Intersection Theory on Abelian-Quotient V-Surfaces and Q-Resolutions

20 Pages·2014·0.59 MB·English

Journal of Singularities
received: 11July2013
Volume 8 (2014), 11-30 inrevisedform: 25February2014
DOI:10.5427/jsing.2014.8b
INTERSECTION THEORY ON ABELIAN-QUOTIENT V-SURFACES AND
Q-RESOLUTIONS
ENRIQUEARTALBARTOLO,JORGEMARTÍN-MORALES,ANDJORGEORTIGAS-GALINDO
Abstract. Inthispaperwestudytheintersectiontheoryonsurfaceswithabelianquotient
singularities and we obtain formulas for its behavior under weighted blow-ups. As appli-
cations, we extend Mumford’s formulas for the intersection theory on normal divisors, we
derivepropertiesforquotientsofweightedprojectiveplanes,andfinally,wecomputeabstract
Q-resolutionsofnormalsurfacesusingJung’smethod.
Introduction
In [6], Fulton developed a general intersection theory for algebraic varieties. For the case of
normalsurfaces,Mumford[10]providedamoredetaileddescriptionusingresolutionofsingular-
ities. In this work we focus on V-surfaces with abelian quotient singularities (cyclic V-surfaces
for short). One of the goals is to prove that the resolution of singularities is not needed for the
description of the intersection theory of these V-surfaces and in fact it can be realized following
the same ideas as in the smooth case, see Definition 3.1. In the latter case, the description is
based on the identification of Weil and Cartier divisors. This identification is no longer true for
V-surfaces, but it becomes true using Q-divisors. The relationship between intersection theory
and Weil divisors for normal surface singularities was already studied by F. Sakai [12].
Mumford’s definition is based on the formulas which relate intersection numbers before and
after a blow-up, namely, the self-intersection of the exceptional component and the relationship
between the intersection number of the divisors and the one of their strict transforms. The
main result in this paper, Theorem 4.3, generalizes these formulas replacing smooth surfaces by
cyclic V-surfaces and the standard blow-ups by weighted blow-ups, i.e., the result of blowing up
theweightfiltrationforanisolatedsingularitywithgoodC∗-action. Thesespaces(eveninhigher
dimension) are V-manifolds with abelian quotient singularities, see [4] (or [1] for a description
closer to the language of this work).
We derive several applications of Theorem 4.3, see [2] for further applications. The first one
is to provide formulas for the intersection theory of normal surfaces in terms of Q-resolutions
instead of standardresolutions, see §2 for definitionsand Theorem 4.5 foran explicit statement.
TheseQ-resolutionsarecombinatoriallylessinvolvedthanstandardresolutionswhilecontaining
the same information; e.g., Veys [15] used them to simplify the computation of the topological
zeta function.
Thesecondoneisthedescriptionoftheintersectiontheoryofthemostwell-knownV-surfaces,
namely the weighted projective planes, and more generally, their cyclic quotients, with explicit
formulas, see §5.
2000 Mathematics Subject Classification. Primary: 32S25;Secondary: 32S45.
Key words and phrases. Quotientsingularity,intersectionnumber,embeddedQ-resolution.
Partially supported by MTM2010-21740-C02-02 and E15 (Aragón). Second named author is partially sup-
portedbyFQM-333(Andalucía)andPRI-AIBDE-2011-0986.
12 E.ARTAL,J.MARTÍN-MORALES,ANDJ.ORTIGAS-GALINDO
Wefinish this work withthe application of cyclic quotient singularities, Q-resolutions and Q-
intersectiontheorytotheimplementationofJungresolutionmethodofnormalsingularities. The
ideaofthismethodisthefollowing. Considerafiniteprojectionπ ofanormalsingularityS onto
(C2,0)withdiscriminant∆andperformanembeddedresolutionσof∆;thenormalizationofthe
pull-backofσandπhappenstobeaQ-resolutionofS;theresolutionofitssingularitiesproduces
a resolution of S. There are two main advantages in using the methods developed in this work
withJungresolution. Forthefirstadvantage,onecanreplaceσ byanembeddedQ-resolutionof
∆ (reducing computation time). For the second one, intersection theory gives a straightforward
way to obtain the final resolution of S with the self-intersections of the exceptional divisors.
We thank J.I. Cogolludo for his fruitful conversations and ideas.
1. V-Manifolds and Quotient Singularities
A V-manifold [13] (or orbifold) of dimension n is a complex analytic space which admits an
open covering U such that U is analytically isomorphic to B /G where B Cn is an open
i i i i i
ball and G is a{fin}ite subgroup of GL(n,C). They have been classified locally b⊂y Prill [11]: it is
i
enough to consider the so-called small subgroups G GL(n,C), i.e., without rotations around
⊂
hyperplanes other than the identity. We fix the notations when G is abelian.
