Cohomology classes of strata of differentials

0.54 MB·

by  Adrien Sauvaget

COHOMOLOGY CLASSES OF STRATA OF DIFFERENTIALS
7
ADRIENSAUVAGET
1
0
2

b
e Abstract. We introduce a space of stable meromorphic differentials with
F
poles of prescribed orders and define its tautological cohomology ring. This
8 space, justasthe space ofholomorphicdifferentials, isstratifiedaccording to
2 thesetofmultiplicitiesofzerosofthedifferential. Themaingoalofthispaper
is to compute the Poincaré-dual cohomology classes of all strata. We prove
] thatalltheseclassesaretautologicalandgiveanalgorithmtocomputethem.
G InasecondpartofthepaperwestudythePicardgroupofthestrata. We
use the tools introduced in the first part to deduce several relations in these
A
Picardgroups.
.
h
t
a Contents
m
1. Introduction 2
[
1.1. Stratification of the Hodge Bundle 2

2 1.2. Stable differentials 2
v
1.3. The tautological ring of M 3
7 g,n
1.4. The tautological ring of PH 4
6 g,n,P
8 1.5. Statement of the results 5
7 1.6. An example 5
0 1.7. Applications and related work 7
.
1 1.8. Plan of the paper 9
0 2. Stable differentials 9
7
2.1. The cone of generalized principal parts 9
1
2.2. The space of stable differentials 11
:
v 2.3. Residues 13
i
X 2.4. Standard coordinates 14
2.5. Dimension of the strata 17
r
a 2.6. Stable differentials on disconnected curves 19
2.7. Fibers of the map p:AR →M 21
g,Z,P g,Z,P
2.8. Unstable base 24
2.9. Pre-stable graphs 25
3. The induction formula 28
3.1. Twisted graphs with level structures 28
3.2. Boundary strata associated to admissible graphs 29
3.3. Description of boundary divisors 32
3.4. Class and multiplicity of a boundary divisor 34
Date:March1,2017.
1991 Mathematics Subject Classification. 14C17,14H10,30F30,32G15.
Keywords andphrases. Modulispacesofcurves,Hodgebundle,tautologicalclasses,strataof
differentials.
1
2 ADRIENSAUVAGET
3.5. Induction formula 36
3.6. Proof of Theorems 1, 2, and 3 41
4. Examples of computation 42
4.1. The class [PH (3)] 42
g
4.2. The class of H (4) 43
3
5. Relations in the Picard group of the strata 43
5.1. Classes defined by admissible graphs 43
5.2. Class defined by residue conditions 45
5.3. Classes defined by intersection 45
References 49
1. Introduction
1.1. Stratification of the Hodge Bundle. Let g ≥ 1. Let M be the space of
g
smooth curves of genus g. The Hodge bundle,
H →M
g g
isthe vectorbundle whosefiberoverapoint[C]ofM is the spaceofholomorphic
g
differentialsonC. ApointofH is thenapair([C],α), whereC is acurveandαa
g
differential on C. We will denote by PH →M the projectivization of the Hodge
g g
bundle.
Notation 1.1. Let Z (for zeros) be a vector (k ,k ,...,k ) of positive integers
1 2 n
satisfying
n
k =2g−2.
i
i=1
X
We will denote by PH (Z) the subspace of PH composed of pairs ([C],α) such
g g
thatαisadifferential(defineduptoamultiplicativeconstant)withzerosoforders
k ,...,k .
1 n
The locus PH (Z) is a smooth orbifold (or a Deligne-Mumford stack), see for
g
instance, [25]. However, neither PH , nor the strata PH (Z) are compact.
g g
The Hodge bundle has a natural extension to the space of stable curves:
H →M .
g g
We recall that abelian differentials over a nodal curve are allowed to have simple
poles at the nodes with opposite residues on the two branches.
ThespacePH iscompactandsmooth,andwecanconsidertheclosuresPH (Z)
g g
of the strata inside this space. Computing the Poincaré-dual cohomology classes
of these strata is our motivating problem. In this paper we solve this problem and
present a more general computation in the case of meromorphic differentials.
