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Lecture Notes
for the Algebraic Geometry course
held by Rahul Pandharipande
Endrit Fejzullahu, Nikolas Kuhn, Vlad Margarint,
Nicolas Muller,¨ Samuel Stark, Lazar Todorovic
July 28, 2014
Contents
0 References 1
1 Affine varieties 1
2 Morphisms of affine varieties 2
3 Projective varieties and morphisms 5
3.1 Morphisms of affine algebraic varieties . . . . . . . . . . . . . . . . . . . . . . . 7
4 Projective varieties and morphisms II 8
5 Veronese embedding 10
n
5.1 Linear maps and Linear Hypersurfaces in P . . . . . . . . . . . . . . . . . . . . 11
5.2 Quadratic hypersurfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
6 Elliptic functions and cubic curves 12
7 Intersections of lines with curves 14
8 Products of varieties and the Segre embedding 14
8.1 Four lines in P3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
9 Intersections of quadrics 17
10 The Grassmannian and the incidence correspondence 18
10.1 The incidence correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
11 Irreducibility 20
12 Images of quasi-projective varieties under algebraic maps 20
13 Varieties defined by polynomials of equal degrees 20
14 Images of projective varieties under algebraic maps 21
I
15 Bezout’s Theorem 24
15.1 The resultant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
16 Pythagorean triples 25
17 The Riemann-Hurwitz formula 26
18 Points in projective space 26
19 Rational functions. 28
20 Tangent Spaces I 28
21 Tangent Space II 31
22 Blow-up 31
23 Dimension I 32
24 Dimension II 32
25 Sheaves 32
26 Schemes I 33
0 References
Good references for the topics here:
1. Harris, Algebraic Geometry, Springer.
2. Hartshorne, Algebraic Geometry, Springer. Chapter I, but also beginning of Chapter II
for schemes.
3. Shafarevich, Basic algebraic geometry I, II.
4. Atiyah, Macdonald Commutative Algebra (for basic commutative algebra).
1 Affinevarieties
In this course we mainly consider algebraic varieties and schemes. It is worth noting that
several definitions related to algebraic varieties are formally similar to those involving C∞-
manifolds. (In algebraic geometry the local analysis of algebraic varieties is by commutative
algebra, whereas in differential geometry the local analysis of C∞-manifolds is by calculus.) In
the following we take the complex field C to be the underlying field (one could also consider
any algebraically closed field instead; the discussion would be identical).
By affine n-space over C we mean Cn. For a subset X of C[x ,...,x ] we denote by
1 n
n
V(X):={x∈C |f(x)=0forall f ∈X} the common zero locus of the elements of X. No-
tice that V (X) = V (hXi). A subset of Cn of the form V (X) for some subset X of C[x ,...,x ]
1 n
is said to be an affine algebraic variety. By Hilbert’s basis theorem C[x ,...,x ] is Noethe-
1 n
rian, so that every affine algebraic variety is in fact the common zero locus of finitely many
n
polynomials. For a subset S of C the ideal of S is the set of polynomials vanishing on S,
I(S) := {f ∈ C[x ,...,x ] | f(x) = 0 for all x ∈ S}.
1 n
1
BothI andV areinclusion-reversing, and are connected by Hilbert’s Nullstellensatz: for ev-
√
ery ideal I of C[x ,...,x ] we have I(V (I)) = I. Hence I and V induce a bijective inclusion-
1 n
reversing correspondence between affine algebraic varieties and radical ideals of C[x ,...,x ].
1 n
n n
The Zariski topology on C is defined by letting the closed subsets of C be the affine alge-
n
braic varieties. The Zariski topology on a variety V in C is the induced (subspace) topology;
a more intrinsic definition relies on the following notion: a subvariety W of V is a variety W
in Cn such that W ⊂ V. Then a subset U of V is open in the Zariski topology of V if and
only if V \ U is a subvariety of V . Let us now consider the sets {Uf} defined by
f∈C[x ,...,xn]
1
Uf := {x ∈ V | f(x) 6= 0}.
Remark 1.1 {U } is a basis of the Zariski topology: let U ⊂ V be an open subset.
f f∈C[x ,...,xn]
1
By definition, Uc = V(I) for some ideal I of C[x ,...,x ]. Since C[x ,...,x ] is noetherian, I
1 n 1 n
is finitely generated , i.e. I = hf ,...,f i for some polynomials. Therefore we have
1 k
c c k c k
U =V(I) =V(hf ,...,f i) = (∩ V(hf i)) = ∪ U ,
1 k i=1 i i=1 fi
so every open subset is a union of just finitely many basic open subsets.
