lecturenotes 5,6 - Lectures on algebraic geometry Jacobs...

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Unformatted text preview: Lectures on algebraic geometry Jacobs 2010 Notes of lectures 5 and 6 Alan Huckleberry February 28, 2010 1 Introduction In the previous lectures we discussed affine curves as ramified covers over the complex plane. With the goal of compactifying such affine varieties we then introduced Grassmannians, in particular projective spaces. In these lectures we look in more detail at low-dimensional projective spaces and begin a study of (algebraic) plane curves, i.e., curves in P 2 ( C ). 2 A closer look at projective space Let V be a complex vector space equipped with a basis { v ,...,v n } and ( z ,...,z n ) be global coordinates which identify V with C n +1 . As usual denote by [ z : ... : z n ] the associated homogeneous coordinates for P n ( C ). The standard associated covering { U i } of coordinate charts is defined by U i := { [ v ]; z i ( v ) 6 = 0 } . Recall that although the function z i is not defined on P n , the condition z i 6 = 0 makes sense. Example. Let us look more closely at P 3 . The standard covering considers of four sets U i , i = 0 ,..., 3. We view U as the basic big cell with its coordinates ( ζ 1 ,ζ 2 ,ζ 3 ) where ζ i = z i z- 1 . Note that the complement H of this open cell may be identifed with P 2 by the map [0 : z 1 : z 2 : z 3 ] → [ z 1 : z 2 : z 3 ]. Thus we think of P 3 as the disjoint union C 3 ˙ ∪ P 2 . For the following reason one often 1 regards H as being the hyperplane at infinity of the affine space C 3 . The reason for this is that every line L = C .v in C 3 closes up to exactly one point in H : If v has affine coordinates ( ζ 1 ,ζ 2 ,ζ 3 ), i.e., homogeneous coordinates [1 : ζ 1 : ζ 2 : ζ 3 ], then we view the closure of L as the image of P 1 given by the map P 1 → P 3 , [ s,t ] 7→ s [1 : 0 : 0 : 0] + t [0 : ζ 1 : ζ 2 : ζ 3 ] := [ s : tζ 1 : tζ 2 : tζ 3 ] . For example, on U one has the coordinate ϕ : [ z : ... : z 3 ] 7→ ( z 1 z , z 2 z , z 3 z ) = ( ζ 1 ,ζ 2 ,ζ 3 ) . If ( η 1 ,η 2 ,η n ) are the analgous coordinates on U 1 , then the change of coordi- nates map ϕ 01 is given by ( ζ 1 ,ζ 2 ,ζ 3 ) 7→ ( ζ- 1 1 ,ζ 2 ζ- 1 1 ,ζ 3 ζ- 1 1 ). 3 Rational functions as ramified coverings One of our original reasons for introducing P 1 was to compactify the complex plane and to understand the behavior of a rational function at infinity. For this the processes of homogenization and dehomogenization of polynomials is of basic importance. Here we begin by discussing this for P ∈ C [ z ]. In this case we regard z = z 1 z a the standard coordinate on U = { [ z : z 1 ] ∈ P 1 ; z 6 = 0 } . If P = a d z d + a d- 1 z d- 1 + ... + a is a polynomial of degree d , then we insert z = z 1 z and clear denominators to obtain P = z- d P h ( z ,z 1 ) where P h ( z ,z 1 ) = a d z d 1 + a d- 1 z d- 1 1 z + ... + a z d is a homogeneous polynomial of degree d . We refer to the process P 7→ P h as having homogenized the polynomial P to obtain the associated homogeneous polynomial P h . Conversely if....
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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.

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lecturenotes 5,6 - Lectures on algebraic geometry Jacobs...

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