1.2. Stable differentials.
Definition 1.2. Let n,m ∈ N and let P (for poles) be a vector (p ,p ,...,p )
1 2 m
of positive integers. The space of stable differentials H is the moduli space
g,n,P
of objects ([C],x ,...,x ,α) where C is a pre-stable curve with n+m marked
1 n+m
points and α is a non-zero meromorphic differential on C, such that
• the differentialα hasnopoles outside the m lastmarkedpoints andnodes;
COHOMOLOGY CLASSES OF STRATA OF DIFFERENTIALS 3
• the poles at the nodes are at most simple and have opposite residues on
the two branches;
• if p > 1 then the pole at the marked point x is of order exactly p ; if
i n+i i
p =1 then x canbe a simple pole, a regularpoint, ora zeroofany order;
i i
• the group of isomorphisms of C preserving α and the marked points is
finite.
Proposition 1.3. The forgetful map H → M is an orbifold cone of
g,n,P g,n+m
rank g−1+m+ p . Its Segre class is given by
i
Pm (pi−1)pi−1 1−λ1+...+(−1)gλg
· .
(p −1)! m (1−(p −1)ψ )
i=1 i i=1 i i
Y
This proposition will be proved inQSection 2.
Notation 1.4. Let P be a vector of positive integers of length m. Let Z be a
vector of nonnegative integers of length n. We denote by A ⊂ H , the
g,Z,P g,n,P
locusofstabledifferentials([C],x, ,...,x ,α)suchthatC issmoothandαhas
1 n+m
zeros exactly of orders prescribed by Z at the first n marked points. The locus
A isinvariantundertheC∗-action. WedenotebyPA theprojectivization
g,Z,P g,Z,P
of A . Moreover, we denote by A (respectively PA ) the closures of
g,Z,P g,Z,P g,Z,P
A (resp. PA ) in the space H (respectively in PH ).
g,Z,P g,Z,P g,n,P g,n,P
Let V be a vector,in this article we will denote by |V| the sumof elements of V
and by ℓ(V) the length of V.
Definition 1.5. Given g and P, we will say that Z is complete if it satisfies
|Z| − |P| = 2g − 2. If Z is complete, we will denote by Z − P the vector
(k ,...,k ,−p ,...,−p ).
1 n 1 m
1.3. The tautological ring of M . Let g and n be nonnegative integers satis-
g,n
fying 2g−2+n > 0. Let M be the space of stable curves of genus g with n
g,n
marked points. Define the following cohomology classes:
• ψ = c (L ) ∈ H2(M ,Q), where L is the cotangent line bundle at the
i 1 i g,n i
ith marked point,
• κ = π (ψm+1) ∈ H2m(M ,Q), where π : M → M is the
m ∗ n+1 g,n g,n+1 g,n
forgetful map,
• λ =c (H )∈H2k(M ,Q), for k =1,...,g.
k k g,n g,n
Definition 1.6. A stable graph is the datum of
Γ=(V,H,g :V →N,a:H →V,i:H →H,E,L)
satisfying the following properties:
• V is a vertex set with a genus function g;
• H isahalf-edgesetequippedwithavertexassignmentaandaninvolutioni;
• E, the edge set, is defined as the set of length 2 orbits of i in H (self-edges
at vertices are permitted);
• (V,E) define a connected graph;
• L is the set of fixed points of i called legs;
• foreachvertexv,the stabilityconditionholds: 2g(v)−2+n(v)>0,where
n(v)=#(a−1(v)) (the cardinal of a−1(v).
The genus of Γ is defined by g(v)+#(E)−#(V)+1.
P
4 ADRIENSAUVAGET
Let v(Γ), e(Γ), and n(Γ) denote the cardinalities of V,E, and L, respectively.
A boundary stratum of the moduli space of curves naturally determines a stable
graph of genus g with n legs by considering the dual graph of a generic pointed
curve parameterized by the stratum. Thus the boundary strata of M are in
g,n
1-to-1 correspondence with stable graphs.
