Sara Muhvić: Racionalne ravninske krivulje
Date of publication: 6. 5. 2013
Algebra seminar
Sreda, 8. maja 2013, ob 10. uri v Plemljevem seminarju, Jadranska 19/III, Ljubljana
SAZETAK
Ovaj rad predstavlja neka osnovna svojstva racionalnih krivulja i jednu metodu odreivanja racionalnosti ravninskih krivulja. Daje se dokaz Lurothovog teorema koji povlai ekvivalenciju dviju standardnih denicija racionalnosti. Kako su te denicije isto teorijske, u nastavku se daje kriterij za racionalnost krivulje koji se temelji na poznavanju jednostavnih singulariteta krivulje i njihovih multipliciteta. Konano, predstavlja se algoritam koji ireducibilnu krivulju nizom kvadratnih transformacija prevodi u krivulju iji su svi singulariteti jednostavni, s ciljem odreivanja racionalnosti polazne krivulje.
SUMARRY
This paper covers some of the basic properties of rational plane curves and a method of determining the rationality of a given curve. The proof of the Luroth theorem is presented, which gives, as a consequence, equivalence between two standard denitions of rational plane curves. Since those denitions have only theoretical importance, the criteria for determining the rationality of a curve, based on knowledge of all the ordinary singularities of a given curve, is also given. Finally, the paper presents the algorithm which, while not changing its rationality, transforms a given curve into a curve with only ordinary singularities.