lecturenotes 3,4 - Lectures on algebraic geometry Jacobs...

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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 z-axis 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 k-planes 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 k-plane might cause confusion, because one might also regard a k-dimensional affine subspace as a k-plane. 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 1-dimensional 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 k-plane 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 k-planes 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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lecturenotes 3,4 - Lectures on algebraic geometry Jacobs...

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