Adjoint Computation for Hypersurfaces Using Formal Desingularizations

0.2 MB·English

by  T. Beck

ADJOINT COMPUTATION FOR HYPERSURFACES USING FORMAL
DESINGULARIZATIONS
TOBIASBECKANDJOSEFSCHICHO
8
0
Abstract. Weshow how touse formaldesingularizations (defined earlierbythe firstauthor)
0
inordertocompute theglobalsections (alsocalledadjoints)oftwistedpluricanonicalsheaves.
2
Thesesectionsdefinemapsthatplayanimportantroleinthebirationalclassificationofschemes.

n
a
J

5
1 Contents

] 1. Introduction 1
G 2. Definition of Formal Desingularizations 2
A 3. Adjoint Differential Forms 3
. 4. Computing Adjoints 5
h
5. The Algorithm 7
t
a 6. Example 8
m
References 10
[

1
v
6 1. Introduction
8
In [1] we have introduced formal desingularizations of schemes and shown how to compute
2
2 them efficiently for surfaces X ⊂ P3. There we have already mentioned as an application the
. computation of invertible sheaves on Y for a certain desingularization π :Y →X. In this report
1
wewanttomakethispreciseusingthepowersofthecanonicalsheafonY. Morepreciselywewant
0
8 to determine the coherent sheaf π∗(ωY⊗m) which can be given by its associated graded module
0
v: Γ∗(π∗(ωY⊗m)):= Γ(X,π∗(ωY⊗m)⊗OX OX(n)),
n∈Z
i M
X see[5,p.118]. Thesheavesπ (ω⊗m)⊗ O (n)arecalledtwisted pluricanonical. OurMainThe-
∗ Y OX X
r orem4.6 givesacriterionforcomputing its homogeneouscomponents. The theoryis independent
a
of the dimension and given for hypersurfaces of some projective space.
The importance of the components of Γ (π (ω⊗m)) stems from the birational classification of
∗ ∗ Y
schemes. Forexample,theyprovideaneffectivewaytocheckCastelnuovo’sCriterionandperform
the Enriques-Manin Reduction of rational surfaces [10].
This report is structured as follows: In Section 2 we start by recalling the definition of formal
desingularizationsfortheconvenienceofthereader. InSection3wedefinethesheafofm-adjoints
on X by a property involving formal prime divisors and show that it is isomorphic to π (ω⊗m).
∗ Y
In particular it is independent of a special Y. In Section 4 we find a super-sheaf of π (ω⊗m) and
∗ Y
show that the defining property of the latter is easily checked using a formal desingularization of
X. This immediately yields Algorithm 1 given in Section 5. We close with an example.
Before we proceedwe recalland fix some notions. Let E be a field of characteristiczero and X
and Y integral E-schemes. All (rational) maps are relative over SpecE. By E(X) and E(Y) we
denote the respective function fields. A rational map π : Y 99K X is given by a tuple (V,π) s.t.
Date:February2,2008.
The authors were supported by the FWF (Austrian Science Fund) in the frame of the research project SFB
F1303(partoftheSpecialResearchProgram“NumericalandSymbolicScientificComputing”).
1
2 TOBIASBECKANDJOSEFSCHICHO
V ⊆Y is open and π :V →X is a regular morphism. Note that we do not restrict to schemes of
finite type here. In particular all regular morphisms are rational maps. Two tuples (V ,π ) and
1 1
(V ,π ) are equivalent, or define the same rational map, if π | =π | .
2 2 1 V1∩V2 2 V1∩V2
Assume that two maps send the generic point of Y to p ∈ X (its image is always defined
for rational maps). Then (V ,π ) and (V ,π ) are equivalent iff the induced inclusions of fields
1 1 2 2
O /m ֒→E(Y) are the same (where m ⊂O is the maximal ideal). In particular if π is
X,p X,p X,p X,p
dense, i.e., p is the generic point of X, we get an inclusion E(X)֒→E(Y) determining π.
