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File: Geometry Pdf 168245 | Alggeom 2002
algebraic geometry andreas gathmann notes for a class taught at the university of kaiserslautern 2002 2003 contents 0 introduction 1 0 1 whatisalgebraic geometry 1 0 2 exercises 6 1 ...

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                       Algebraic Geometry
                         Andreas Gathmann
                          Notes for a class
                  taught at the University of Kaiserslautern 2002/2003
                                                                            CONTENTS
                                      0.  Introduction                                                                           1
                                      0.1.  Whatisalgebraic geometry?                                                            1
                                      0.2.  Exercises                                                                            6
                                      1.  Affinevarieties                                                                         8
                                      1.1.  Algebraic sets and the Zariski topology                                              8
                                      1.2.  Hilbert’s Nullstellensatz                                                           11
                                      1.3.  Irreducibility and dimension                                                        13
                                      1.4.  Exercises                                                                           16
                                      2.  Functions, morphisms, and varieties                                                   18
                                      2.1.  Functions on affine varieties                                                        18
                                      2.2.  Sheaves                                                                             21
                                      2.3.  Morphismsbetweenaffinevarieties                                                      23
                                      2.4.  Prevarieties                                                                        27
                                      2.5.  Varieties                                                                           31
                                      2.6.  Exercises                                                                           32
                                      3.  Projective varieties                                                                  35
                                      3.1.  Projective spaces and projective varieties                                          35
                                      3.2.  Cones and the projective Nullstellensatz                                            39
                                      3.3.  Projective varieties as ringed spaces                                               40
                                      3.4.  Themaintheoremonprojective varieties                                                44
                                      3.5.  Exercises                                                                           47
                                      4.  Dimension                                                                             50
                                      4.1.  Thedimension of projective varieties                                                50
                                      4.2.  Thedimension of varieties                                                           54
                                      4.3.  Blowing up                                                                          57
                                      4.4.  Smoothvarieties                                                                     63
                                      4.5.  The27lines on a smooth cubic surface                                                68
                                      4.6.  Exercises                                                                           71
                                      5.  Schemes                                                                               74
                                      5.1.  Affineschemes                                                                        74
                                      5.2.  Morphismsandlocally ringed spaces                                                   78
                                      5.3.  Schemesandprevarieties                                                              80
                                      5.4.  Fiber products                                                                      82
                                      5.5.  Projective schemes                                                                  86
                                      5.6.  Exercises                                                                           89
                                      6.  First applications of scheme theory                                                   92
                                      6.1.  Hilbert polynomials                                                                 92
                                              ´
                                      6.2.  Bezout’s theorem                                                                    96
                                      6.3.  Divisors on curves                                                                 101
                                      6.4.  Thegroupstructure on a plane cubic curve                                           104
                                      6.5.  Plane cubic curves as complex tori                                                 108
                                      6.6.  Wheretogofromhere                                                                  112
                                      6.7.  Exercises                                                                          117
                                      7.  Moreaboutsheaves                                                                     120
                                      7.1.  Sheaves and sheafification                                                           120
                                      7.2.  Quasi-coherent sheaves                                                             127
                                      7.3.  Locally free sheaves                                                               131
                                      7.4.  Differentials                                                                      133
                                      7.5.  Line bundles on curves                                                             137
                                      7.6.  TheRiemann-Hurwitzformula                                                          141
                                      7.7.  TheRiemann-Rochtheorem                                                             143
                                      7.8.  Exercises                                                                          147
                                      8.  Cohomologyofsheaves                                                                  149
                                      8.1.  Motivation and definitions                                                          149
                                      8.2.  Thelongexact cohomology sequence                                                   152
                                      8.3.  TheRiemann-Rochtheoremrevisited                                                    155
                                      8.4.  Thecohomologyoflinebundles on projective spaces                                    159
                                      8.5.  Proof of the independence of the affine cover                                       162
                                      8.6.  Exercises                                                                          163
                                      9.  Intersection theory                                                                  165
                                      9.1.  Chowgroups                                                                         165
                                      9.2.  Proper push-forward of cycles                                                      171
                                      9.3.  Weil and Cartier divisors                                                          176
                                      9.4.  Intersections with Cartier divisors                                                181
                                      9.5.  Exercises                                                                          185
                                      10.   Chern classes                                                                      188
                                      10.1.   Projective bundles                                                               188
                                      10.2.   Segre and Chern classes of vector bundles                                        191
                                      10.3.   Properties of Chern classes                                                      194
                                      10.4.   Statement of the Hirzebruch-Riemann-Roch theorem                                 200
                                      10.5.   Proof of the Hirzebruch-Riemann-Roch theorem                                     203
                                      10.6.   Exercises                                                                        209
                                      References                                                                               211
                                                                                   0.   Introduction                                             1
                                                                                0. INTRODUCTION
                                               Inaveryroughsketchweexplainwhatalgebraicgeometryisaboutandwhatitcanbe
                                               used for. We stress the many correlations with other fields of research, such as com-
                                               plex analysis, topology, differential geometry, singularity theory, computer algebra,
                                               commutative algebra, number theory, enumerative geometry, and even theoretical
                                               physics. The goal of this section is just motivational; you will not find definitions or
                                               proofs here (and probably not even a mathematically precise statement).
