19th
century
-
significant
work
on
functions
on
algebraic
curves
Riemann-Roch
Theorem,
Abel-Jacobi
Theorem
major
contributions
using
divisors
(algebraic
cycles
of
codimension
1)
In
higher
dimensions,
behavior
of
varieties
becomes
more
complicated
and
these
theorems
don’t
nicely
extend
HistoryandMotivation
19th
century
-
significant
work
on
functions
on
algebraic
curves
Riemann-Roch
Theorem,
Abel-Jacobi
Theorem
major
contributions
using
divisors
(algebraic
cycles
of
codimension
1)
In
higher
dimensions,
behavior
of
varieties
becomes
more
complicated
and
these
theorems
don’t
nicely
extend
HistoryandMotivation
19th
century
-
significant
work
on
functions
on
algebraic
curves
Riemann-Roch
Theorem,
Abel-Jacobi
Theorem
major
contributions
using
divisors
(algebraic
cycles
of
codimension
1)
In
higher
dimensions,
behavior
of
varieties
becomes
more
complicated
and
these
theorems
don’t
nicely
extend
HistoryandMotivation
19th
century
-
significant
work
on
functions
on
algebraic
curves
Riemann-Roch
Theorem,
Abel-Jacobi
Theorem
major
contributions
using
divisors
(algebraic
cycles
of
codimension
1)
In
higher
dimensions,
behavior
of
varieties
becomes
more
complicated
and
these
theorems
don’t
nicely
extend
AlgebraicCycles
Let X be any variety defined over a field k.
Definition
A
cycleZofcodimensionr
on
X
is
an
element
of
the
free
abelian
group
generated
by
the
closed
irreducible
subvarieties
of
codimension
r
of
X.
It
is
a
finite
formal
sum∑ni[Vi]
where
ni
are
integers
and
Vi
are
subvarieties.
We denote by Cr(X) the group of all cycles of codimension r on
X.
AlgebraicCycles
Let X be any variety defined over a field k.
Definition
A
cycleZofcodimensionr
on
X
is
an
element
of
the
free
abelian
group
generated
by
the
closed
irreducible
subvarieties
of
codimension
r
of
X.
It
is
a
finite
formal
sum
∑ni[Vi]
where
ni
are
integers
and
Vi
are
subvarieties.
We denote by Cr(X) the group of all cycles of codimension r on
X.
AlgebraicCycles
Let X be any variety defined over a field k.
Definition
A
cycleZofcodimensionr
on
X
is
an
element
of
the
free
abelian
group
generated
by
the
closed
irreducible
subvarieties
of
codimension
r
of
X.
It
is
a
finite
formal
sum
∑ni[Vi]
where
ni
are
integers
and
Vi
are
subvarieties.
We denote by Cr(X) the group of all cycles of codimension r on
X.
AlgebraicCycles
Let X be any variety defined over a field k.
Definition
A
cycleZofcodimensionr
on
X
is
an
element
of
the
free
abelian
group
generated
by
the
closed
irreducible
subvarieties
of
codimension
r
of
X.
It
is
a
finite
formal
sum
∑ni[Vi]
where
ni
are
integers
and
Vi
are
subvarieties.
We denote by Cr(X) the group of all cycles of codimension r on
X.
AlgebraicCycles
Cr(X)
is
naturally
quite
large
Use
equivalence
relations
to
develop
more
intuition
on
the
geometry
of
X.
AlgebraicCycles
Cr(X)
is
naturally
quite
large
Use
equivalence
relations
to
develop
more
intuition
on
the
geometry
of
X.
ANiceExample
Let r = 1. A cycle of codimension one is a divisor.
Definition
Two
divisors
D1
and
D2
on
X
are
linearly
equivalent
if
there
exists
a
rational
function
on
X
such
that
D1− D2 = (f )0− (f )∞,
where
(f )0
denotes
the
divisor
of
zeros
and
(f )∞
denotes
the
divisor
of
poles.
Example
Let
Clin1(X)
denote
the
group
of
divisors
linearly
equivalent
to
0.
Then,
the
quotient
group
C1(X)∕Clin1(X)
is
Pic X,
the
group
of
linear
equivalence
classes
of
divisors
on
X.
ANiceExample
Let r = 1. A cycle of codimension one is a divisor.
Definition
Two
divisors
D1
and
D2
on
X
are
linearly
equivalent
if
there
exists
a
rational
function
on
X
such
that
D1− D2 = (f )0− (f )∞,
where
(f )0
denotes
the
divisor
of
zeros
and
(f )∞
denotes
the
divisor
of
poles.
Example
Let
Clin1(X)
denote
the
group
of
divisors
linearly
equivalent
to
0.
Then,
the
quotient
group
C1(X)∕Clin1(X)
is
Pic X,
the
group
of
linear
equivalence
classes
of
divisors
on
X.
ANiceExample
For X = ℙn, C1(X)∕Clin1(X)ℤ.
RationalEquivalence
Same
as
linear
equivalence
in
codim
1
Definition
Two
cycles
Z1
and
Z2
of
codimension
r
on
X
are
rationallyequivalent
if
there
is
a
cycle
Z
on
X × ℙ1,
which
intersects
each
fiber
X ×{t}
in
something
of
codimension
r,
and
such
that
Z1
and
Z2
are
obtained
respectively
by
intersecting
Z
with
the
fibers
X ×{0}
and
X ×{1}.
