A stratification of the moduli space of vector bundles on curves
Abstract
Let
E
be a vector bundle of rank
r≥2
on a smooth projective curve
C
of genus
g≥2
over an algebraically closed field
K
of arbitrary characteristic. For any integer with
1≤k≤r−1
we define
{\se}_k(E):=k°E-r\max°F.
where the maximum is taken over all subbundles
F
of rank
k
of
E
. The
s
k
gives a stratification of the moduli space
M(r,d)
of stable vector bundles of rank
r
and degree on
d
on
C
into locally closed subsets ${\calM}(r,d,k,s)$ according to the value of
s
and
k
. There is a component
M
0
(r,d,k,s)
of
M(r,d,k,s)
distinguish by the fact that a general
E∈
M
0
(r,d,k,s)
admits a stable subbundle
F
such that
E/F
is also stable. We prove: {\it For
g≥
r+1
2
and
0<s≤k(r−k)(g−1)+(r+1)
,
s≡kdmodr,
M
0
(r,d,k,s)
is non-empty,and its component
M
0
(r,d,k,s)
is of dimension}
dim
M
0
(r,d,k,s)={
(
r
2
+
k
2
−rk)(g−1)+s−1
s<k(r−k)(g−1)
if
r
2
(g−1)+1
s≥k(r−k)(g−1)