dc.contributor.advisor | Šťovíček, Jan | |
dc.creator | Till, Daniel | |
dc.date.accessioned | 2021-08-03T09:09:41Z | |
dc.date.available | 2021-08-03T09:09:41Z | |
dc.date.issued | 2021 | |
dc.identifier.uri | http://hdl.handle.net/20.500.11956/128186 | |
dc.description.abstract | V tejto práci dokážeme, že na každej nesingulárnej kubickej ploche nad al- gebraicky uzavretým telesom charakteristiky rôznej od dvoch existuje práve 27 rôznych priamok. Najprv sa budeme venovať afinným algebraickým varietám a ich ideálom. Dokážeme si Hilbertovu vetu o nulácha a zavedieme morfizmy medzi afinnými algebraickými varietami. Potom sa presunieme k projektívnym alge- braickým varietám a ich ideálom. Zavedieme morfizmy medzi projektívnymi va- rietami a názvoslovia pre vybrané typy projektívnych variet. Dokážeme pomocné tvrdenia o prieniku dvoch rôznych priamok na projektívnej rovine, respektíve priamky a roviny v P3 K. Taktiež definujeme pojmy ako dotyčnicový priestor k variete v danom bode, singularita nadplochy a ireducibilná varieta. Následne sa presunieme do P3 K, kde dokážeme existenciu 27 rôznych priamok na ľubovoľnej nesingulárnej kubickej ploche. Tento dôkaz urobíme tak, že najprv dokážeme, že na takejto ploche existuje priamka a potom skonštruujeme všetkých 27 priamok vzájomnými vzťahmi. 1 | cs_CZ |
dc.description.abstract | In this work we will prove there are exactly 27 different lines on each non- singular cubic surface over an agebraically closed field not of characteristic two. Firstly, we will focus on affine algebraic varieties and their ideals. We will prove Hilbert's Nullstellensatz and introduce morphisms between affine algebraic va- rieties. Then we move on to projective algebraic varieties and their ideals. We introduce morphisms between projective varieties and nomenclature for selected types of projective varieties. We will prove auxiliary statements about intersection of two distinct lines in a projective plane, respectively a line and a plane in P3 K. We also define concepts such as a tangent space to variety at a given point, sin- gularity of a hypersurface and irreducible variety. Then we move to P3 K, where we will prove the existence of 27 different lines on any nonsingular cubic surface. We will firstly prove that there is a line on such a surface and then we construct all 27 lines by mutual relations. 1 | en_US |
dc.language | Slovenčina | cs_CZ |
dc.language.iso | sk_SK | |
dc.publisher | Univerzita Karlova, Matematicko-fyzikální fakulta | cs_CZ |
dc.subject | projektívna varieta|nesingulárna nadplocha|priamka|rovina | cs_CZ |
dc.subject | projective algebraic variety|nonsingular hypersurface|line|plane | en_US |
dc.title | Veta o 27 priamkach | sk_SK |
dc.type | bakalářská práce | cs_CZ |
dcterms.created | 2021 | |
dcterms.dateAccepted | 2021-07-08 | |
dc.description.department | Department of Algebra | en_US |
dc.description.department | Katedra algebry | cs_CZ |
dc.description.faculty | Faculty of Mathematics and Physics | en_US |
dc.description.faculty | Matematicko-fyzikální fakulta | cs_CZ |
dc.identifier.repId | 206527 | |
dc.title.translated | The theorem about 27 lines | en_US |
dc.title.translated | Věta o 27 přímkách | cs_CZ |
dc.contributor.referee | Příhoda, Pavel | |
thesis.degree.name | Bc. | |
thesis.degree.level | bakalářské | cs_CZ |
thesis.degree.discipline | Obecná matematika | cs_CZ |
thesis.degree.discipline | General Mathematics | en_US |
thesis.degree.program | Mathematics | en_US |
thesis.degree.program | Matematika | cs_CZ |
uk.thesis.type | bakalářská práce | cs_CZ |
uk.taxonomy.organization-cs | Matematicko-fyzikální fakulta::Katedra algebry | cs_CZ |
uk.taxonomy.organization-en | Faculty of Mathematics and Physics::Department of Algebra | en_US |
uk.faculty-name.cs | Matematicko-fyzikální fakulta | cs_CZ |
uk.faculty-name.en | Faculty of Mathematics and Physics | en_US |
uk.faculty-abbr.cs | MFF | cs_CZ |
uk.degree-discipline.cs | Obecná matematika | cs_CZ |
uk.degree-discipline.en | General Mathematics | en_US |
uk.degree-program.cs | Matematika | cs_CZ |
uk.degree-program.en | Mathematics | en_US |
thesis.grade.cs | Velmi dobře | cs_CZ |
thesis.grade.en | Very good | en_US |
uk.abstract.cs | V tejto práci dokážeme, že na každej nesingulárnej kubickej ploche nad al- gebraicky uzavretým telesom charakteristiky rôznej od dvoch existuje práve 27 rôznych priamok. Najprv sa budeme venovať afinným algebraickým varietám a ich ideálom. Dokážeme si Hilbertovu vetu o nulácha a zavedieme morfizmy medzi afinnými algebraickými varietami. Potom sa presunieme k projektívnym alge- braickým varietám a ich ideálom. Zavedieme morfizmy medzi projektívnymi va- rietami a názvoslovia pre vybrané typy projektívnych variet. Dokážeme pomocné tvrdenia o prieniku dvoch rôznych priamok na projektívnej rovine, respektíve priamky a roviny v P3 K. Taktiež definujeme pojmy ako dotyčnicový priestor k variete v danom bode, singularita nadplochy a ireducibilná varieta. Následne sa presunieme do P3 K, kde dokážeme existenciu 27 rôznych priamok na ľubovoľnej nesingulárnej kubickej ploche. Tento dôkaz urobíme tak, že najprv dokážeme, že na takejto ploche existuje priamka a potom skonštruujeme všetkých 27 priamok vzájomnými vzťahmi. 1 | cs_CZ |
uk.abstract.en | In this work we will prove there are exactly 27 different lines on each non- singular cubic surface over an agebraically closed field not of characteristic two. Firstly, we will focus on affine algebraic varieties and their ideals. We will prove Hilbert's Nullstellensatz and introduce morphisms between affine algebraic va- rieties. Then we move on to projective algebraic varieties and their ideals. We introduce morphisms between projective varieties and nomenclature for selected types of projective varieties. We will prove auxiliary statements about intersection of two distinct lines in a projective plane, respectively a line and a plane in P3 K. We also define concepts such as a tangent space to variety at a given point, sin- gularity of a hypersurface and irreducible variety. Then we move to P3 K, where we will prove the existence of 27 different lines on any nonsingular cubic surface. We will firstly prove that there is a line on such a surface and then we construct all 27 lines by mutual relations. 1 | en_US |
uk.file-availability | V | |
uk.grantor | Univerzita Karlova, Matematicko-fyzikální fakulta, Katedra algebry | cs_CZ |
thesis.grade.code | 2 | |
uk.publication-place | Praha | cs_CZ |
uk.thesis.defenceStatus | O | |