible curve. Then calculations in the framework of deformation theory show that such a curve varies into a smooth rational curve of degree d. Thus, the set of reducible curves of degree d of the above type lies in the closure of a unique component of the Hilbert scheme of smooth rational curves of degree d on X. From this fact and the irreducibility of the Hilbert scheme of smooth rational curves of degree d on the Grassmannian G(1, 4) one deduces the irreducibility of the Hilbert scheme of smooth rational curves of degree d on a general Fano threefold X. |
| first_indexed | 2024-04-10T02:25:32Z |
| format | Article |
| id | doaj.art-f9ca878f8714402bb36fad8120f401a7 |
| institution | Directory Open Access Journal |
| issn | 1818-1015 2313-5417 |
| language | English |
| last_indexed | 2024-04-10T02:25:32Z |
| publishDate | 2013-06-01 |
| publisher | Yaroslavl State University |
| record_format | Article |
| series | Моделирование и анализ информационных систем |
| spelling | doaj.art-f9ca878f8714402bb36fad8120f401a72023-03-13T08:07:31ZengYaroslavl State UniversityМоделирование и анализ информационных систем1818-10152313-54172013-06-012039910710.18255/1818-1015-2013-3-99-107192Families of Smooth Rational Curves of Small Degree on the Fano Variety of Degree 5 of Main SeriesM. S. Omelkova0Костромской государственный университет им. Н. А. НекрасоваIn this paper we consider some families of smooth rational curves of degree 2, 3 and 4 on a smooth Fano threefold X which is a linear section of the Grassmanian G(1, 4) under the Pl¨ucker embedding. We prove that these families are irreducible. The proof of the irreducibility of the families of curves of degree d is based on the study of degeneration of a rational curve of degree d into a curve which decomposes into an irreducible rational curve of degree d−1 and a projective line intersecting transversally at a point. We prove that the Hilbert scheme of curves of degree d on X is smooth at the point corresponding to such a reducible curve. Then calculations in the framework of deformation theory show that such a curve varies into a smooth rational curve of degree d. Thus, the set of reducible curves of degree d of the above type lies in the closure of a unique component of the Hilbert scheme of smooth rational curves of degree d on X. From this fact and the irreducibility of the Hilbert scheme of smooth rational curves of degree d on the Grassmannian G(1, 4) one deduces the irreducibility of the Hilbert scheme of smooth rational curves of degree d on a general Fano threefold X.https://www.mais-journal.ru/jour/article/view/198многообразия фаноконструкция серрасхемы гильберта кривых |
| spellingShingle | M. S. Omelkova Families of Smooth Rational Curves of Small Degree on the Fano Variety of Degree 5 of Main Series Моделирование и анализ информационных систем многообразия фано конструкция серра схемы гильберта кривых |
| title | Families of Smooth Rational Curves of Small Degree on the Fano Variety of Degree 5 of Main Series |
| title_full | Families of Smooth Rational Curves of Small Degree on the Fano Variety of Degree 5 of Main Series |
| title_fullStr | Families of Smooth Rational Curves of Small Degree on the Fano Variety of Degree 5 of Main Series |
| title_full_unstemmed | Families of Smooth Rational Curves of Small Degree on the Fano Variety of Degree 5 of Main Series |
| title_short | Families of Smooth Rational Curves of Small Degree on the Fano Variety of Degree 5 of Main Series |
| title_sort | families of smooth rational curves of small degree on the fano variety of degree 5 of main series |
| topic | многообразия фано конструкция серра схемы гильберта кривых |
| url | https://www.mais-journal.ru/jour/article/view/198 |
| work_keys_str_mv | AT msomelkova familiesofsmoothrationalcurvesofsmalldegreeonthefanovarietyofdegree5ofmainseries |