Let Γ be a stable graph. Define the moduli space M by the product
Γ
M = M ,
Γ g(v),n(v)
v∈V
Y
and let ζ :M →M be the natural morphism.
Γ Γ g,n
Definition1.7. Atautological class isalinearcombinationofclassesβ oftheform
β =ζ ( P ),
Γ∗ v
v∈V
Y
whereΓisastablegraphandP isapolynomialinκ,λandψclassesonM .
v g(v),n(v)
Proposition 1.8. The tautological classes form a subring of H∗(M ,Q).
g,n
See [15] for the description of the product of tautological classes.
Definition 1.9. This subring is called the tautological ring of M and denoted
g,n
by RH∗(M ).
g,n
Remark 1.10. Actually the classes α as above that do not involve λ-classes span
the tautological ring. However it will be more convenient for us to use this larger
set of generators.
1.4. The tautological ring of PH . Let P be a vector of positive integers.
g,n,P
From now on, unless specified otherwise, we will denote by π : M → M
g,n+1 g,n
the forgetful map and by p : H → M the projection from the space
g,n,P g,n+m
of differentials to M . Moreover we will use the same notation p : PH →
g,n g,n,P
M for the projectivized cone. Let
g,n+m
L=O(1)→PH
g,n,P
be the dual of the tautological line bundle of PH , and let
g,n,P
ξ =c (L).
1
Definition1.11. ThetautologicalringofPH isthesubringofthecohomology
g,n,P
ring H∗(PH ,Q) generated by ξ and the pull-back of RH∗(M ) under p.
g,n,P g,n+m
We denote it by RH∗(H ).
g,n,P
Remark 1.12. We have ξd =0 for d>dim(PH ). Therefore the tautological
g,n,P
ring of stable differentials is a finite extension of the tautological ring of stable
curves. In particular,in absence of poles, the Hodge bundle is a vector bundle and
we have
RH∗(PH )=RH∗(M )[ξ]/(ξg +λ ξg−1+...+λ ).
g,n g,n 1 g
COHOMOLOGY CLASSES OF STRATA OF DIFFERENTIALS 5
1.5. Statement of the results. We have now all elements to state the main
theorems of this article.
Theorem 1. For any vectors Z and P, the class PA introduced in Nota-
g,Z,P
tion 1.4, lies in the tautological ring of PH and is explicitly computable.
g,n,P (cid:2) (cid:3)
The main ingredient to prove this theorem will be the induction formula of
Theorem 5.
Restricting ourselves to the holomorphic case and applying the forgetful map of
the marked points we obtain the following corollary.
Theorem 2. For any vector Z, the class PH (Z) introduced in Notation 1.1 lies
g
in the tautological ring of PH and is explicitly computable.
g (cid:2) (cid:3)
Remark 1.13. As a guideline for the reader, it will be important to understand
that the holomorphic case in Theorem 1 cannot be proved without using strictly
meromorphic differentials. Thus Theorem 2 is a consequence of a specific case of
Theorem 1 but one cannot avoid to prove Theorem 1 in its full generality.
The secondimportantcorollaryis obtainedby forgettingthe differentialinstead
ofthemarkedpoints. LetP =(p ,...,p )beavectorofpolesandZ =(k ,...,k )
1 m 1 n
be a complete vector of zeros. We define H (Z −P) ⊂ M as the locus of
g g,n+m
points (C,x ,...,x ) that satisfy:
1 n
n m
ω − k x + p x ≃O .
C i i j n+j C
 
i=1 j=1
X X
 
We denote by H (Z−P) the closure of H (Z−P) in M .
g g g,n+m
Theorem 3. For any vectors Z and P, the class H (Z−P) lies in the tautolog-
g
ical ring of M and is explicitly computable.
g,n+m (cid:2) (cid:3)
Remark 1.14. Theorems 1, 2 and 3 are stated for the Poincaré-dual rational
cohomology classes. However, all the results of this paper remains remain valid in
the Chow groups.