Note, however, that not all such field inclusions yield rational maps under our assumption
since we have not yet restricted to schemes of finite type over E. E.g., let X := SpecE[x],
Y := SpecE[x] and π : Y → X be the morphism induced by localization. Then π induces an
hxi
isomorphism of function fields E(X) ∼= E(Y). Nevertheless π has no rational inverse. A rational
map with inverse is called birational (or also a birational transformation).
Further it is easy to see that dense rational maps may be composed. A rational map has a
domain of definition, which is the maximal open set on which it can be defined (equivalently, the
union of all such open sets).
2. Definition of Formal Desingularizations
From now on X and Y will denote separated, integral schemes of finite type over E and they
will have the same dimension l. Let (A,m) be a valuation ring of E(X) over E (where m is the
maximal ideal). If A is discrete of rank 1 and the transcendence degree of A/m over E is n−1
then it is called a divisorial valuation ring of E(X) over E or a prime divisor of E(X) (see, e.g.,
[11, Def. 2.6]). It is an essentially finite, regular, local E-algebra of Krull-dimension 1 (i.e., the
localization of a finitely generated E-algebra at a prime ideal, see [12, Thm. VI.14.31]).
Let (A,m) be a divisorial valuation ring of E(X) over E. By [5, Lem. II.4.4.] the inclusion
A⊂E(X)definesauniquemorphismSpecQ(A)→X andthereforearationalmapSpecA99KX
sendinggenericpointtogenericpoint. Composingthiswiththemorphismobtainedbythem-adic
completion A→A we get a rational map SpecA99KX in a natural way.
Definition 2.1 (FormalPrimeDivisor). Let (A,m) be a divisorial valuation ring of E(X) over E.
b b
Assume that the rational map SpecA 99K X (as above) is actually a morphism ϕ : SpecA → X
(i.e., defined also at the closed point). Then ϕ is a representative for a class of schemes up to
X-isomorphism. This class (and, bybabuse of notation, any representative) will be called abformal
prime divisor on X.
HencewemaycomposearepresentativeϕwithanisomorphismSpecB →SpecAtogetanother
representative for the same formal prime divisor. By the Cohen Structure Theorem (see, e.g., [3,
Thm. 7.7] with I = 0) we know that A ∼= FϕJtK with Fϕ := A/m ∼= A/mA. Thberefore we will
sometimes assume that ϕ is of the form SpecF JtK→X.
ϕ
Formalprime divisorsprovideanalgobrithmicwayfordealing withcerbtainbvaluations; Aformal
prime divisors yields aninclusionof function fields E(X)֒→F ((t)). Vice versa,by whatwas said
ϕ
above, ϕ is determined by this inclusion. So it is this piece of information one has to compute
(see Remark 2.7 below). Composing this inclusion with the order function ord :F ((t)) →Z we
t ϕ
get the corresponding divisorial valuation (see Definition 4.4 below).
We want to single out a special class of formal prime divisors.
Definition 2.2 (Realized Formal Prime Divisors). Let p ∈ X be a regular point of codimension
1. The formal prime divisor
SpecO[ →X
X,p
(given by composing the canonic morphism SpecO → X with the morphism induced by the
X,p
completion O →O[) is called realized.
X,p X,p
IfX is normal thenallgenericpoints ofclosedsubsetsofcodimension1 arenecessarilyregular
[5, Thm II.8.22A]. Therefore there is a one-one correspondence of realized formal prime divisors
and prime Weil divisors. Another important fact is that we can match the formal prime divisors
of birationally equivalent schemes under certain conditions.
ADJOINT COMPUTATION FOR HYPERSURFACES USING FORMAL DESINGULARIZATIONS 3
Lemma 2.3 (Pullback along Proper Morphisms). Let π : Y → X be a proper, birational mor-
phism. A formal prime divisor ϕ : SpecF JtK → X lifts to a unique formal prime divisor
ϕ
π∗ϕ : SpecF JtK → Y s.t. π ◦ (π∗ϕ) = ϕ. Vice versa, a formal prime divisor on Y extends
ϕ
to a unique formal prime divisor on X, hence π∗ is a bijection.