Ford:=t(d ...d )wedenoteµ :=µ µ afiniteabeliangroupwrittenasaproduct
of finite cyclic1grouprs, that is, µ dis thed1cy×c·li·c·×grodurp of d -th roots of unity in C. Consider a
di i
matrix of weight vectors
A:=(a ) =[a a ] Mat(r n,Z), a :=t(a ...a ) Mat(r 1,Z),
ij i,j 1 n j 1j rj
| ··· | ∈ × ∈ ×
and the action
(µ µ ) Cn Cn, ξ :=(ξ ,...,ξ ),
(1) d1 ×···× dr × −→ d d1 dr
ξ ,x (ξa11 ... ξar1x , ... ,ξa1n ... ξarnx ), x:=(x ,...,x ).
d (cid:55)→ d1 · · dr 1 d1 · · dr n 1 n
Note that t(cid:0)he i-t(cid:1)h row of the matrix A can be considered modulo d . The set of all orbits Cn/G
i
is called (cyclic) quotient space of type (d;A) and it is denoted by
d a a
1 11 1n
X(d;A):=X ... ... ·.·.·. ... .
d a a
 r r1 ··· rn 
 
The orbit of an element x Cn under this action is denoted by [x] and the subindex is
(d;A)
∈
omitted if no ambiguity seems likely to arise. Using multi-index notation the action takes the
simple form
µ Cn Cn,
d
× −→
(ξ ,x) ξ x:=(ξa1x ,...,ξanx ).
d (cid:55)→ d· d 1 d n
The quotient of Cn by a finite abelian group is always isomorphic to a quotient space of
type (d;A) but different types (d;A) can give rise to isomorphic quotient spaces. Using [1,
Lemma 1.8] we can prove the following lemma which restricts the number of possible factors of
the abelian group in terms of the dimension.
Lemma 1.1. The space X(d;A) = Cn/µ can always be represented by an upper triangular
d
matrix of dimension (n 1) n. More precisely, there exist a vector e=(e ,...,e ), a matrix
1 n−1
− ×
INTERSECTION THEORY AND Q-RESOLUTIONS 13
B =(b ) , and an isomorphism [(x ,...,x )] [(x ,...,xk)] for some k N such that
i,j i,j 1 n (cid:55)→ 1 n ∈
e b b b
1 1,1 1,n−1 1,n
X(d;A)∼=X ... ... ·.·.·. ... ... =X(e;B).
e 0 b b
 n−1 ··· n−1,n−1 n−1,n 
 
Remark 1.2. For n=2 it is enough to consider cyclic quotients. Nevertheless, in order to avoid
cumbersome statements, we will allow if necessary quotients of non-cyclic groups.
We say that a type (d;A) is normalized if the action is free on (C∗)n and µ is small as
d
subgroup of GL(n,C), i.e., if the stabilizer subgroup of P is trivial for all P Cn with exactly
∈
n 1 coordinates different from zero. If n=2, then a normalized type is always cyclic.
−
In the cyclic case the stabilizer of a point as above (with exactly n 1 coordinates different
−
from zero) has order gcd(d,a ,...,a ,...,a ).
1 i n
Definition 1.3. The index of a quotient X(d;A) of C2 equals d for X(d;A) = X(d;a,b)
∼
(cid:98)
normalized.
Example 1.4. Following Lemma 1.1, all quotient spaces for n = 2 are cyclic. The space
X(d;a,b) is written in a normalized form if and only if gcd(d,a) = gcd(d,b) = 1. If this is not
the case, one uses the isomorphism (assuming gcd(d,a,b)=1)
X(d;a,b) X d ; a , b ,
−→ (d,a)(d,b) (d,a) (d,b)
(cid:16) (cid:17)
(x,y) (x(d,b),y(d,a))
(cid:55)→
to convert it into a normalize(cid:2)d one(cid:3)where (u(cid:2),v) stands for(cid:3)gcd(u,v).
Weighted projective spaces are canonical examples of compact V-manifolds, see [3]. Let
ω :=(q ,...,q ) be a weight vector, that is, a finite set of coprime positive integers. There is a
0 n
natural action of the multiplicative group C∗ on Cn+1 0 given by
\{ }
(x ,...,x ) (tq0x ,...,tqnx ).