In a second part of the Paper (Section 5) we will consider the rational Picard
groupof the space H (Z−P). We will define severalnaturalclassesin this Picard
g
groupandapplythetoolsdevelopedinthefirstpartofthepapertodeduceaseries
of relations between these classes: see Theorem 6.
1.6. An example. Here we illustrate the general method used in this article by
computing the class of differentials with a double zero PH (2,1,...,1) . This
g
computation was carried out by D. Zvonkine in an unpublished note [27] and was
(cid:2) (cid:3)
the starting point of the present work.
Webeginbymarkingapoint,i.e. westudythespacePH oftriples(C,x ,[α])
g,1 1
composedof a stable curve C with one markedpoint x andan abeliandifferential
1
α modulo a multiplicative constant. Recall that PA ⊂ PH is the closure of
g,(2) g,1
the locus of smooth curves with a double zero at the marked point. In order to
compute [PA ], we consider the line bundle
g,(2)
L⊗L ≃Hom(L∨,L )
1 1
6 ADRIENSAUVAGET
over PH . (Recall that L∨ is the tautological line bundle of the projectivization
g,1
PH and L is the cotangent line bundle at the marked point x .) We construct
g,1 1 1
a natural section s of this line bundle,
1
s :L∨ → L
1 1
α 7→ α(x ).
1
Namely,anelementofL∨ isanabeliandifferentialonC,andwetakeitsrestriction
to the marked point.
The section s vanishes if and only if the marked point is a zero of the abelian
1
differential. Thus we have the following identity in H2(PH ):
g,1
[PA ]=[{s =0}]=c (L⊗L )=ξ+ψ .
g,(1) 1 1 1 1
Now we restrict ourselves to the locus {s =0} and consider the line bundle
1
L⊗L⊗2.
1
We build a sections of this new line bundle. An element of L∨ is an abelian
2 |{s1=0}
differential with at least a simple zero at the marked point x . Its next leading
1
term at the marked point is an element of L⊗2 (we can verify this assertion using
1
a local coordinate at x ).
1
As before, s is equalto zero if and only if the markedpoint is atleast a double
2
zerooftheabeliandifferential. However,{s =0}iscomposedofthreecomponents:
2
• PA ;
g,(2)
• thelocusα wherethemarkedpointliesonanelliptic componentattached
e
to the rest of the stable curve at exactly one point and the abelian differ-
ential vanishes identically on the elliptic component;
• the locus α where the marked point lies on a “rational bridge”, that is, a
r
rationalcomponentattachedtotwocomponentsofthestablecurvethatare
not connected except by this rational component (in this case the abelian
differential automatically vanishes on the rational bridge).
We deduce the following formula for [PA ]:
g,(2)
[PA ] = [{s =0}]−[α ]−[α ]
g,(2) 2 e r
= (ξ+ψ )(ξ+2ψ )−[α ]−[α ]
1 1 e r
Remark 1.15. We make a series of remarks on this result.
• Totransformtheaboveconsiderationsintoanactualproofweneedtocheck
that the vanishing multiplicity of s along all three components equals 1.
2
We will prove this assertion and its generalization in Section 3.
• Denote by π : M → M the forgetful map then we have, by δ the
g,1 g sep
boundarydivisorofcomposedofcurveswithaseparatingnode,andδ
nonsep
the boundarydivisorofcurveswithaseparatingnode. Applying the push-
forward of π to the above expression of [PA ] we get
g,(2)
[PH(2,1,...,1)]=π [PA ]=(3κ ξ+2κ )−δ −0,
∗ g,(2) 0 1 sep
The terms π (ξ2) and π (α ) vanish by the projection formula. Using the
∗ ∗ e
relations κ = 2g − 2, and κ = 12λ −δ − δ on M (see, for
0 1 1 sep nonsep g
example, [1], chapter 17), we have
[PH(2,1,...,1)]=(6g−6)ξ+24λ −3δ −2δ .
1 sep nonsep
COHOMOLOGY CLASSES OF STRATA OF DIFFERENTIALS 7
This formula was first proved by Korotkin and Zograf in 2011 using an
analysis of the Bergmantau function [20]. Dawei Chen gave another proof
of this result in 2013 using test curves [6].