Proof. See [1, Cor. 2.4]. (cid:3)
We will apply the operator π∗ also to sets of formal prime divisors.
Definition 2.4 (Center and Support). Let ϕ : SpecF JtK → X be a formal prime divisor. We
ϕ
define its center, in symbols center(ϕ) ∈ X, to be the image of the closed point. Further the
support of a finite set of formal prime divisors S is defined as supp(S) := {center(ϕ)|ϕ∈S},
i.e., the closure of the set of all centers.
Now we are in the situation to define formal desingularizations.
Definition 2.5 (Formal Description of a Desingularization). Let π : Y → X be a desingulariza-
tion, i.e., π is proper, birational and Y is regular. Let S be a finite set of formal prime divisors
on X. We say that S is a formal description of π iff
(1) all divisors in π∗S are realized,
(2) π−1(supp(S))=supp(π∗S) and
(3) the restricted morphism Y \supp(π∗S)→X\supp(S) is an isomorphism.
ThesetS itselfconsistsofformalprimedivisorsonX andmakesnoreferencetothe morphism
π. By another definition we can avoid mentioning any explicit π.
Definition 2.6 (Formal Desingularization). Let S be a finite set of formal prime divisors on X.
Then S is called a formal desingularization of X iff there exists some desingularization π s.t. S
is a formal description of it.
In [1, Thm. 2.9] it is shown that such π is unique up to X-isomorphism if X is a surface. In
this case we also have an efficient algorithm to compute a set S. Informally speaking S makes it
possible to work with invertible sheaves on Y, although Y is not constructed explicitely.
Remark 2.7 (Formal Desingularizationsin Higher Dimensions). This paper deals with projective
hypersurfaces of arbitrary dimension l. Also in this case, formal desingularizations exist but are
not so easy to compute. We want to indicate how a formal desingularization could be obtained by
an ad hoc method (modulo a means to represent algebraic power series).
First compute a desingularization π : Y → X, for example, by Villamayor’s algorithm [4].
Let Z ⊂X be the singular locus of X. The algorithm will produce π s.t. π−1(Z)⊂Y is a normal
crossing divisor and π restricts to an isomorphism on Y \π−1(Z). Let {p ,...,p } ⊂ Y be the
1 r
finitely many generic points of the irreducible components of π−1(Z).
Next we have to compute isomorphisms O[ → F JtK where F := O /m . Therefore
Y,pi i i Y,pi Y,pi
let Ui ∼= SpecE[xi,1,...,xi,mi]/hfi,1,...,fi,nii be an affine neighborhood of pi. Constructing the
isomorphism involves finding certain “minimal” algebraic power series X ,...,X ∈ F JtK
i,1 i,mi i
(compare [1, Cor. A.2]) that simultaneously solve f ,...,f , essentially, computing a Taylor
i,1 i,ni
expansion. ThesepowerseriestogetherwithπandtheinclusionsU ֒→Y canbeusedtorepresenta
i
formal prime divisor ϕ via, for example, the inducedembedding of function fields E(X)֒→F ((t)).
i i
Set S :={ϕ ,...,ϕ }.
1 r
This approach of course is not practical. It suffers from the huge computational overhead that
the general resolution machinery involves. Also we are not very flexible with regard to the repre-
sentation of the blown up schemes, thus annihilating the benefits of formal descriptions.
3. Adjoint Differential Forms
We write
Ωm :=(Ω∧l )⊗m
X,rat E(X)|E
4 TOBIASBECKANDJOSEFSCHICHO
for the m-fold tensor power of the rational differential l-forms (which is an 1-dimensional E(X)-
vectorspaceby[3,Thm.16.14]and[8,Prop.XIX.1.1&Cor.XVI.2.4]andcanaswellbeconsidered
a constant sheaf of O -modules).