0 n 0 n
(cid:55)−→
Thesetoforbits Cn+1\{0} underthisactionisdenotedbyPn (orPn(ω)incaseofcomplicated
C∗ ω
weight vectors) and it is called the weighted projective space of type ω. The class of a nonzero
element (x ,...,x ) Cn+1 is denoted by [x :...:x ] and the weight vector is omitted if no
0 n 0 n ω
∈
ambiguity seems likely to arise.
Consider the decomposition Pn = U U , where U is the open set consisting of all
ω 0 ∪···∪ n i
elements [x :...:x ] with x =0. The map
0 n ω i
(cid:54)
ψ :Cn U , ψ (x , ,x ):=[1:x :...:x ]
0 0 0 1 n 1 n ω
−→ ···
defines an isomorphism ψ if we replace Cn by X(q ; q ,...,q ). Analogously,
0 0 1 n
(cid:101) (cid:101)
X(q ; q ,...,q ,...,q )=U
i 0 i n ∼ i
under the obvious analytic map.
Weights can be normalized as follows. Let d(cid:98):=gcd(q ,...,q ,...,q ) and denote
i 0 i n
e :=d ... d ... d
i 0 i n
· · · · (cid:98)
and p := qi. The following map is an isomorphism:
i ei (cid:98)
Pn q ,...,q Pn(p ,...,p )
0 n 0 n
−→
[x(cid:0)0 :...:xn(cid:1)] (cid:55)→ xd00 :...:xdnn .
(cid:2) (cid:3)
14 E.ARTAL,J.MARTÍN-MORALES,ANDJ.ORTIGAS-GALINDO
Remark 1.5. One can always assume the weight vector is normalized, i.e., it satisfies
gcd(q ,...,q ,...,q )=1,
0 i n
for i = 0,...,n. In particular, P1(q ,q ) = P1 and for n = 2 we can take (q ,q ,q ) relatively
0 1 ∼ 0 1 2
prime numbers. (cid:98)
2. Abstract and Embedded Q-Resolutions
An embedded resolution of f = 0 Cn is a proper map π : X (Cn,0) from a smooth
{ } ⊂ →
varietyX satisfying,amongotherconditions,thatπ−1( f =0 )isanormalcrossingdivisor. To
{ }
weaken the condition on the preimage of the singularity we allow the new ambient space X to
containabelianquotientsingularitiesandthedivisorπ−1( f =0 )tohavenormalcrossings over
{ }
thiskindofvarieties. ThisnotionofnormalcrossingdivisoronV-manifoldswasfirstintroduced
by Steenbrink in [14].
LetM =Cn+1/µ beanabelianquotientspacenotnecessarilycyclicorwritteninnormalized
d
form. Consider H M an analytic subvariety of codimension one.
⊂
Definition 2.1. An embedded Q-resolution of (H,0) (M,0) is a proper analytic map
⊂
π :X (M,0)
→
such that:
(1) X is a V-manifold with abelian quotient singularities.
(2) π is an isomorphism over X π−1(Sing(H)).
(3) π−1(H) is a Q-normal crossi\ng hypersurface on X (i.e., it has only normal crossings in
the sense of Steenbrink).
In the same way we define abstract Q-resolutions.
Definition 2.2. Let (X,0) be a germ of singular point. An abstract good Q-resolution is a
proper birational morphism π : Xˆ (X,0) such that Xˆ is a V-manifold with abelian quotient
singularities, π is an isomorphism→outside Sing(X), and π−1(Sing(X)) is a Q-normal crossing
divisor.
Note that one can pass from a Q-resolution to a standard resolution by solving the abelian
quotient singularities. These singularities were solved by Fujiki [5]; the solution of the surface
wasobtainedmuchearlieranditisknownastheJung-Hirzebruchmethod,see[7]foranexplicit
description.
In the surface case, an embedded Q-resolution is obtained as a composition of the so-called
weighted blow-ups. Let us describe this classical notion in terms of charts.
2.3. Blow-up of X(d;a,b) with respect to ω := (p,q). Let X = X(d;a,b) assumed to be
normalized. Let
(cid:92)
π :=π : X(d;a,b) X(d;a,b)
(d;a,b),ω ω −→
be the weighted blow-up at the origin of X(d;a,b) with respect to ω =(p,q), i.e.,
X(cid:92)(d;a,b) := ((x,y),[u:v] ) C2 P1 xqvp =ypuq µ
ω { ω ∈ × ω | } d
(cid:46)
where the action of µ is the natural extension of the one defining X(d;a,b) and π is induced
d
(cid:92)
by the first projection. Then, X(d;a,b) is covered by
ω
p 1 q q p 1
U1∪U2 =X pd −a pb qa ∪X qd qa pb −b
(cid:18) − (cid:19) (cid:18) − (cid:19)
(cid:98) (cid:98)
INTERSECTION THEORY AND Q-RESOLUTIONS 15
and the charts are given by
1st chart X p −1 q U [(x,y)] [((xp,xqy),[1:y] )] .
pd a pb−qa −→ 1 (cid:55)→ ω (d;a,b)
2nd chart X(cid:16)qqd (cid:12)(cid:12)(cid:12)qa−ppb−b1(cid:17) −→ U(cid:98)2 [(x,y)] (cid:55)→ [((xyp,yq),[x:1]ω)](d;a,b).