• In general, to prove Theorem 1 we will work by induction. Let Z =
(k ,k ,...,k ) and P be vectors. Let Z′ = (k ,...,k +1,...,k ). Then
1 2 n 1 i n
we will show that
PAg,Z′,P =(ξ+(ki+1)ψi) PAg,Z,P − boundary terms.
The(cid:2)computa(cid:3)tion of these bounda(cid:2)ry terms(cid:3)is the crucial part of the proof.
1.7. Applications and related work.
Classes in the Picard group of M . Scott Mullane and Dawei Chen gave a closed
g
formula for the class of π PH (Z) in the rational Picard group of M for all
∗ g g
Z of length g −2 (see [5] and [21]). They used test curves and linear series to
(cid:2) (cid:3)
compute this formula. This result has the advantage of giving explicit expressions,
however it has the drawback of not keeping track of the positions of the zeros and
of being restricted to the vectors Z of length g−2 (see Section 4.2 for an example
of computation).
Incidence variety compactification. The problem of the compactification of the
strataisextensivelystudiedfromdifferentapproachesinajointworkofBainbridge,
Chen,Gendron,Grushevsky,andMoeller(see[2]and[14]). Theircompactification
(calledincidence variety compactification) isslightly differentfromthe one thatwe
use here. We will recall their definitions in Section 3.2 since we will make use of
some of their results.
Moduli space of twisted canonical divisors and Double Ramification Cycle. In [12],
Farkas and Pandharipande proposed another compactification of the strata. Let
g,n such that 2g−2+n > 0. Let µ = (k ,...,k ) be a partition of 2g−2. We
1 n
recallthatH (µ)⊂M isthelocusofsmoothcurvessuchthatω (−k x −...−
g g,n C 1 1
k x )≃‰ andthatwedenote byH (µ)its closureinM . In[12],Farkasand
n x C g g,n
Pandharipande defined the space of twisted canonical divisors denoted by H˜(µ).
The space of twisted canonical divisors is a singular closed subspace of M such
g,n
that H(µ) is one of the irreducible components of H˜(µ).
We assume that µ contains at least one negative value. In the appendix of [12],
FarkasandPandharipandedefinedaclassH (µ)inA (M )(orH2g(M )): this
g g g,n g,n
class is a weighted sum over the classes of irreducible components.
Conjecturalexpression ofH (µ)andDoubleRamification Cycle. Letrbeapositive
g
integerand (C,x ,...,x ) be a smooth curve with markings. A r-spin structure is
1 n
a line bundle L such that L⊗r ≃ ω (−k x −...−k x ). We denote the moduli
C 1 1 n n
spaceofr-spinstructuresbyM1/r. Thisspaceadmitsastandardcompactification
g,µ
1/r 1/r 1/r
by twisted r-spin structures: M . We denote by π : C →M the universal
g,µ g,µ g,µ
1/r
curves and by Ł → C the universal line bundle. The moduli space of twisted
g,µ
1/r
r-spin structures has a natural forgetful map ǫ:M →M ; the map ǫ is finite
g,µ g,n
of degree r2g−1. We consider Rπ (Ł) the image of Ł in the derived category of
∗
8 ADRIENSAUVAGET
1/r
M . The following diagram sums up the notation:
g,µ
Ł
))
1/r,µ
C
g,n
(cid:15)(cid:15)π
(cid:15)(cid:15)
1/r,µ ǫ //
Rπ (Ł) M M .
∗ g,n g,n
Ifµhasatleastonenegativevalue,weconsiderclasscr(µ)d=ef Rπ (Ł)∈A (M1/r).
g ∗ g g,µ
If all values of µ are positive we consider a different class, namely Witten’s class:
cr ∈(µ)A (M1/r). There areseveralequivalentdefinitions ofofWitten’s class,
W g−1 g,µ
allofwhichrequireseveraltechnicaltoolsthatwewillnotdescribehere(see[26],[7]
or [4]). Instead, we consider the two following functions:
P , PW :N∗ → A (M )
g,µ g,µ ∗ g,n
r 7→ rǫ (cr(µ)), rǫ (cr (µ)).