X
Let ϕ:A→X be a formal prime divisor. We define
Ωm :=A⊗ (Ω∧l )⊗m (canonically included in Ωm :=Q(A)⊗ (Ω∧l )⊗m),
Ab|bE A A|E Q(Ab)|E A A|E
which is the m-fold tensor power of the universally finite module of l-differentials of A (see [7,
e b e b
Cor. 12.5]). Note that we have a derivation d:Q(A)→Q(A)⊗ Ω .
A A|E
This module is independent of the choice of a representative of ϕ up to X-isomorpbhism be-
cause Ωm can be defined by universal propertiesb. So webcan again substitute A by F JtK. If
Ab|E ϕ
s ,...,s ∈F isatranscendencebasisoverEthen{ds ,...,ds ,dt}isafreebasisof
ϕ,1 ϕ,l−1 ϕ ϕ,1 ϕ,l−1
Ω e by[3,Thm.16.14]and[9,Thm.25.1]. ThereforealsoΩm isfreeofrabnk1generated
Fϕ[t]hti|E FϕJtK|E
by (ds ∧...∧ds ∧dt)⊗m. (Note that the module of K¨ahler differentials Ω would
ϕ,1 ϕ,l−1 FϕJtK|E
not be finitely generated.) e
We have seen above that a formal prime divisor ϕ : SpecA → X induces an embedding
ϕ# : E(X) → F ((t)) of function fields. This again induces embeddings ϕ# : Ωm → Ωm
ϕ X,rat Fϕ((t))|E
in the obvious way (i.e., df 7→dϕ#(f)). b
e
Definition 3.1 (Regularity of Forms at Formal Prime Divisors). Let ϕ : SpecF JtK → X be a
ϕ
formal prime divisor. We say that η ∈Ωm is regular at ϕ iff ϕ#(η)∈Ωm .
X,rat FϕJtK|E
This manner of speaking is justified by Lemma 3.5 below.
e
Definition 3.2 (Sheaf of Adjoint Forms). The map
U 7→{η ∈Ωm |η is regular at all formal prime divisors on X centered in U}
X,rat
for all open subsets U ⊆X defines a subsheaf which we call the sheaf of m-adjoint forms (or just
m-adjoints), in symbols Ωm .
X,adj
By what we have worked out so far we immediately find a nice property of adjoint forms.
Corollary 3.3 (Covariance of Adjoint Forms). The sheaves of adjoints are covariants under
proper, birational morphisms: If π : Y → X is a proper, birational morphism then π (Ωm ) =
∗ Y,adj
Ωm as subsheaves of Ωm .
X,adj X,rat
Proof. Since π is birational we get a vector space isomorphism π# : Ωm → Ωm . With this
X,rat Y,rat
identificationπ (Ωm )becomesasubsheafofΩm . Therestfollowsfromtheabovedefinitions
∗ Y,adj X,rat
and Lemma 2.3. (cid:3)
By Ωm := ( O dO ) ⊂ Ωm we denote the subsheaf of regular forms, i.e., all
X,reg X X X,rat
forms locally expressible by sections of O . More precisely, if Ω is the usual sheaf of K¨ahler
X X|E
N V
differentials then we mean its image under the naturalmapι:(Ω∧l )⊗m →Ωm . Note that we
X|E X,rat
do not have (Ω∧l )⊗m ∼= Ωm in general; At singular points the K¨ahler differentials need not
X|E X,reg
be torsion free (and neither their exterior and tensor products) whereas Ωm ⊂ Ωm always
X,reg X,rat
is. At a regular point p, on the contrary, ((Ω∧l )⊗m) is free of rank 1 (see [5, Thm. II.8.15]).
X|E p
Therefore ι is locally an inclusion at p. In the next three lemmas we wantto explore in detail the
relation between the concepts of adjointness and regularity (at points or formal prime divisors)
for forms in Ωm .
