| Abstract | U ovom radu bavili smo se Riemann-Rochovim teoremom za algebarske krivulje. Uveli smo nužne pojmove iz algebarske geometrije, dokazali teorem i naveli neke njegove primjene. Prvo smo dali kratki uvod u algebarsku geometriju gdje smo definirali afine i projektivne mnogostrukosti, te preslikavanja između njih. Dalje smo se fokusirali specifično na glatke krivulje, tj. nesingularne projektivne mnogostrukosti dimenzije 1. Uveli smo funkcijska polja koja čine kategoriju koja je ekvivalentna kategoriji glatkih krivulja. Pokazali smo kako se tvrdnje o funkcijskim poljima i o krivuljama prevode iz jednog konteksta u drugi. Definirali smo grupu divizora koja je pridružena krivulji i ustanovili osnovna svojstva divizora. Svakom divizoru je pridružen Riemann-Rochov prostor koji je sačinjen od racionalnih funkcija sa zadanim nultočkama i polovima. Definirali smo genus krivulje i ustanovili nekoliko rezultata o Riemann-Rochovim prostorima. Dokazali smo Riemannovu nejednakost koja nam daje relaciju između dimenzije Riemann-Rochovog prostora, stupnja divizora i genusa krivulje. Onda smo uveli prstene adela i Weilove diferencijale da okarakteriziramo indeks specijalnosti i dokažemo Riemann-Rochov teorem koji pojačava Riemannovu nejednakost. Naveli smo primjene Riemann-Rochovog teorema: odredili smo svojstva kanonskog divizora, karakterizirali krivulje genusa 0 s racionalnom točkom do na izomorfizam, dokazali da je svaka eliptička krivulja izomorfna nekoj krivulji zadanoj Weierstrassovom jednadžbom, definirali smo grupni zakon na eliptičkoj krivulji, dokazali smo neke tvrdnje o specijalnim divizorima uključujući Cliffordov teorem. |
| Abstract (english) | In this thesis, we explore the Riemann-Roch theorem for algebraic curves. We introduced the necessary notions from algebraic geometry, proved the Riemann-Roch theorem, and stated some of its applications. We started with a brief introduction to algebraic geometry, in which we defined affine and projective varieties, and the maps between them. Later we focused specifically on smooth curves, i.e. nonsingular projective varieties of dimension 1. We introduced function fields that form a category, that is equivalent to the category of smooth curves. We demonstrated how statements about function fields and curves can be translated from one context to another. For each curve, we defined the associated divisor group and established some of the basic properties of divisors. To each divisor, there is an associated Riemann-Roch space. It is made up of rational functions with prescribed zeros and poles. We defined the genus of a curve and proved a couple of results about Riemann-Roch spaces. We proved the Riemann inequality, which relates the dimension of a Rimeann-Roch space, the degree of a divisor, and the curve genus. Then, we introduced adele rings and Weil differentials to characterize the index of speciality and prove the Riemann-Roch theorem which improves the Riemann inequality. We stated some applications of the Riemann-Roch theorem: we determined the properties of the canonical divisor, we characterized smooth curves of genus 0 containing a rational point up to isomorphism, we proved that each elliptic curve is isomorphic to a curve defined by Weierstrass equation, we defined the group law on the elliptic curve, and we proved some results about special divisors, including Clifford's theorem. |