∗ g ∗ W
Both P and PW are polynomials for large values of r. This result is due to
g,µ g,µ
Aaron Pixton for P (see [17]), and Felix Janda for PW (see [24]). We denote
g,µ g,µ
by P˜ and P˜W the asymptotic polynomials. The two following conjectures have
g,µ g,µ
been proposed:
Conjecture A. (see [12]) If µ has at least one negative value then the equality
H (µ)=P˜ (0) holds in A (M ).
g g,µ g g,n
Conjecture B. (see [24]) If µ has only positive values then the equality [H (µ)]=
g
P˜W(0) holds in A (M ).
g,µ g−1 g,n
These two conjectures are the analogous for differentials of the formula for the
so-called Double Ramification cycles: a Double Ramification cycle is the push-
foward to the moduli space of curves of the virtual fundamental class of a moduli
space of rubber maps to P1 with prescribed singularities at 0 and ∞ (see [17]).
As a consequence of Theorem 3, we know that the classes H (µ) and [H (µ)] are
g g
tautologicaland we have an algorithmto check the validity of the conjectures case
by case (see Section 4.2 for examples of computations).
Compactification via log-geometry. Jérémy Guéré has defined a moduli space of
"rubber" differentials using log geometry. He proves that this space is endowed
with a perfect obstruction theory. Moreover, if µ has negative values, this moduli
space surjects onto the moduli space of twisted canonical divisors and the class
H (µ) is the push-forward of the virtual fundamental cycle (see [16]).
g
If µ has only positive values, Dawei Chen and Qile Chen have also used log
geometry to define a compactification of the strata H (µ) (see [8]). They also
g
proved that this compactification is endowed with a perfect obstruction theory;
howeverthevirtualdimensionoftheircompactificationisequaltodim(H (µ))−1.
g
Induction formula for singularities in families. The central result of the present
work is the induction formula of Section 3. A similar formula has been proved
by Kazarian, Lando and Zvonkine for classes of singularities in families of genus 0
COHOMOLOGY CLASSES OF STRATA OF DIFFERENTIALS 9
stablemaps(see[18]). Theirformulacontainsonlythegenus0partofourinduction
formula.
Theygiveaninterpretationoftheinductionformulaingenus0asageneralization
of the completed cycle formula of Okounkov and Pandharipande (see [23]). For,
now it is not clear if this generalized completed cycle formula has an extension to
higher genera.
Computation oftheLyapunovexponentsofstrata. Ourcomputationofcohomology
classes of strata in the space of differentials could be useful for the study of the
dynamicsofflatsurfaces. Thisideaisdevelopedforexamplein[20]and[5]basedon
theworkofKontsevichandZorich[10](seeSection5.3.3foradetailedpresentation).
1.8. Plan of the paper. InSection2wedescribethegeometryoftheconeofsta-
bledifferentialsandcomputeitsSegreclass. InSection3wedescribetheboundary
of the strata of differentials with prescribed orders of zeros and we establish the
formula for the boundary terms in the induction; we use this result to prove The-
orems 1, 2 and 3. In Section 4, we give some examples of explicit computations
in the projectivized Hodge bundle or in the moduli space of curves. In Section 5,
we introduce several classes in the Picard group of the strata and prove several
relations between these classes by using the induction formula.
Acknowledgments. I am very grateful to my PhD. advisor Dimitri Zvonkine who
proposed me this interesting problem, supervised my work with great implication
and brought many beneficial comments.
IwouldalsoliketothankJérémyGuéré,CharlesFougeron,SamuelGrushevsky,
Rahul Pandharipande, Dawei Chen, Qile Chen and Anton Zorich, for helpful con-
versations. They have enriched my understanding of the subject in many ways. I
am grateful to Felix Janda for the computations that he made and for the long
discussions we had regarding the present work.
Finally I also would like to thank the organizers of the conference "Dynamics
in the Teichümuller Space" at the CIRM (Marseille, France) for their invitation as
well as the people I have met for the first time at this conference : Pascal Hubert,
Erwan Lanneau, Samuel Lelièvre, Anton Zorich, Quentin Guendron and Martin
Möller.