It follows from this also that Cn is quasi-compact.
Example 1.2 As every nonzero polynomial f ∈ C[x] has at most finitely many zeros, the
Zariski topology on the affine line C1 is exactly the cofinite topology. Notice that in this
topology every injection f : C → C is continuous, in contrast to the standard (euclidean)
topology.
2 Morphisms of affine varieties
n
Let V ⊂ C an affine algebraic variety, U ⊂ V an open subset.
Definition 2.1 f : U → C is an algebraic (regular) function if for each p ∈ C there exist an
open subset W ⊂ U containing p and g ,h ∈ C[x ,...,x ] such that
p p p 1 n
1. 0 6∈ g (W ).
p p
h
2. f| = p.
Wp g
p
Theorem 2.2 A map f : Cn → C is algebraic if and only if f is a polynomial function on Cn.
Proof. The “if” part is trivial.
For the “only if” part, let f : Cn → C be an algebraic map. Then for every p ∈ C there is an
open neighborhood W of p in Cn and polynomial functions h and g such that f| =hp and
p p p W
p gp
g never vanishes on W . Inside W we can find a basic open set p ∈ U ⊂W,where
p p p rp p
U ={ρ∈Cn|r (ρ)6=0}.
rp p
h
It follows that for all points p ∈ C we have a r ∈ C[x ,...,x ] such that f| = p and
p 1 n U
rp gp
n
∀q ∈ C : g (q) = 0 → r (q) = 0.
p p p
With Hilbert’s Nullstellensatz it follows that r ∈ (g ) and therefore rk = α g for some
p p p p p
α ∈C[x ,...,x ]. On U we have that r ,g ,α are never 0.
p 1 n r p p p
p
On U we have
rp
h α h α
f| = p p = p p
U
r
p g α rk
p p p
2
ˆ k n
Define h =h α and rˆ = r . Then U = U k, and for every p ∈ C there is r with r (p) 6= 0
p p p p r r p p
p p p
and f| =hp (removing the hats again).
Ur
p r
p
It follows that finitely many U suffice to cover Cn. Take p ,...,p such that U , . . . , U
r 1 m r r
p p p
1 m
cover Cm and write r instead of r . We have (because of the cover) (r ,...,r ) = (1) and
i pi 1 m
therefore m
∃s ,...,s ∈C[x ,...,x ] : Xs r = 1
1 m 1 n i i
i=1
h
Wehave f| = i and U ∩U =U . OnU we have
Ur r r r r r r
i ri i j i j i j
h h
i = j,h r = h r ,h r −h r = 0 on U
i j j i i j j i r r
r r i j
i j
n
r r (h r −h r ) = 0 everywhere on C (∗)
i j i j j i
|{z}|{z}
6=0 6=0
h r −h r =0∈C[x ,...,x ]
i j j i 1 n
Wehave to prove f ∈ C[x ,...,x ]. Let
1 n
m
F =Xh s
k k
k=1
F is a polynomial. We want to show F = f everywhere.
m m m
r F = Xrh s =Xr hs =h Xs r =h,
i i k k k i k i k k i
k=1 k=1 k=1
because of (∗). And on U we have
r
i
h h
f| = i,F| = i
U U
ri r ri r
i i
If V ⊂ Cn is an affine algebraic variety and f : V → C is an algebraic function, f is
restriction of a polynomial (see exercises). If f ,f : V → C are algebraic and the same
1 2
function ⇔ f −f ∈ I(V). We denote by Γ(V) the ring of algebraic functions V → C.
1 2
Proposition 2.3
C[x ,...,x ]
Γ(V) ∼ 1 n
= I(V)
Proof. By
f ∈ C[x ,...,x ] 7→ f| : V → C
1 n V
we have C[x ,...,x ]
1 n ⊂Γ(V)
I(V)
So, to prove the proposition, we need to show that this is surjective. Go through the proof of
Theorem 2.2 step by step. It is the same, except for one step.
h h
i j ˆ 2
Again, we have U ∩U = U . On U we have = . Let h = hr,rˆ = r . And we
r r r r r r i i i i i
i j i j i j r r
i j
have h r −r h = 0 on U . And
i j i j r r
i j
r r (h r −r h ) = 0 everywhere on V
i j i j i j
(see exercise 1.3 on the exercise sheets)
3
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