Theexceptionaldiviso(cid:16)rE(cid:12)(cid:12)(cid:12)=π(−d1;a,b),(cid:17)ω(0)isi(cid:98)dentifiedwiththequotientspaceP1ω(d;a,b):=P1ω/µd
which is isomorphic to P1 under the map
P1(d;a,b) P1,
ω −→
[x:y] [xdq/e :ydp/e],
ω,(d;a,b)
(cid:55)→
where e:=gcd(d,pb qa).
−
Remark 2.4. Letusshowhowtoconvertaspaceoftype(p a b intoitscyclicform. Bysuitable
q| c d
multiplications of the rows, we can assume p = q = r: X(r a b . For the second step we add
r| c(cid:1)d
a third row by adding the first row multiplied by α and the second row multiplied by β, where
(cid:1)
αa+βc=m and m:=gcd(a,c) (note that gcd(α,β)=1):
r a b r 0 βad−bc
− m r 0 ad−bc
X r c d =X r 0 αad−bc =X m .
 (cid:12)   (cid:12) m  r m αb+βd
r(cid:12)m αb+βd r(cid:12)m αb+βd (cid:18) (cid:12) (cid:19)
(cid:12) (cid:12) (cid:12)
 (cid:12)   (cid:12)  (cid:12)
Let t := gcd(r,ad−(cid:12)bc). Then, our space is(cid:12)of type (r;m,(αb+βd)r(cid:12)) and normalization follows
m(cid:12) (cid:12) t
by taking gcd’s. The isomorphism is [(x,y)] [(x,yrt)](r;m,(αb+βd)r).
(cid:55)→ t
Let us apply the previous remark to the preceding charts. Assume the type (d;a,b) is nor-
malized. To normalize these quotient spaces, note that
e=gcd(d,pb qa)=gcd(d, q+βpb)=gcd(pd, q+βpb)=gcd(qd,p qaµ),
− − − −
where βa µb 1 mod d. Then another expressions for the two charts are given below.
≡ ≡
1st chart X pd;1,−q+βpb U [(xe,y)] [((xp,xqy),[1:y] )] .
e e −→ 1 (cid:55)→ ω (d;a,b)
2nd chart X(cid:16)qd;−p+µqa,1(cid:17) U [(x,ye)] [((xyp,yq),[x:1] )] .
e e −→ (cid:98)2 (cid:55)→ ω (d;a,b)
Both quotient spaces(cid:16)are now writte(cid:17)n in their normalized form. The equation of the charts will
(cid:98)
be useful to compute multiplicities, see Remark 4.4.
For an irreducible germ of function in (C2,0), only a weighted blow-up is needed for each
Puiseux pair in order to compute an embedded Q-resolution, and the weight is determined by
the Puiseux pairs. In the reducible case, one has to consider the weighted blow-ups associated
withthePuiseuxpairsofeachirreduciblecomponentandaddalsoweightedblow-upsassociated
with the contact exponents for each pair of branches. There is another longer way to get this
Q-resolution: performastandardembeddedresolutionandcontractanyexceptionalcomponent
having at most two singular points in the divisor, cf. [15].
3. Rational Intersection Number on V-Surfaces
RationalintersectionmultiplicitywasfirstintroducedbyMumfordfornormalsurfaces,see[10,
Pag. 17]. A general intersection theory is developed in [6]. In this section we give an alternative
description of Mumford’s definition restricted to V-surfaces. Moreover the use of Q-resolutions
allows us to give an alternative description of Mumford’s definition for normal surfaces which
does not involve a resolution. In particular self-intersection numbers of the exceptional divisors
of weighted blow-ups can be computed directly, see Theorem 4.3.
16 E.ARTAL,J.MARTÍN-MORALES,ANDJ.ORTIGAS-GALINDO
In the smooth case it is possible to define the intersection number W D for divisors W,D
·
provided W D is finite or W is compact; this definition can be extended to the singular case
∩
when W is a Weil divisor and D is a Cartier divisor. It is well known that V-manifolds are
Q-factorial, i.e., rational Cartier and Weil divisors coincide. Hence a Q-divisor refer to both
notions and the corresponding vector space is denoted by Q-Div(X).
