Contemporary Mathematics
Volume 403, 2006
Moduli Spaces of Curves with Effective r-Spin Structures
A. Polishchuk
ABSTRACT.
We introduce the moduli stack of pointed curves equipped with
effective r-spin structures: these are effective divisors
D
such that
r D
is a
canonical divisor modified at marked points. We prove that this moduli space
is smooth and describe its connected components. We also prove that it always
contains a component that projects birationally to the locus S
0
in the moduli
space of r-spin curves consisting of r-spin structures L such that h0 (L)
f.
0.
Finally, we study the relation between the locus S
0
and Witten's virtual top
Chern class.
1. Introduction
Let us fix integers
g
~
1,
r
~
2,
n
~
0 and a vector m
non-negative integers such that
m1
+ ... +
mn
+
rd
=
2g - 2
for some integer
d
~
0. Consider the moduli space
M!,~,eff
of effective r-spin
structures parametrizing collections
(C,
D,p1, ... ,pn), where
C
is a (connected)
smooth complex projective curve of genus g, D
C
Cis an effective divisor of degree
d,
and Pl, ... ,
Pn
are (distinct) marked points on
C
such that
(1.1) Oc(rD
+
m1P1
+ ... +
mnPn) ~we
(see section 2 for the precise definition of the moduli stack).
Our main result is the following theorem.
THEOREM
1.1. (a) The stack
M~,~,e1f
is smooth of dimension 2g- 2
+
d
+
n.
(b) If d
~
0 then there exists a point (C, D,p1
, •••
,pn) in
M~,~,eff
such that
h0 (D)
=
1.
The proof of this theorem will occupy sections 2 and 3. The idea of the proof of
part (a) is very simple. We show that the dimension of
M~,~,eff
at every point is
at least 2g- 2
+
d
+
n,
representing
M~.~,eff
as a degeneracy locus. Then we prove
that the dimension of the tangent space to
M~,~,eff
at every point is no greater
2000 Mathematics Subject Classification. Primary: 14H10 Secondary: 14N35.
Supported in part by an NSF grant.
@2006 American Mathematical Society
http://dx.doi.org/10.1090/conm/403/07592
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