Lecture 7
Morphisms of projective algebraic
varieties
Following the denition of a morphism of a ne algebraic varieties we can dene a
morphism f : X ! Y of two projective algebraic varieties as a set of maps fK :
X (K ) ! Y (K ) dened for each k -algebra K
Lecture 6
Bzout theorem and a group law
e
on a plane cubic curve
We begin with an example. Consider two concentric circles:
2
2
C : Z1 + Z2 = 1 ,
2
2
C 0 : Z1 + Z2 = 4 .
Obviously, they have no common points in the a ne plane A2 (K ) no matter in
which al
48LECTURE 6. BEZOUT THEOREM AND A GROUP LAW ON A PLANE CUBIC CURVE
the line T0 + T1 + T2 = 0 contains a given point (resp. two distinct points)
is a two-dimensional (resp. one-dimensional) linear subspace of k 3 . Thus the set
of lines T0 + T1 + T2 = 0 c
50LECTURE 6. BEZOUT THEOREM AND A GROUP LAW ON A PLANE CUBIC CURVE
Then we extend the denition to all polynomials by linearity over k requiring that
@ (aP + bQ)
@P
@Q
=a
+b
@ Zj
@ Zj
@ Zj
for all a, b 2 k and any monomials P, Q. It is easy to check that
55
T0 T1 T2 = 0 which contains the points (1, 0, ), (1, , 0). The set x1 , . . . , x8
is the needed conguration. One easily checks that the nine points x1 , . . . , x9
are the inection points of the cubic curve C (by Remark 6.5 we expect exactly 9
inectio
53
x
y
xy
o
x
y
Figure 6.1:
line to X at e. If y = x, take for L1 the tangent line at y . We claim that this
construction denes the group law on X (K ).
Clearly
y x = x y,
i.e., the binary law is commutative. The point e is the zero element of the law. If
44
LECTURE 5. PROJECTIVE ALGEBRAIC VARIETIES
In future we will always assume that a projective variety X is given by a
system of equations S such that the ideal (S ) is saturated. Then I = (S ) is
dened uniquely and is called the homogeneous ideal of X an
43
Clearly I sat is a homogeneous ideal in k [T ] containing the ideal I (Check it !) .
Proposition 5.14. Two homogeneous systems S and S 0 dene the same projective variety if and only if (S )sat = (S 0 )sat .
Proof. Let us show rst that for any k -algebr
33
the eld of fractions: it is the set of equivalence classes of pairs (m, s) 2 M
S with the equivalence relation: (m, s) (m0 , s0 ) () 9s00 2 S such that
s00 (s0 m sm0 ) = 0. The equivalence class of a pair (m, s) is denoted by m . The
s
equivalence cla
Lecture 5
Projective algebraic varieties
Let A be a commutative ring and An+1 (n
0) be the Cartesian product
equipped with the natural structure of a free A-module of rank n + 1. A free
submodule M of An+1 of rank 1 is said to be a line in An+1 , if M = A
35
Proof. Let Matn (A) be the ring of n n matrices with coe cients in a commutative ring A. For any ideal I in A we have a natural surjective homomorphism
of rings Matn (A) ! Matn (A/I ), X 7! X , which obtained by replacing each
entry of a matrix X with
37
A is a eld this is a familiar duality between lines in a vector space V and
hyperplanes in the dual vector space V .
A set cfw_fi i2I of elements from A is called a covering set if it generates
the unit ideal. Every covering set contains a nite coverin
41
Example 5.12. Let X be given by aT0 + bT1 + cT2 = 0, a projective subvariety
of the projective plane P2 . It is equal to the projective closure of the line L A2
k
k
given by the equation bZ1 + cZ2 + a = 0. For every K the set X (K ) has a
unique point
39
Denition 5.4. A polynomial F (T0 , . . . , Tn ) 2 k [T0 , . . . , Tn ] is called homogeneous of degree d if
X
X
i
i
F (T0 , . . . , T n ) =
ai0 0,.,in 0 T00 Tnn =
ai T i
i0 ,.,in
i
with |i| = d for all i. Here we use the vector notation for polynomials
23
Proof. Let us prove the rst part. If V is irreducible, then the assertion is obvious.
Otherwise, V = V1 [ V2 , where Vi are proper closed subsets of V . If both of
them are irreducible, the assertion is true. Otherwise, one of them, say V1 is
reducible
15
Proof. Let f : X ! Y be a morphism. It denes a homomorphism
fO(X ) : Homk (O(X ), O(X ) ! Homk (O(Y ), O(X ).
The image of the identity homomorphism idO(X ) is a homomorphism : O(Y ) !
O(X ). Let us show that ( ) = f . Let 2 X (K ) = Homk (O(X ), K ).
17
algebras: O(Y ) = k [T1 ] ! O(X ). Every such homomorphism is determined by
its value at T1 , i.e. by an element of O(X ). This gives us one more interpretation
of the elements of the coordinate algebra O(X ). This time they are morphisms
from X to A1
19
X ! Y of algebraic varieties denes a map fK : X (K ) = V ! Y (K ) =
W of the algebraic sets. So it is natural to take for the denition of regular
maps of algebraic sets the maps arising in this way. We know that f is given
by a homomorphism of k -algeb
Lecture 4
Irreducible algebraic sets and
rational functions
We know that two a ne algebraic k -sets V and V 0 are isomorphic if and only if
their coordinate algebras O(V ) and O(V 0 ) are isomorphic. Assume that both of
these algebras are integral domains
Lecture 3
Morphisms of a ne algebraic
varieties
In Lecture 1 we dened two systems of algebraic equations to be equivalent if
they have the same sets of solutions. This is very familiar from the theory of
linear equations. However this notion is too strong
11
Next we shall show that the set of algebraic k -subsets in K n can be used
to dene a unique topology in K n for which these sets are closed subsets. This
follows from the following:
Proposition 2.7. (i) The intersection \s2S Vs of any family cfw_Vs s2S
Lecture 1
Systems of algebraic equations
The main objects of study in algebraic geometry are systems of algebraic equations and their sets of solutions. Let k be a eld and k [T1 , . . . , Tn ] = k [T ] be
the algebra of polynomials in n variables over k .
3
Example 1.1. 1. The system S = cfw_0 k [T1 , . . . , Tn ] denes an a ne algebraic variety denoted by An . It is called the a ne n-space over k . We have, for
k
any k -algebra K ,
Sol(cfw_0; K ) = K n .
2. The system 1 = 0 denes the empty a ne algebraic
5
1, . . . , m(r), be the polynomials from I which have the highest coe cient equal
to ai,r . Next, we consider the union J of the ideals Jr . By multiplying a
polynomial F by a power of Tn we see that Jr Jr+1 . This immediately implies
that the union J i
9
k as a subeld. Note that A is nitely generated as a k -algebra (because k [T ]
is). Suppose we show that A is an algebraic extension of k . Then we will be
able to extend the inclusion k K to a homomorphism A ! K (since K is
algebraically closed), the c
6
LECTURE 1. SYSTEMS OF ALGEBRAIC EQUATIONS
Exercises.
1. For which elds k do the systems
n
X
S = cfw_ i (T1 , . . . , Tn ) = 0i=1,.,n , and S = cfw_
Tji = 0i=1,.,n
0
j =1
dene the same a ne algebraic varieties? Here i (T1 , . . . , Tn ) denotes the eleme