Robert Garbary, Pure Mathematics, University of Waterloo
“Jac(C) over any (perfect) field”
Let
C
be
a
smooth
projective
curve
of
genus
g
≥
1.
Then
the
Riemann-Roch
theorem
says
that
for
any
C-divisor
D
satisfying
deg(D)
≥
2g
−
1,
we
have
l(D)
=
deg(D)
+
1
−
g.
Let
N
=
2g−1
-
we
define
Jac(C)
to
be
the
set
of
all
degree
N
linear
systems
on
C,
which
is
the
same
as
SynN(C)
modulo
the
relation
of
linear
equivalence.
We
will
equip
Jac(C)
with
a
group
law,
show
that
it
naturally
bijects
Pic0(C),
show
that
C
embeds
into
Jac(C),
and
prove
the
universal
property
it
satisfies.
The
only
black
box
is
why
Jac(C)
is
a
variety
and
why
the
group
law
is
a
morphism
-
coming
later
to
a
seminar
near
you!