Abstract
We prove that for a sufficiently ample line bundle
L
on a surface
S
, the number of
δ
–nodal curves in a general
δ
–dimensional linear system is given by a universal polynomial of degree
δ
in the four numbers
L
2
,
L
.
K
S
,
K
2
S
and
c
2
(
S
)
.
The technique is a study of Hilbert schemes of points on curves on a surface, using the BPS calculus of Pandharipande and the third author [J. Amer. Math. Soc. 23 (2010) 267–297] and the computation of tautological integrals on Hilbert schemes by Ellingsrud, Göttsche and Lehn [J. Algebraic Geom. 10 (2001) 81–100].
We are also able to weaken the ampleness required, from Göttsche’s
(
5
δ
−
1
)
–very ample to
δ
–very ample.
L
on a surface
S
, the number of
δ
–nodal curves in a general
δ
–dimensional linear system is given by a universal polynomial of degree
δ
in the four numbers
L
2
,
L
.
K
S
,
K
2
S
and
c
2
(
S
)
.
The technique is a study of Hilbert schemes of points on curves on a surface, using the BPS calculus of Pandharipande and the third author [J. Amer. Math. Soc. 23 (2010) 267–297] and the computation of tautological integrals on Hilbert schemes by Ellingsrud, Göttsche and Lehn [J. Algebraic Geom. 10 (2001) 81–100].
We are also able to weaken the ampleness required, from Göttsche’s
(
5
δ
−
1
)
–very ample to
δ
–very ample.
| Original language | English |
|---|---|
| Pages (from-to) | 397–406 |
| Number of pages | 10 |
| Journal | Geometry and Topology |
| Volume | 15 |
| Issue number | 1 |
| DOIs | |
| Publication status | Published - 2011 |