GRR theorem.pdf - GROTHENDIECK-RIEMANN-ROCH THEOREM ANDERS SKOVSTED BUCH 1 The topic This is a proposal for a first topic in Intersection Theory The


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GROTHENDIECK-RIEMANN-ROCH THEOREM ANDERS SKOVSTED BUCH 1. The topic This is a proposal for a first topic in Intersection Theory. The goal in the topic is to understand the Grothendieck-Riemann-Roch theorem and Prof. William Fulton’s proof of it. The topic has been worked out under Prof. Madhav Nori’s supervision. In doing the topic I have read F.A.C. [3], most of chapters 1-3 in Hartshorne’s book [2], and chapters 1-8 plus chapter 15 in Fulton’s book [1]. Furthermore I used Borel and Serre’s article on Grothendieck-Riemann-Roch theorem [4], and chapter 5 in Altman and Kleiman’s book on Grothendieck duality [5]. Of these, Fulton’s book has been the main reference. During the topic I have done a number of exercises in Hartshorne’s book. Fulton’s book does not contain exercises, however it has taken a lot of work to understand and verify most of the examples. I also plan to do exercises from J. Harris’ book [6] to see more examples of algebraic varieties. 2. Intersection Theory A very simple problem in Intersection Theory is the following: If f ( X ) C [ X ] is a nonzero polynomial of degree d , then how many solutions a C exist to the equation f ( a ) = 0 ? The answer is simple: If you count properly, then there are d solutions. The above problem has a natural generalization to several variables. If f 1 , . . . , f n C [ X 1 , . . . , X n ] are polynomials of degrees d 1 , . . . , d n , then how many solutions a = ( a 1 , . . . , a n ) A n = C n exist to the set of equations f i ( a ) = f i ( a 1 , . . . , a n ) = 0 for 1 i n ? Given that the number of solutions is finite, the answer to this question is almost as simple as in the above case. If you include solutions in the enlargement P n of A n and furthermore count properly, then there are exactly Q i d i . This is a special case of B´ ezout’s Theorem. Date : April 13, 2001. 1
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2 ANDERS SKOVSTED BUCH Intersection Theory is a branch of Algebraic Geometry, of which ezout’s Theorem is a particularly nice example. The basic question in Intersection Theory is what do you get when you intersect two sub- varieties of an algebraic variety. 3. The group of cycle classes on a scheme Let k be a field. In the following a scheme will mean a Noetherian scheme of finite type over k .
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  • Fall '08
  • Staff
  • Algebraic geometry, Chern class, line bundle, ANDERS SKOVSTED BUCH

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