Definition 3.1. Let X be a V-manifold of dimension 2 and consider D ,D Q-Div(X). The
1 2
∈
intersection number is defined as
1
D D := (k D k D ) Q,
1 2 1 1 2 2
· k k · ∈
1 2
where k ,k Z are chosen so that k D is Weil, k D is Cartier and either the divisor D is
1 2 1 1 2 2 1
∈
compact or D D is finite [6, Ch. 2].
1 2
∩
Analogously, it is defined the local intersection number at P D D , if the condition
1 2
D (cid:42)D is satisfied. ∈ ∩
1 2
For later use we make explicit some properties of this intersection multiplicity. Their proofs
are omitted since they are well known for the classical case (i.e., without tensorizing with Q),
cf.[6],andourgeneralizationisbasedonextendingtheclassicaldefinitiontorationalcoefficients.
Theorem 3.2. Let F : Y X be a proper morphism between two irreducible V-surfaces, and
D ,D Q-Div(X). →
1 2
∈
(1) The cardinal of F−1(P), P X being generic, is a finite constant. This number is denoted
∈
by deg(F).
(2) If D D is defined, then so is the number F∗(D ) F∗(D ). In such a case
1 2 1 2
· ·
F∗(D ) F∗(D )=deg(F)(D D ).
1 2 1 2
· ·
(3) If (D D ) is defined for some P X, then so is (F∗(D ) F∗(D )) , Q F−1(P), and
1 2 P 1 2 Q
· ∈ · ∀ ∈
(F∗(D ) F∗(D )) =deg(F)(D D ) .
Q∈F−1(P) 1 · 2 Q 1· 2 P
(cid:80) 4. Intersection Numbers and Weighted Blow-ups
Previouslyweightedblow-upswereintroducedasatoolforcomputingembeddedQ-resolutions.
To obtain information about the corresponding embedded singularity, an intersection theory on
V-manifolds has been developed. Here we calculate self-intersection numbers of exceptional
divisors of weighted blow-ups on analytic varieties with abelian quotient singularities, see The-
orem 4.3.
The first step in this computation is to explicitly write the exceptional divisor as a rational
Cartier divisor applying the procedure described in [1, §4.3].
Let X be a surface with abelian quotient singularities. Let π : X X be the weighted
→
blow-up at a point of type (d;a,b) with respect to ω =(p,q). In general, the exceptional divisor
E :=π−1(0)=P1(d;a,b) is a Weil divisor on X which does not corresp(cid:98)ond to a Cartier divisor.
∼ ω
Let us write E as an element in CaDiv(X) ZQ.
Asin2.3,assumeπ :=π :X(cid:92)(d;a,b⊗) (cid:98) X(d;a,b). Assumealsothatgcd(p,q)=1and
(d;a,b),ω (cid:98) ω →
(d;a,b) is normalized. Using the notation introduced in 2.3, the space X is covered by U U
1 2
∪
and the first chart is given by
(cid:98) (cid:98) (cid:98)
Q :=X pd;1,−q+βpb U ,
(2) 1 e e −→ 1
(cid:16) (xe,y)(cid:17) ((xp,xqy),[1:y] ) ,
(cid:55)→ (cid:98) ω (d;a,b)
where e:=gcd(d,pb qa). (cid:2) (cid:3) (cid:2) (cid:3)
−
INTERSECTION THEORY AND Q-RESOLUTIONS 17
In the first chart, E is the Weil divisor x = 0 Q . Note that the type representing the
1
space Q is in a normalized form and hence{the co}rr⊂esponding subgroup of GL(2,C) is small.
1
The divisor {x = 0} ⊂ Q1 is written as an element in CaDiv(Q1)⊗Z Q like ped{(Q1,xped)},
which is mapped to ped{(U1,xd)}∈CaDiv(U1)⊗ZQ under the isomorphism (2).
Analogously E in the second chart is e (U ,yd) . Finally one writes the exceptional divisor
(cid:98) qd{(cid:98) 2 }
of π as claimed,
e e (cid:98) e
(3) E = (U ,xd),(U ,1) + (U ,1),(U ,yd) = (U ,xdq),(U ,ydp) .
1 2 1 2 1 2
dp dq dpq
(cid:8) (cid:9) (cid:8) (cid:9) (cid:8) (cid:9)
We state some p(cid:98)relimina(cid:98)ry lemmas sep(cid:98)arately(cid:98)so that the proo(cid:98)f of the m(cid:98)ain result of this
section becomes simpler.
