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Math 431 ALGEBRAIC GEOMETRY Homework 3 Solution Key Ali Sinan Sert¨oz 7 January 2003 1) Show that for any g ≥ 0, there is a compact Riemann surface of genus g . In particular show that for any g > 2 there exist a hyperelliptic and a non-hyperelliptic compact Riemann surface of genus g . You may not be able to give the proofs in detail but you can quote some technical results with full references which imply the existence of the required surfaces. For g = 0 and g = 1 we have the projective line and the torus respectively. To construct a hyperelliptic curve of genus g ≥ 2 we quote your text book page 141; for any given 2 g + 2 mutually distinct points a 1 ,...,a 2 g +2 in C we ﬁrst consider the curve C = ( y 2 - 2 g +2 Y i =1 ( x - a i ) = 0 ) ∪ [0 : 0 : 1] ⊂ P 1 . The normalization C 0 of this curve is a hyperelliptic curve of genus g . Now the remark on page 144 of your textbook says that two such hyperelliptic curves are isomorphic if and only if the associated points a 1 ,...,a
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