Lectures on algebraic geometry
Jacobs 2010
Notes of lectures 7 and 8
Alan Huckleberry
March 9, 2010
Lecture 7 was primarily devoted to answering questions concerning the linear
families of curves that were discussed int Lecture 6. Here we begin by review
this material and then going on to the new material on conics which was
presented in Lecture 8. We have also added two references to the literature.
1
Linear families
Previously we considered two relatively prime homogeneous polynomials
P
0
and
P
1
of degree
d
in
P
2
which define smooth curves
C
0
and
C
1
. The linear
family
F
:=
{
([
ζ
0
:
ζ
1
]
, z
)
∈
P
1
×
P
2
;
ζ
0
P
0
(
z
) +
ζ
1
P
(
z
) = 0
}
.
is viewed as
connecting
C
0
and
C
1
.
Letting
π
:
F →
P
1
be defined by
the projection on the first factor, we noted that there exist smooth real
curves
γ
:
I
→
P
1
defined on the interval
I
= [0
,
1] with
γ
(0) = [1 : 0]
and
γ
(1) = [0 : 1] so that the pullback
π
:
F
γ
→
I
is a smooth trivial
fiber bundle.
This shows in particular that
C
0
and
C
1
are diffeomorphic.
In fact they are diffeomorphic to any of the intermediate curves
C
t
.
The
C
t
are smooth complex algebraic subvarieties of
P
2
.
There are are special
cases where they do not vary complex analytically or, equivalently, algebraic
geometrically.
However, as we will see in the next lectures, for
d
≥
3 one
should expect that these structures vary nontrivially. This type of variation
of structure is a major theme in algebraic geometry.
1
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One goal of the above construction was to prove the weaker result that
e
top
(
C
0
) =
e
top
(
C
1
).
Using an explicit calculation for the Fermat curve
C
0
this resulted in the genus formula
g
(
C
) =
(
d

1)(
d

2)
2
(1)
for smooth curves
C
of degree
d
.
Observe that the family
F
is defined for any two relatively prime polynomials
and that the map
F
γ
→
P
2
defined by projection on the second factor can
be regarded as a 3cycle in singular homology theory. The boundary of this
3cycle is
C
0

C
1
. This proves the following observation.
Proposition 1.1.
Any two curves of degree
d
in
P
2
are homologous.
In
particular, any curve of degree
d
is homologous to a smooth curve of genus
g
=
(
d

1)(
d

2)
2
.
Hence we may think of the righhand side of (1) as a homology invariant
and therefore it carries its own name,
virtual genus
π
(
C
).
Independent of
the embedding in
P
2
, we can regard this as the genus of any smooth curve
homologous to
C
.
2
Conics
The simplest curves in
P
2
are of degree
d
= 1. These are the curves defined
by linear functions and as a consequence it is an easy matter to parameterize
them.
Exercise.
Let
V
be a 3dimensional complex vector space. Show that the
projective space
P
(
V
*
) of the dual space
V
*
parameterizes the curves
C
of
degree
d
= 1 in
P
(
V
).
A curve of degree two in
P
2
is called a
conic
. If we think of
z
= [
z
0
:
z
1
:
z
2
]
as a (homogeneous) column vector, then every homogeneous polynomial
P
of degree
d
= 2 can be uniquely written as
P
(
z
) =
z
t
Sz
where
S
is a 3
×
3 symmetric matrix. Thus we may regard the 5dimensional
projective space
P
(Sym
3
×
3
) of the vector space of all symmetric matrices as
the space of all conics in
P
2
.
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