Robert Garbary, Pure Mathematics University of Waterloo
“Quotient Spaces”
Suppose
we
have
an
algebraic
group
G
acting
algebraicly
on
a
variety
X,
ie
for
each
g
∈
G
the
associated
map
X
→
X
is
a
morphism.
A
quotient
of
X
by
G
is
defined
to
be
a
variety
Y
and
a
morphism
π
:
X
→
Y
satisfying
(1)
π−1(π(x))
=
Gx
for
all
x
∈
X.
(2)
For
any
variety
Z
and
G-invariant
morphism
X
→
Z,
there
is
a
unique
factorization
through
Y.
Quotients
do
not
always
exist
-
for
example,
consider
C∗
acting
on
C
(or
more
generally,
any
toric
variety).
One
necessary
but
insufficient
condition
is
that
all
orbits
be
closed.
I’m
going
to
prove
that
if
G
is
finite
then
the
quotient
always
does
exist.
The
main
ingredient
is
a
classical
result
of
Hilbert
from
Invariant
Theory.
In
particular,
this
proves
that
the
set
SymN(X)
is
actually
a
variety,
which
is
the
beginning
of
a
proof
that
Jac(C)
is
a
variety.