2. Stable differentials
In this section, we study the space of stable differentials. We prove that it is a
cone overM and compute the Segre class. We also define stable differentials on
g,n
disconnected curves.
2.1. The cone of generalized principal parts. We follow here the approachof
[9]. Let X be a projective DM stack.
Definition 2.1. An orbifold cone is a finitely generated sheaf of graded ‰ -
X
algebras S =S0⊕S1⊕S2⊕... such that S0 =‰ .
X
Remark 2.2. This definition of cone is weaker than the classical definition of
Fulton(see[13])becausewedonotaskthatS begeneratedbyS1. However,wecan
assignto this objectanorbifoldC =Spec(S)andits projectivizationPC =Proj(S)
which is also an orbifold (see example below). The projectivization PC comes with
a natural orbifold line bundle ‰(1), the dual of the tautological line bundle. We
10 ADRIENSAUVAGET
denotep:PC =Proj(C)→X andξ =c1(O(1)). LetC →X beapure-dimensional
cone and r the rank of the cone defined as dim(C)−dim(X). The i-th Segre class
of C is defined as
s =p (ξr+i−1)∈H2i(X,Q).
i ∗
Example 2.3. Let us consider the graded algebra C[x,y,z] such that x is an
element of weight 2, y is an element of weight 3 and z is an element of weight 1.
Thisgradedalgebrawhichisnotgeneratedbyitsdegree1elements. Theassociated
projectivizedconeoverapoint is the weightedprojectivespace P(2,3)whichis the
quotient of (C3)∗ by C∗ with the action:
λ·(x,y,z)=(λ2x,λ3y,λz).
Definition2.4. Letpbeanintegergreaterthan1. Aprincipal partoforderpata
smoothpointofacurveisanequivalenceclassofgermsofmeromorphicdifferentials
with a pole of order p ; two germs f ,f are equivalent if f −f is a meromorphic
1 2 1 2
differential with at most a simple pole.
First, we parametrize the space of principal parts at a point. Let z be a local
coordinate at 0 ∈C. A principal part at 0 of order p is given by:
u p−1 u p−2 u dz
+a +...+a
1 p−2
z z z z
(cid:20)(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)(cid:21)
with u6=0. Howeverthe choice of (u,a ,...,a ) is not unique. Indeed there are
1 k−2
p−1 choices of u given by the ζℓ ·u (with ζℓ = exp(2iπ·ℓ), for 0 ≤ ℓ ≤ p−1).
p−1
Therefore the coordinates (u,a ,...,a ) parametrize a covering of the space of
1 p−1
principalparts. The groupof automorphismsofthe coveringis Z (p−1)Z. It acts
by
(cid:14)
ζ·(u,a ,...,a )=(ζu,ζa ,...,ζp−1a ).
1 p−1 1 p−1
Moreover, the natural action of C∗ on the space of principal parts in coordinates
(u,a ,...,a ) is given by
1 p−2
1 1 p−1
λ·(u,a1,...,ap−1)=(λpu,λpa1,...,λ p ap−1).
This is well defined up to the action of Z (p−1)Z. From now on, we will use the
nonstandard notation Z = Z nZ. Set the weight of u to 1 and the weights of
n (cid:14) p−1
the a to j . The graded algebra
j p−1 (cid:14)
S =C[u,a ,...,a ]Zp−1,
1 p−2
is called the algebra of generalized principal parts and P =Spec(S) is the space of
generalized principal parts. We denote by A the subspace defined by {up−1 = 0}.
It is the Cartier divisor obtained as the image of the Weyl divisor {u=0}⊂Cp−1
inthe quotientofCp−1 by the actionofZ . The spaceofprincipalpartsembeds
p−1
into P as the complementary of A.
Lemma 2.5. A change of local coordinate z induces an isomorphism of S that
preserves the grading and acts trivially on the quotient algebra S/I , where I is
u u
the ideal of mononomials containing u.