Affiliation:
1. Beijing National Center for Applied Mathematics , Academy for Multidisciplinary Studies , Capital Normal University , 100048 , Beijing , P. R. China
Abstract
Abstract
Let
M
(
d
,
χ
)
{M(d,\chi)}
, with
(
d
,
χ
)
=
1
{(d,\chi)=1}
, be the moduli space of semistable sheaves on
ℙ
2
{\mathbb{P}^{2}}
supported on curves of degree d and with Euler characteristic χ. The cohomology ring
H
*
(
M
(
d
,
χ
)
,
ℤ
)
{H^{*}(M(d,\chi),\mathbb{Z})}
of
M
(
d
,
χ
)
{M(d,\chi)}
is isomorphic to its Chow ring
A
*
(
M
(
d
,
χ
)
)
{A^{*}(M(d,\chi))}
by Markman’s result. Pi and Shen have described a minimal generating set of
A
*
(
M
(
d
,
χ
)
)
{A^{*}(M(d,\chi))}
consisting of
3
d
-
7
{3d-7}
generators, which they also showed to have no relation in
A
≤
d
-
2
(
M
(
d
,
χ
)
)
{A^{\leq d-2}(M(d,\chi))}
. We compute the two Betti numbers
b
2
(
d
-
1
)
{b_{2(d-1)}}
and
b
2
d
{b_{2d}}
of
M
(
d
,
χ
)
{M(d,\chi)}
, and as a corollary we show that the generators given by Pi and Shen have no relations in
A
≤
d
-
1
(
M
(
d
,
χ
)
)
{A^{\leq d-1}(M(d,\chi))}
,
but do have three linearly independent relations in
A
d
(
M
(
d
,
χ
)
)
{A^{d}(M(d,\chi))}
.
Funder
National Natural Science Foundation of China
Subject
Applied Mathematics,General Mathematics
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