RationalEquivalence
Same
as
linear
equivalence
in
codim
1
Definition
Two
cycles
Z1
and
Z2
of
codimension
r
on
X
are
rationallyequivalent
if
there
is
a
cycle
Z
on
X × ℙ1,
which
intersects
each
fiber
X ×{t}
in
something
of
codimension
r,
and
such
that
Z1
and
Z2
are
obtained
respectively
by
intersecting
Z
with
the
fibers
X ×{0}
and
X ×{1}.
RationalEquivalenceisanEquivalenceRelation
Reflectivity:
Z1∼ratZ1.
Take
Z = Z1× ℙ1
Symmetry:
Z1∼ratZ2Z2∼ratZ1.
Apply
automorphism
of
ℙ1
that
interchanges
0
and
1.
Transivity:
Z1∼ratZ2,Z2∼ratZ3Z1∼ratZ3.
Let
Z ⊆ X × ℙ1
give
Z1∼ratZ2
and
Z′⊆ X × ℙ1
give
Z2∼ratZ3.
Then,
Z + Z′− Z2× ℙ1
gives
Z1∼ratZ3.
RationalEquivalenceisanEquivalenceRelation
Reflectivity:
Z1∼ratZ1.
Take
Z = Z1× ℙ1
Symmetry:
Z1∼ratZ2Z2∼ratZ1.
Apply
automorphism
of
ℙ1
that
interchanges
0
and
1.
Transivity:
Z1∼ratZ2,Z2∼ratZ3Z1∼ratZ3.
Let
Z ⊆ X × ℙ1
give
Z1∼ratZ2
and
Z′⊆ X × ℙ1
give
Z2∼ratZ3.
Then,
Z + Z′− Z2× ℙ1
gives
Z1∼ratZ3.
RationalEquivalenceisanEquivalenceRelation
Reflectivity:
Z1∼ratZ1.
Take
Z = Z1× ℙ1
Symmetry:
Z1∼ratZ2Z2∼ratZ1.
Apply
automorphism
of
ℙ1
that
interchanges
0
and
1.
Transivity:
Z1∼ratZ2,Z2∼ratZ3Z1∼ratZ3.
Let
Z ⊆ X × ℙ1
give
Z1∼ratZ2
and
Z′⊆ X × ℙ1
give
Z2∼ratZ3.
Then,
Z + Z′− Z2× ℙ1
gives
Z1∼ratZ3.
AlgebraicEquivalence
Use
same
construction
for
rational
equivalence
Definition
Let
C
be
an
irreducible
curve,
and
a,b ∈ C
be
any
two
points.
Two
cycles
Z1
and
Z2
of
codimension
r
on
X
are
algebraicallyequivalent
if
there
is
a
cycle
Z
on
X × C,
which
intersects
each
fiber
X ×{t}
in
something
of
codimension
r,
and
such
that
Z1
and
Z2
are
obtained
respectively
by
intersecting
Z
with
the
fibers
X ×{a}
and
X ×{b}.
RemarkCratr(X) ⊂ Calgr(X) ⊂ Cr(X)
AlgebraicEquivalence
Use
same
construction
for
rational
equivalence
Definition
Let
C
be
an
irreducible
curve,
and
a,b ∈ C
be
any
two
points.
Two
cycles
Z1
and
Z2
of
codimension
r
on
X
are
algebraicallyequivalent
if
there
is
a
cycle
Z
on
X × C,
which
intersects
each
fiber
X ×{t}
in
something
of
codimension
r,
and
such
that
Z1
and
Z2
are
obtained
respectively
by
intersecting
Z
with
the
fibers
X ×{a}
and
X ×{b}.
RemarkCratr(X) ⊂ Calgr(X) ⊂ Cr(X)
AlgebraicEquivalence
Use
same
construction
for
rational
equivalence
Definition
Let
C
be
an
irreducible
curve,
and
a,b ∈ C
be
any
two
points.
Two
cycles
Z1
and
Z2
of
codimension
r
on
X
are
algebraicallyequivalent
if
there
is
a
cycle
Z
on
X × C,
which
intersects
each
fiber
X ×{t}
in
something
of
codimension
r,
and
such
that
Z1
and
Z2
are
obtained
respectively
by
intersecting
Z
with
the
fibers
X ×{a}
and
X ×{b}.
ℤ.
Z2 ∼ratZ1.
Apply
automorphism
of
ℙ1
that
interchanges
0
and
1. 
Z1 ∼ratZ3.
Let
Z ⊆ X × ℙ1
give
Z1 ∼ratZ2
and
Z′⊆ X × ℙ1
give
Z2 ∼ratZ3.
Then,
Z + Z′− Z2 × ℙ1
gives
Z1 ∼ratZ3.
Z2 ∼ratZ1.
Apply
automorphism
of
ℙ1
that
interchanges
0
and
1.
Z1 ∼ratZ3.
Let
Z ⊆ X × ℙ1
give
Z1 ∼ratZ2
and
Z′⊆ X × ℙ1
give
Z2 ∼ratZ3.
Then,
Z + Z′− Z2 × ℙ1
gives
Z1 ∼ratZ3.
Z2 ∼ratZ1.
Apply
automorphism
of
ℙ1
that
interchanges
0
and
1.
Z1 ∼ratZ3.
Let
Z ⊆ X × ℙ1
give
Z1 ∼ratZ2
and
Z′⊆ X × ℙ1
give
Z2 ∼ratZ3.
Then,
Z + Z′− Z2 × ℙ1
gives
Z1 ∼ratZ3.