X,rat
Lemma 3.4 (Adjoint Forms and Regular Forms). Let p ∈ X be a point. Then (Ωm ) ⊆
X,reg p
(Ωm ) as subsheaves of Ωm .
X,adj p X,rat
Proof. Assumeη ∈(Ωm ) orequivalentlyη ∈ ( Γ(U,O )dΓ(U,O ))forsomeopenneigh-
X,reg p X X
borhood U of p. We have η ∈ Γ(U,Ωm ) iff ϕ#(η) ∈ Ωm = ( F JtKdF JtK) for
X,adj N V FϕJtK|E ϕ ϕ
all formal prime divisors ϕ : SpecF JtK → X centered in U. But for such ϕ we must have
ϕ N V
ϕ#(Γ(U,O ))⊆F JtK and hence this condition is trivially fulefilled. A fortiori η ∈(Ωm ) . (cid:3)
X ϕ X,adj p
ADJOINT COMPUTATION FOR HYPERSURFACES USING FORMAL DESINGULARIZATIONS 5
Lemma 3.5 (Regularity and Realized Formal Prime Divisors). Let ϕ be a realized formal prime
divisor on X, p:=center(ϕ) and η ∈Ωm . Then η ∈(Ωm ) iff η is regular at ϕ.
X,rat X,reg p
Proof. By Lemma 3.4 it remains to show that regularity at ϕ implies η ∈ (Ωm ) . Since ϕ is
X,reg p
realized it is of the form SpecO[ → X. Let γ ∈ (Ω∧l )⊗m be a generator. Write η = a/bγ
X,p OX,p|E
with a,b ∈ O . Then ϕ#(η) ∈ Ωm implies b|a in O[, in other words a ∈ bO[. But
X,p O\X,p|E X,p X,p
bO[ ∩O = bO by [9, Thm. 7.5(ii)] (because completion is faithfully flat [9, Thm. 7.2 &
X,p X,p X,p e
Thm. 8.8]). Therefore b|a in O and η ∈(Ωm ) . (cid:3)
X,p X,reg p
Lemma 3.6 (AdjointFormsatRegularPoints). Let p∈X be a regular point. Then (Ωm ) =
X,reg p
(Ωm ) as subsheaves of Ωm .
X,adj p X,rat
Proof. By Lemma 3.4 it remains to show (Ωm ) ⊆ (Ωm ) . Assume indirectly that η ∈
X,adj p X,reg p
(Ωm ) but η 6∈ (Ωm ) . Regularity is an open property and Ωm is free of rank 1 in a
X,adj p X,reg p X,reg
neighborhood of p. Since p is regular (in particular normal) we must have η 6∈(Ωm ) for some
X,reg q
point q s.t. q is regular, of codimension 1 and p ∈ q (see [3, Cor. 11.4]). Consider the realized
formal prime divisor ϕ : SpecO[ → X. Lemma 3.5 above implies that η is not regular at ϕ.
X,q
But q =center(ϕ) is contained in any open neighborhood of p contradicting η ∈(Ωm ) . (cid:3)
X,adj p
Now assume that Y is regular. In this situation one has (Ω∧l )⊗m ∼= Ωm . In terms of the
Y|E Y,reg
canonical sheaf this means ω⊗m ∼= Ωm = Ωm by the above lemmas. Finally using Corol-
Y Y,reg Y,adj
lary3.3wegetanalternativecharacterizationofthesheafofm-adjoints,infact,theusualdefinition
when working in a category of desingularizable schemes (e.g., for our case of characteristic zero).
Corollary 3.7 (Alternative CharacterizationofAdjoints). If π :Y →X is any desingularization
then Ωm ∼=π (ω⊗m).
