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Unformatted text preview: Lectures on algebraic geometry Jacobs 2010 Notes of lectures 9 and 10 Alan Huckleberry March 16, 2010 In the previous lecture we discussed curves of degree two, i.e., conics. Here we begin by considering curves of degree three which are presented as dou ble covers of P 1 . Interesting families of such elliptic curves are discussed. Methods for dealing with singularities are sketched. 1 Elliptic curves Let us begin by considering affine curves of the form C = { ( z, w ); w 2 = P ( z ) } where P ∈ C [ z ] is a polynomial in a single variable. Families of singular curves It is important to emphasize that singular curves arise as above. For example, let P t := z 2 ( z t ). Computing the differential of the defining equation Q t ( z ) = w 2 P t ( z ) = 0 we observe that every curve in the family C t = { Q t ( z ) = 0 } , t ∈ C , is singular at the point (0 , 0). Consider the linear C *action on C 2 defined by λ ( z, w ) = ( λ 2 z, λ 3 w ) . The associated action on the polynomials Q t is given by Q t ( z ) 7→ Q t ( λ 1 z ) = λ 6 ( w 2 z 2 ( z λt ) . 1 This fixes the curve defined by w 2 z 3 and acts transitively on the rest. In particular we see that for t 6 = s the curves C t and C s are isomorphic. We would like to point out that C and, e.g., C 1 are not isomorphic, even locally at their singularities. For this we first consider C 1 near the origin where we view its defining equation as w 2 z 2 u ( z ) = 0 where u ( z ) = z 1 can be written as u ( z ) = v ( z ) 2 . Changing coordinates by replacing z by zv ( z ), the equation takes on the simple form w 2 z 2 = ( w z )( w + z ) =. In other words, C 1 is locally near (0 , 0) the union of two lines. Now consider C and observe that there is a globally defined regular map N : C = b C → C , ζ 7→ ( ζ 2 , ζ 3 ) . It is even a homeomorphism! Example. Show that C is (locally at the origin) not even homeomorphic to C 1 . By homogenizing the defining polynomials we now consider C and C 1 to be curves in P 2 which are smooth outside of the origin [1 : 0 : 0]. It should be observed that the regular map N above can be extended to a regular map N : P 1 → C which is a homeomorphism and is nonsingular over all points except the origin where C is singular. Note that our considerations which revolved around the genus formula show that, since the defining polynomial of C in P 2 is a cubic, C is homologous to a torus. On the other hand the map N shows that it is homeomorphic to a sphere! Exercise. Show that there is a regular map N : P 1 → C 1 , → P 2 which is everywhere of maximal rank with N ([1 : 0]) = N ([0 : 1]) = [1 : 0 : 0] and with N : P 1 \ { [1 : 0] , [0 : 1] } → C 1 \ { [1 : 0 : 0] } an isomorphism, i.e., a biregular map of algebraic varieties....
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This note was uploaded on 05/18/2010 for the course MATHEMATIC 120102 taught by Professor Xxxxxxxxxyyyyyy during the Spring '10 term at Jacobs University Bremen.
 Spring '10
 xxxxxxxxxyyyyyy
 Algebra, Geometry

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