Lemma 4.1. Let X be an analytic surface with abelian quotient singularities and let π :X X
be a weighted blow-up at a point P X. Let C be a Q-divisor on X and E the except→ional
∈
divisor of π. Then, E π∗(C)=0. (cid:98)
·
Proof. The proof uses the same ideas as in the smooth case. After multiplying by an integer
we can assume that C is a Cartier divisor. In a neighborhood of P its associated line bundle is
trivial, and hence it is also the case for the associated line bundle of π∗(C) in a neighborhood
of E, and hence the result follows. (cid:3)
Lemma 4.2. Let X be a V-surface locally irreducible at P X, and a Q-divisor C . Consider
X
∈
a weighted blow-up π :X X at P. Denote by E the exceptional divisor of π , and C the
X X X X
→
strict transform of C .
X
Let Y be another V-su(cid:98)rface locally irreducible at Q Y and a proper morphism h : Y(cid:98) X
∈ →
such that h−1(P)=Q and the map π in the diagram
Y
H (cid:47)(cid:47)
Y X
πY(cid:98)(cid:15)(cid:15) # (cid:98)(cid:15)(cid:15)πX
(cid:47)(cid:47)
Y X
h
is a weighted blow-up at Q; the exceptional divisor of π is denoted by E . Let us suppose that
Y Y
there exist two rational numbers, e and ν such that
(a) H∗(E )=eE , (b) π∗(h∗(C ))=H∗(C )+νE .
X Y Y X X Y
Then the following equalities hold: (cid:98)
(1) π∗ (C )=C + νE , (2) E C = −eν E2, (3) E2 = e2 E2.
X X X e X X · X deg(h) Y X deg(h) Y
Proof. For (1) n(cid:98)ote the total transform π∗(cid:98)(C ) can always be written as C +mE for some
X X X X
m Q. Considering its pull-back under H∗ one obtains two expressions for the same Q-divisor
∈
on Y, (cid:98)
H∗(π∗ (C ))dia=gramπ∗(h∗(C ))(=b)H∗(C )+νE ,
(cid:98) X X Y X X Y
H∗(C +mE )=H∗(C )+mH∗(E )(=a)H∗(C )+meE .
X X X X (cid:98) X Y
It follows that m= ν.
e
(cid:98) (cid:98) (cid:98)
18 E.ARTAL,J.MARTÍN-MORALES,ANDJ.ORTIGAS-GALINDO
For(2)firstnotethatdeg(H)=deg(h). FromLemma4.1,onehasthatE π∗(h∗(C ))=0.
Y · Y X
On the other hand, H being proper, Theorem 3.2(2) can be applied thus obtaining
deg(h)(E C )=H∗(E ) H∗(C )(a)=-(b)eE π∗(h∗(C )) νE = eνE2.
X · X X · X Y · Y X − Y − Y
Analogously deg(h)E2 =H∗(E )2 =e2E2 and (3) follo(cid:2)ws. (cid:3) (cid:3)
(cid:98) X X (cid:98)Y
Now we are ready to present the main result of this section.
Theorem4.3. LetX beananalyticsurfacewithabelianquotientsingularitiesandletπ :X X
→
bethe(p,q)-weightedblow-upatapointP X oftype(d;a,b). Assumegcd(p,q)=1and(d;a,b)
∈
is normalized, i.e., gcd(d,a)=gcd(d,b)=1. Also write e=gcd(d,pb qa). (cid:98)
Consider two Q-divisors C and D on X. As usual, denote by E the−exceptional divisor of π,
and by C (resp. D) the strict transform of C (resp. D). Let ν and µ be the (p,q)-multiplicities
of C and D at P, i.e., x (resp. y) has (p,q)-multiplicity p (resp. q). Then there are the following
equalitie(cid:98)s: (cid:98)
(1) π∗(C)=C+ νE. (3) E2 = e2 .
e
eν −dpq
(2) E C = . νµ
· d(cid:98)pq (4) C D =C D .
· · − dpq
(cid:98) µ2
In addition, if D has compact support then D2 =D2(cid:98) (cid:98) .
− dpq
Proof. The item (4), and final conclusion, are a(cid:98)n easy consequence of (1)-(3) and the fact that
π∗(C) π∗(D)=C D.
