Moduli of curves with a root via graph theory
Mattia Galeotti
Institut des Mathématiques de Jussieu - Paris Rive Gauche. Université Pierre et Marie Curie
Introduction
Enriched dual graph
Mg = the moduli of genus g stable curves.
Consider C 0(Γ, Z/`) and C 1(Γ, Z/`) the cochains
groups on vertices and edges of Γ, with the natural
differential
1
Mg is of general type for g ≥ 24 (HarrisMumford, 1982). The Kodaira dimension of Mg
at the threshold g = 23 is still an open problem
(see Farkas, 1999).
We attack this problem by studying some classes
of finite covers of Mg .
0
∂ : C (Γ, Z/`) → C (Γ, Z/`).
An `th root of ω ⊗s is a line bundle L on a curve
C such that
⊗`
L
∂M (v) = s · deg(ωCv )
A twisted curve is a Deligne-Mumford stack C
whose coarse space is a stable curve C and with
Z/`-stabilizers on all nodes.
We contract the edges where M has 0 value. The
contracted graph is noted Γ0(C).
with action ζ · (x, y) = (ζx, ζ −1y) ∀ζ `th root of
the unit.
1
AutC (C, L) ,→ C (Γ, Z/`).
Age
a=
Figure 3: In a tree-like graph every circuit is a loop.
a1
am
Diag(ζn , . . . , ζn ),
then we define
An automorphism a ∈ GL(C ) is a quasireflection (QR) if it fixes a hyperplane, it is a junior
automorphism if age(a) < 1.
m
Figure 2: Contraction of the edge e of a graph.
The moduli space
s
Rg,`
s
Rg,`,
We note
the moduli space of twisted curves C
of genus g plus an `th root L of ω ⊗s.
• [C, L]
s
Rg,`
is a smooth point iff Aut(C, L) is
∈
generated by quasireflections;
s
• [C, L] ∈ Rg,` is a non-canonical singularity if
Aut(C, L) contains a junior automorphism (up to
QR).
s
Rg,`
We have a natural projection π :
→ Mg :
s
it maps a point [C, L] in Rg,` to its coarse curve
[C].
Local structure. Consider the finite group
∗ ∼
Aut(C, L) = {s ∈ Aut(C)| s L = L}.
For any point [C, L] of
hood isomorphic to
s
Rg,`,
there exists a neighbor-
C3g−3/Aut(C, L).
Figure 1: The dual graphs Γ(C) of the general curves in strata
∆i and ∆0 of Mg , with the annotations of nodes stabilizers.
For example if ` is a prime number, any point
s
[C, L] of Rg,` is smooth iff Γ0(C) is a tree-like
graph.
m
1X
ai.
age(a) :=
n i=1
The local picture at nodes is
[{xy = 0}/Z/`] ,
We note AutC (C, L) the group of ghost automorphisms acting as the identity on the coarse curve C.
There exists a canonical injection
Given any automorphism a ∈ GL(Cm) of finite order n, if its diagonalization is
⊗s
∼
= ωC .
With scheme theoretic curves we “lose”
some roots on the nodal case. We need a
more general class of curve.
Some of the automorphisms of (C, L) are invisible
from the point of view of the coarse curve C.
The characterization of new singularities comes completely from graph
theoretical data.
An `th root comes with a Z/` marking M ∈
1
C (Γ; Z/`) on the edges of the dual graph Γ(C).
This must respect the relation
for any irreducible component Cv .
Roots on stable curves
Ghosts
Towards
G
Rg
=



G










yF
F
→
C
is
an
admissible

↓  :
cover of a stable curve C
C




G-
In my thesis I work the case G = S3, where the
J-locus is empty.
Therefore every pluricanonical form is extendable
over the singular locus, i.e. for a desingularization
G
R̃g and m sufficiently big and divisible,
H
0
S3,reg
⊗m
(Rg
,K )
=H
0
S3
⊗m
(R̃g , K ).
Theorem (G.) - arXiv:1504.00568. The
s
locus of non-canonical singularities of Rg,` is
Sing
s
• Tg,`
nc
nc
s
Rg,`
s
Rg,`
=
−1
s
Tg,`
∪
s
Jg,`.
nc
= Sing
∩ π Sing Mg : it is the
locus of old singularities, coming from Mg ;
s
• Jg,`
the locus of curves with a junior ghost
automorphism, this is the locus of new
singularities.
Chiodo-Farkas (2012) proved that for ` ≤ 6 and
0
` 6= 5, the J-locus is empty in Rg,`.




G
Rg
To prove
being of generale type, we evaluate
its canonical bundle and show it to be big.
For this, it is key a generalization of Chiodo formula for Grothendieck Riemann Roch.
chdRπ∗V =
X vi(j)Bd+1(j/|ξi|) d
vBd+1
· κd −
ψi + · · ·
(d + 1)!
(d + 1)!
i,j