Math 431 ALGEBRAIC GEOMETRYHomework 3 Solution KeyAli Sinan Sert¨oz7 January 20031)Show that for anyg≥0, there is a compact Riemann surface of genusg. In particular showthat for anyg >2 there exist a hyperelliptic and a non-hyperelliptic compact Riemannsurface of genusg. You may not be able to give the proofs in detail but you can quote sometechnical results with full references which imply the existence of the required surfaces.Forg= 0 andg= 1 we have the projective line and the torus respectively.To construct a hyperelliptic curve of genusg≥2 we quote your text book page 141; for anygiven 2g+ 2 mutually distinct pointsa1, . . . , a2g+2inCwe first consider the curveC=(y2-2g+2Yi=1(x-ai) = 0)∪[0 : 0 : 1]⊂P1.The normalizationC0of this curve is a hyperelliptic curve of genusg.Now the remark onpage 144 of your textbook says that two such hyperelliptic curves are isomorphic if and onlyif the associated pointsa1, . . . , a2g+2
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