lecturenotes 1,2 - Lectures on algebraic geometry Jacobs...

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Lectures on algebraic geometry Jacobs 2010 Notes of lectures 1 and 2 Alan Huckleberry February 6, 2010 1 Introduction We will study the geometry of algebraic varieties X which are at least lo- cally defined to be the common zero-set { P 1 = . . . = P k = 0 } of a number polynomials in n variables. At least at the beginning the discussion will be carried out over the field C . A good starting point is one polynomial of a single variable. 2 The fundamental theorem of algebra Since the field of complex numbers is really an analytic object, the name of this theorem may be regarded as a misnomer. Here is the most familiar formulation. Theorem 2.1. If P C [ w ] a polynomial of degree d , then there are uniquely determined complex numbers c C * and w 1 , . . . , w k C and positive integers n 1 , . . . , n k with n j = d so that P = c ( w - w 1 ) n 1 · . . . · ( w - w k ) n k . We would like to discuss this theorem in detail, but first let us prove it using several ideas which are useful in complex geometry. 1
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First, instead of regarding P as an element of some ring, we interpret it as a map P : C C . Such a map has a graph C = { ( z, w ) C 2 ; P ( w ) - z = 0 } which we regard as a complex curve in C 2 . We may identify the domain of definition of P with C and consquently P : C C is defined by the projection ( z, w ) z . Thus the proof Theorem 2.1 is reduced to proving the following. Theorem 2.2. The map P : C C is surjective. To prove this we will show that Im( P ) is both open and closed. For the latter property we recall that a map F : X Y between topological spaces is called proper if for all compact subsets K of X the image F ( K ) is compact. Here we have assumed that X and Y are Hausdorff topological spaces. Note properness can also be defined by requiring divergent sequences to be mapped to divergent sequences. Proposition 2.3. A nonconstant polynomial P C [ z ] defines a proper map P : C C . In particular, Im( P ) is closed. Proof. If P ( w ) = a d w d + O ( w d - 1), then | P ( w ) | ≥ | a d || w | d - O ( | w | d - 1 ) . As | w | → ∞ the term | w | d dominates and therefore lim | w |→∞ | P ( w ) | = . Now we turn to proving the second property, i.e., to proving the openness of the image. Exercise. Decompose P into real and imaginary parts, P = u + iv , and view P as mapping P : R 2 R 2 . Compute the Jacobian determinant J ( P ) of this real mapping and show that it is J ( P ) = | P ( w ) | 2 . Here the complex derivative is computed as expected: If P = a k w k , then P = ka k w k - 1 . 2
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Application of the C -inverse mapping theorem. As a result one knows that P is locally invertible as a smooth map at every point which is not in the ramification set R := { w ; P ( w ) = 0 } . Therefore the set B = F ( R ) of branch points will play a key role our discussion. The following completes the proof of Theorem 2.1 Proposition 2.4. The image Im( P ) is an open subset of C . Proof. We must only show that every b B is contained in an open subset which is itself contained in Im( P ). For this note that B is finite and let Δ be a disk at b which contains no other branch point. Define Δ * := Δ \ { b } and let S := Im( P ) Δ * . Since S contains no branch points, it is open and since Im( P ) is closed, it is also closed. Finally, since Δ
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