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Unformatted text preview: Lectures on algebraic geometry Jacobs 2010 Notes of lectures 3 and 4 Alan Huckleberry February 13, 2010 1 Introduction As we have seen in the last lectures it is convenient to regard an affine algebraic curve C = { ( z,w ); P ( z,w ) = 0 } , defined by a polynomial P ( z,w ) ∈ C [ z,w ], as a ramified cover of the complex plane by the projection map π : C → C , ( z,w ) 7→ z . Often one can restrict to the case of polynomials P ( z,w ) = w d + a d 1 ( z ) w d 1 + ... + a ( z ) where the coefficients a j ( z ) are polynomials of one variable. In that way we have a family of polynomials parameterized by the zaxis and on the curve itself we have an interesting family of linearly equivalent divisors. Note that the degree of these divisors is constant, namely d , and that as z → ∞ in the complex plane there is no pathological behavior. Thus one can be optimistic that the family can be compatified in some reasonable way. Our goal here is to introduce a number of compact manifolds which appear throughout algebraic geometry which turn out to be optimal compactifications of C n . In the following lectures we will extend the ramified covers to compactifications and begin a study of curves in the (compact) complex projective plane. 1 2 Grassmannians Let V be a complex vector space and k be an integer with 0 < k ≤ n := dim C ( V ). Define the Grassmannian of kplanes in V by Gr k ( V ) := { W < V ; dim( W ) = k } . We write W < V to indicate that W is a vector subspace of V . It should be remarked that referring to W as a kplane might cause confusion, because one might also regard a kdimensional affine subspace as a kplane. However, we hope that it will be clear from the contexts considered that we are dealing with subspaces. Example. Let C be a smooth affine curve as defined above. For every point p ∈ C the tangent space T p C is the subspace of vectors v in the tangent space T p C 2 with h∇ ( P )( p ) ,v i = 0 where h , i is the standard complex bilinear form on C 2 . By Euclidean parallel translation we my consider these spaces to be centered at the origin in the ambient space, i.e., in the ambient vector space C 2 . Thus for every p ∈ C we consider the tangent space T p C as a complex 1dimensional subspace of C 2 , i.e., as a point in Gr 1 ( C 2 ). The map G : C → Gr 1 ( C 2 ), p 7→ T p C , which is called the Gauss map, is a holomorphic map of the curve into the Grassmannian which tells us a great deal about the embedding of C in C 2 . 2.1 Coordinate charts Up to this point a Grassmannian Gr k ( V ) is just a set. Now, given a kplane P ∈ Gr k ( V ) we introduce subsets of Gr k ( V ) which contain P and which will define a local basis of open sets at P . For this we let E be a complementary subspace so that V = P ⊕ E and let U E ( P ) be the subset of Gr k ( V ) consisting of those kplanes which arise as graphs of linear mappings T : P → E , i.e., U I ( P ) is identified with Hom( P,E ) by the map T 7→ Graph( T ). It makes)....
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 Spring '10
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 Algebra, Geometry

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