                                      0.1. What is algebraic geometry? To start from something that you probably know, we
                                      can say that algebraic geometry is the combination of linear algebra and algebra:
                                              • In linear algebra, we study systems of linear equations in several variables.
                                              • In algebra, we study (among other things) polynomial equations in one variable.
                                      Algebraic geometry combines these two fields of mathematics by studying systems of
                                      polynomial equations in several variables.
                                          Given such a system of polynomial equations, what sort of questions can we ask? Note
                                      that we cannot expect in general to write down explicitly all the solutions: we know from
                                      algebra that even a single complex polynomial equation of degree d > 4 in one variable
                                      can in general not be solved exactly. So we are more interested in statements about the
                                      geometric structure of the set of solutions. For example, in the case of a complex polyno-
                                      mial equation of degree d, even if we cannot compute the solutions we know that there are
                                      exactly d of them (if we count them with the correct multiplicities). Let us now see what
                                      sort of “geometric structure” we can find in polynomial equations in several variables.
                                      Example0.1.1. Probably the easiest example that is covered neither in linear algebra nor
                                      in algebra is that of a single polynomial equation in two variables. Let us consider the
                                      following example:
                                                                             2    2                                         2
                                                         C ={(x,y)∈C ; y =(x−1)(x−2)···(x−2n)}⊂C ,
                                                           n
                                      where n ≥ 1. Note that in this case it is actually possible to write down all the solutions,
                                      because the equation is (almost) solved for y already: we can pick x to be any complex
                                      number, and then get two values for y — unless x ∈ {1,...,2n}, in which case we only get
                                      one value for y (namely 0).
                                          Soitseemsthatthesetofequationslooksliketwocopiesofthecomplexplanewiththe
                                      two copies of each point 1,...,2n identified: the complex plane parametrizes the values
                                      for x, and the two copies of it correspond to the two possible values for y, i.e. the two roots
                                      of the number (x−1)···(x−2n).
                                          This is not quite true however, because a complex non-zero number does not have a
                                      distinguished first and second root that could correspond to the first and second copy of
                                      the complex plane. Rather, the two roots of a complex number get exchanged if you run
                                      around the origin once: if we consider a path
                                                                          iϕ
                                                                  x =re           for 0 ≤ ϕ ≤ 2π and fixed r > 0
                                      around the complex origin, the square root of this number would have to be defined by
                                                                                    √       √ iϕ
                                                                                       x =    re2
                                      which gives opposite values at ϕ = 0 and ϕ = 2π. In other words, if in C we run around
                                                                                                                              n
                                      oneofthepoints1,...,2n, we go from one copy of the plane to the other. The way to draw
                                      this topologically is to cut the two planes along the lines [1,2],...,[2n−1,2n], and to glue
                                      the two planes along these lines as in this picture (lines marked with the same letter are to
                                      be identified):
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...Algebraic geometry andreas gathmann notes for a class taught at the university of kaiserslautern contents introduction whatisalgebraic exercises afnevarieties sets and zariski topology hilbert s nullstellensatz irreducibility dimension functions morphisms varieties on afne sheaves morphismsbetweenafnevarieties prevarieties projective spaces cones as ringed themaintheoremonprojective thedimension blowing up smoothvarieties thelines smooth cubic surface schemes afneschemes morphismsandlocally schemesandprevarieties fiber products first applications scheme theory polynomials bezout theorem divisors curves thegroupstructure plane curve complex tori wheretogofromhere moreaboutsheaves sheacation quasi coherent locally free differentials line bundles theriemann hurwitzformula rochtheorem cohomologyofsheaves motivation denitions thelongexact cohomology sequence rochtheoremrevisited thecohomologyoflinebundles proof independence cover intersection chowgroups proper push forward cycles weil carti...

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