· · (cid:92)
Fortherestoftheproof,oneassumesthatπ :=π :X(d;a,b) X(d;a,b)istheweighted
X ω −→
blow-upattheoriginofX(d;a,b)withrespecttoω =(p,q). NowtheideaistoapplyLemma4.2
to the commutative diagram
Y :=C2 H (cid:47)(cid:47)X(cid:92)(d;a,b) =:X
ω
(cid:98) πY(cid:98) # πX (cid:98)
(cid:15)(cid:15) (cid:15)(cid:15)
Y :=C2 (cid:47)(cid:47)X(d;a,b)=:X
h
where H and h are the morphisms defined by
((x,y),[u:v]) H [((xp,yq),[up :vq]) ] ;
ω (d;a,b)
(cid:55)−→
(x,y) h [(xp,yq)] ,
(d;a,b)
(cid:55)−→
andπ istheclassicalblowing-upattheorigin. InthissituationE2 = 1. Theclaimisreduced
Y Y −
to the calculation of deg(h) and the verification of the conditions (a)-(b) of Lemma 4.2.
The degree is deg(h) = pq deg pr:C2 X(d;a,b) = dpq. For (a), first recall the decom-
· →
positions
(cid:2) (cid:3)
(4) X(cid:92)(d;a,b) =U U , C2 =U U .
ω 1∪ 2 1∪ 2
One has already written in (3) the exceptional divisor of π as
(cid:98) (cid:98) (cid:98) X
e
E = (U ,xdq),(U ,ydp) .
X 1 2
dpq
(cid:110) (cid:111)
(cid:98) (cid:98)
INTERSECTION THEORY AND Q-RESOLUTIONS 19
Hence its pull-back under H, computed by pulling back the local equations, is
e
H∗(E )= (U ,xdpq),(U ,ydpq) =e (U ,x),(U ,y) =eE .
X 1 2 1 2 Y
dpq
(cid:110) (cid:111) (cid:110) (cid:111)
Finallyfor(b)oneuseslocalequationstocheckπ∗(h∗(C))=H∗(C)+νE . Supposethedivisor
Y Y
C is locally given by a meromorphic function f(x,y) defined on a neighborhood of the origin
of X(d;a,b); note that ν = ord (f). The charts associated wit(cid:98)h the decompositions (4) are
(p,q)
described in detail in 2.3. As a summary we recall here the first chart of each blowing-up:
π : Q :=X p −1 q U , (x,y) ((xp,xqy),[1:y] ) .
X 1 pd a pb−qa −→ 1 (cid:55)→ ω
πY : (cid:16) (cid:12)(cid:12) C(cid:17)2 U(cid:98)1, (cid:2) (x,y(cid:3)) (cid:2)((x,xy),[1:y]). (cid:3)
(cid:12) −→ (cid:55)→
Note that H respects the decompositions and takes the form (x,y) [(x,yq)] in the first chart.
(cid:55)→
Then one has the following local equations for the divisors involved:
Divisor Equation Ambient space
h∗(C) f(xp,yq)=0 C2
π∗(h∗(C)) f(xp,xqyq)=0 C2 =U
Y ∼ 1
f(xp,xqy)
C =0 Q =U
xν 1 ∼ 1
f(xp,xqyq)
H∗(cid:98)(C) =0 C2 =U(cid:98)
xν ∼ 1
E x=0 C2 =U
Y(cid:98) ∼ 1
From these local equations (b) is satisfied and now the proof is complete. (cid:3)
Remark 4.4. Inordertocomputemultiplicitieswhenlookingatmulticharts (forquotientspaces)
wemustbecarefulwiththeexpressionsincoordinatesincasethespaceisrepresentedbyanon-
normalizedtype. Forinstance,ifadivisorislocallygivenbythefunctionxmd :X p −1q C,
d 1 0 →
its multiplicity is m.
(cid:0) (cid:12) (cid:1)
(cid:12)
For a sequence of weighted blow-ups we can adapt Mumford’s approach [10]. Let us fix
X :=X(d;a,b) and let us consider π :Xˆ X a sequence of weighted blow-ups. Let ,...,
1 r
→ E E
bethesetofexceptionalcomponentsinXˆ andletA:=( ) betheintersectionmatrix
Ei·Ej 1≤i,j≤r
in Xˆ, which is a negative definite matrix with rational coefficients. We restrict X to a small
neighborhood of the origin. An Xˆ-curvette γ of is a Weil divisor obtained by considering a
i i
E
disk transversal to a point of and δ =π(γ ) is called an X-curvette of ; the index
Ei\ j(cid:54)=iEj i i Ei
d(γ ):=d(δ ) is the order of the cyclic group associated with γ . We say that (γ ,γ(cid:48)) form
i i (cid:83) i∩Ei i j
a pair of Xˆ-curvettes for ( , ) if they are disjoint curvettes for each divisor; in that case their
i j
E E
images in X form a pair (δ ,δ(cid:48)) X-curvettes.
i j
Theorem 4.5. Let B := A−1 =(b ) . Let (δ ,δ(cid:48)) be a pair of X-curvettes for ( , ).
