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Unformatted text preview: Lectures on algebraic geometry Jacobs 2010 Notes of lectures 20-22 Alan Huckleberry May 9, 2010 1 Introduction and review These lectures are mainly devoted to a very rough description the classi- fication theory for smooth projective algebraic manifolds of dimension one and two. We focus on curves in P 2 and surfaces in P 3 where the adjunction formula takes on its simplest form. Since we have already discussed curves on many occasions, we spend a majority of our time on the 2-dimensional case with a sketch of the Enriques-Kodaira classification scheme. Typical examples in each Kodaira class are explained. Let us recall one of the tools which is used over and over again, namely adjunction: For X a 1-codimensional submanifold of a complex manifold Y it follows that K X = ( K Y ⊗ [ X ]) | X . Here [ X ] denotes the line bundle defined by regarding X as a divisor in Y . Recall that K X denotes the canonical bundle of X , i.e., the determinant bundle of TX * . For X = P n it has been shown that K = H- ( n +1) . In other words sections s ∈ Γ( P n ) ,K- 1 ) of the anticanonical bundle correspond to homogeneous polynomials of degree n + 1. Note that if V 1 ,...,V n are global holomorphic vector fields which are independent at some point p ∈ P n , then V 1 ∧ ... ∧ V n is a nontrivial section of the anticanonical bundle K- 1 P n . Exercise. Define sl n +1 to be the subvector space of End( C n +1 ) of trace- free operators. For each A ∈ sl n +1 define a 1-parameter group t 7→ e tA in 1 Sl n +1 ( C ). Show that differentiation along orbits of this group defines a vector field V ( A ) on P n , i.e., V ( A )( f )( p ) := d dt t =0 f ( e tA .p ) . Show that the map defined by Λ n sl n +1 ( C ) → Γ( P n ,K- 1 ) , A 1 ∧ ... ∧ A n 7→ V ( A 1 ) ∧ ... ∧ V ( A n ) , is an isomorphism. 2 Remarks on curves Here we refer to a (connected) 1-dimensional compact complex manifold X as a curve . As we have indicated every such can be realized as an algebraic curve in P 3 ; so the notation is not farfetched. In this case the canonical bundle K is the cotangent bundle itself. As a sheaf it is often denoted by Ω 1 or simply Ω. Let us consider the case where X can be embedded as a curve C of degree d in P 2 . In that case we see that K X = ( H- 3 ⊗ H d ) | X = H d- 3 | X . This already has several interesting consequences. If d = 1 , 2, K- 1 X = H 3- d inherits sections of K- 1 P 2 by restriction. Exercise. For d = 1 , 2 consider the restriction map R : Γ( P 2 ,H 3- d ) → Γ( X,K- 1 X ) . Show that in both cases R is surjective and that in the case of d = 1 it has a 1-dimensional kernel. Hints: We know that as a complex manifold X is just P 1 . Furthermore K- 1 X is the tangent bundle of X ; so Γ( X,K- 1 X ) is just the space of global holomorphic vector fields on X . From beginning considerations in complex analysis we know that this space is sl 2 ( C )....
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