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