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
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
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