− ij 1≤i,j≤r i j Ei Ej
Then, δ δ(cid:48) = bij .
i· j d(δi)d(δj(cid:48))
Proof. Let γ(cid:48) be a generic Xˆ-curvette. Since γ(cid:48) and d(γ )γ are equivalent Weil divisors, we can
i i i i
assume that d(γ )=1. We have π∗(δ )=γ + n c . Note that γ =δ (δ being the
i i i j=1 ijEj i·Ej ij ij
Kronecker delta).
(cid:80)
20 E.ARTAL,J.MARTÍN-MORALES,ANDJ.ORTIGAS-GALINDO
For a generic γ(cid:48) we have δ(cid:48) δ =π∗(δ(cid:48)) π∗(δ )=γ(cid:48) π∗(δ )=c . Since
j j · i j · i j · i ij
n n
δ =γ = π∗(δ ) c = (δ δ(cid:48))( ),
ik i·Ek i − ijEj ·Ek − i· j Ej ·Ek
j=1 j=1
(cid:0) (cid:88) (cid:1) (cid:88)
we deduce the result. (cid:3)
Example 4.6. Assume gcd(p,q) = gcd(r,s) = 1 and p < r. Let f = (xp +yq)(xr +ys) and
q s
considerC = xp+yq =0 andC = xr+ys =0 . Afterasequenceoftwoweightedblow-ups
1 2
one obtains Fig{ure 1 repre}senting an e{mbedded Q}-resolution of f = 0 C2. We start with
{ } ⊂
a (q,p)-blow-up over a smooth point; the exceptional divisor ˜ has self-intersection −1. We
E1 pq
continue with an (s,qr ps)-blow-up over a point of type (q; 1,p). We denote by the strict
1
− − E
transform of ˜ and by the exceptional divisor.
1 2
E E
(s; 1,r) rq ps s q
− Q=X − −
rq ps r p
(cid:18) − − (cid:19)
C C
2 1
p(q+s)
1
E
s(p+r) Q (p;q, 1)
2
E −
Figure 1. Embedded Q-resolution of (xp+yq)(xr+ys)=0 C2.
{ }⊂
ThepointQisalsooftype(rq ps;ar+bs, 1)whereap+bq =1. Infact,itisinnormalized
− −
form, since gcd(rq ps,ar+bs) = 1. Since ˜ has multiplicity s the self-intersection of is
1 1
−1 s = − r . The self-intersectiEon of is q . The intersection matrEix A
pq − q(rq−ps) −p(rq−ps) E2 −s(rq−ps)
and its opposite inverse B are
1 r 1 pq ps
A= −p , B = .
rq ps 1 q ps sr
− (cid:18) −s(cid:19) (cid:18) (cid:19)
Example 4.7. Let us consider the following divisors on C2,
C = ((x3 y2)2 x4y3)=0 , C = x3 y2 =0 ,
1 2
{ − − } { − }
C = x3+y2 =0 , C = x=0 , C = y =0 .
3 4 5
{ } { } { }
Weshallseethatthelocalintersectionnumbers(C C ) ,i,j 1,...,5 ,i=j,areencoded
i j 0
· ∈{ } (cid:54)
in the intersection matrix associated with any embedded Q-resolution of C = 5 C .
i=1 i
Let π : C2 C2 be the (2,3)-weighted blow-up at the origin. The new space has two
1 (2,3) → (cid:83)
cyclic quotient singular points of type (2;1,1) and (3;1,1) located at the exceptional divisor .
1
E
The local equation of the total transform in the first chart is given by the function
x29 ((1 y2)2 x5y3) (1 y2) (1+y2) y : X(2;1,1) C,
− − − −→
where x = 0 is the equation of the exceptional divisor and the other factors correspond in the
same order to the strict transform of C , C , C , C (denoted again by the same symbol). To
1 2 3 5
study the strict transform of C one needs the second chart, the details are left to the reader.
4
Hence has multiplicity 29 and self-intersection number 1; the divisor intersects transver-
E1 −6
sally C , C and C at three different points, while it intersects C and C at the same smooth
3 4 5 1 2
point P, different from the other three. The local equation of the divisor C C at this
1 2 1
E ∪ ∪
point P is x29y(x5 y2)=0, see Figure 2 below.
−