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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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