lecturenotes 11,12 - Lectures on algebraic geometry Jacobs...

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Unformatted text preview: Lectures on algebraic geometry Jacobs 2010 Notes of lectures 11 and 12 Alan Huckleberry March 20, 2010 Here we begin an introductory study of hyperelliptic curves of degree larger than three. The procedure of blowing up a point in order to desingularize them is introduced. Generalities on 2:1 covers X → P 1 , in particular the unicity of the hyperelliptic involution, are proved. A space parameterizing all such manifolds is constructed. 1 The blowup Bl p ( Z ) Let us begin by underlining our motivation for considering blowups. Singular hyperelliptic curves Let C = C b be the affine hyperelliptic curve defined by w 2- d Y j =1 ( z- b j ) = 0 . As usual we assume that b i 6 = b j for i 6 = j . Dehomogenizing by z = z 1 z- 1 and w = z 2 z- 1 we obtain z 2 2 z d- 2- Y ( z 1- b j z ) = 0 as the equation of the associated projective curve which we also denote by C . Observe that there is only one point at infinity, i.e., in C ∩ { z = 0 } , 1 namely b = [0 : 0 : 1]. The projection of the affine curve C → C defined by ( z,w ) → z extends to the restriction of the projection π b : P 2 → P 1 of the ambient projective space. In order to understand the nature of the projective curve near b we write its defining equation in the standard affine coordinates ζ := z z- 1 2 and η := z 1 z- 1 2 of U 2 = { z 2 6 = 0 } : ζ d- 2- Y ( η- b j ζ ) = 0 . Here we are most interested in the case where d ≥ 4 so that in particular b is a singularity of C . Below we show how to blowup the ambient projective space P 2 in order to desingularize C . Blowing up a point Our goal here is to construct the blowup Bl p ( Z ) of a point p in a 2-dimensional smooth variety (or complex manifold) Z . Since the procedure is local, it is enough to let p = 0 be the origin in Z = C 2 and define Bl ( C 2 ) := { (( x 1 ,x 2 ) , [ z 1 : z 2 ]) ∈ C 2 × P 1 ; x 1 z 2- x 2 z 1 = 0 } . Extending our standard notation to this case, we let U i := { p ∈ Bl ( C 2 ); z i 6 = } , i = 1 , 2. If z := z 2 z- 1 1 in U 1 , then the defining equation for Bl ( C 2 ) in U 1 is x 2 = zx 1 . Similarly we let w := z 1 z- 1 2 in U 2 so that the defining equation is reversed to x 1 = wx 2 . In either open set it is immediate that Bl ( C 2 ) is smooth. Exercise. Show that Bl ( C 2 ) can be regarded as the graph of the meromor- phic function m = x 1 x- 1 2 . Let π : Bl ( C 2 ) → C 2 be the regular map defined by projection on the first factor and define E := π- 1 (0). Observe that E = { ((0 , 0) , [ z 1 : z 2 ]) ∈ C 2 × P 1 } so that it is naturally identified with P 1 . If x = ( x 1 ,x 2 ) is not the origin then π- 1 ( x ) consists of a single point and one checks that π | (Bl ( C 2 ) \ E ) : Bl ( C 2 ) \ E → C 2 \ { } 2 is a biregular (biholomorphic) map. Given any 2-dimensional complex man- ifold Z and a point p ∈ Z , we choose an open coordinate neighborhood V = V ( p ) which can be regarded as an open neighborhood of p = 0 in C 2 ....
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lecturenotes 11,12 - Lectures on algebraic geometry Jacobs...

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