X,adj ∗ Y
4. Computing Adjoints
Now let X ⊂ Pl+1 be a projective hypersurface with defining homogeneous equation F ∈
E
E[x ,...,x ]ofdegree d(not equalto a coordinatehyperplane). For 0≤i≤l+1 we define the
0 l+1
open sets U ⊂X obtained by intersection with the standard open covering sets x 6=0 of Pl+1.
i i E
Definition 4.1 (DualizingSheaf). By ω0 ⊆Ω1 we denote the dualizingsheaf. It is invertible
X X,rat
and generated on U by the form
i
−1
∂F/∂x x x x x
j 0 i j l+1
γ :=σ d ∧···∧d ∧···∧d ∧···∧d
i i,j xdi−1 ! xi xi xi xi
d d
for any choice of j 6=i where
(−1)i+j if j <i,
σ :=
i,j ((−1)i+j+1 if j >i.
(The hats here mean that the corresponding terms in the exterior product are to be excluded.)
Using the rules of calculus and the fact that
F ∂F/∂x x
k k
0=d = d
xdi 0≤k≤l+1,k6=i xdi−1 xi
X
holds in Ω1 one provesthat the definition is indeed independent of the choice of j. Because of
X,rat
local freeness we also have (ω0 )⊗m ⊆Ωm (meaning the natural map is an embedding).
X X,rat
Lemma 4.2 (Properties of the Dualizing Sheaf). For the dualizing sheaf we have:
• ωX0 ∼=OX(d−l−2)
• (ω0 ) =(Ω1 ) =(Ω1 ) at all regular points p
X p X,reg p X,adj p
• Ωm ⊆(ω0 )⊗m as subsheaves of Ωm
X,adj X X,rat
6 TOBIASBECKANDJOSEFSCHICHO
Proof. To prove the first statement one just shows, using the rules of calculus, that γ =
i1
(x /x )d−l−2γ . The same relation is fulfilled by the local generators xd−l−2 and xd−l−2.
i1 i2 i2 i1 i2
We check the second statement for points p∈U . The forms
i
x x x x
0 i j l+1
d ∧···∧d ∧···∧d ∧···∧d
x x x x
i i i i
with j 6= i, which are all Γ(U ,O )-muldtiples of γ ,dclearly generate Γ(U ,Ω1 ). Therefore we
i X i i X,reg
have (Ω1 ) ⊆ (ω0 ) at all points. Assuming moreover that p is a regular point, there must
X,reg p X p
be j 6=i s.t. (∂F/∂x )/xd−1 6∈m andtherefore (∂F/∂x )/xd−1 is invertible in O . Choosing
j i X,p j i X,p
this j in Definition 4.1 one immediately sees that γ is regular at p.
i
For the last statement we consider a generic projection X →Z to a hyperplane Z ⊆Pl+1. We
E
may assume that Z is given by x =0 and that F is monic in x . In this situation one can define
0 0
a trace σ : Ωm → Ωm obtained from the trace of the field extension E(Z) ⊆ E(X). By
X|Z X,rat Z,rat
[6, Satz 2.14] and Lemma 3.6 we know that (ω0 )⊗m = σ−1 (Ωm )= σ−1 (Ωm ). It remains
X X|Z Z,reg X|Z Z,adj
to show that if α∈Ωm then σ (α)∈Ωm .
X,adj X|Z Z,adj
Let R ⊂ E(Z) be a divisorial valuation ring. Further let S ⊂ E(X) (for i in a finite index
i
set) be the extensions,i.e., divisorialvaluationrings dominating R. Using Definition 3.2(and the
fact that completion is faithfully flat) we have to show that α ∈ ( S dS ) for all i implies
i i
σ (α)∈ ( RdR). By [2, Prop.VI.8.6.6]the S are localizations ofthe integralclosureof R
X|Z i
N V
in E(X). Then the statement follows from [6, Satz 2.15]. (cid:3)
N V
We want to see that, under certain additional assumptions, checking for adjointness involves
only finitely many formal prime divisors.
Lemma 4.3 (Adjointness by Formal Desingularizations). Let S be a formal desingularization of
X and U ⊂X an open subset. For η ∈Ωm the following are equivalent:
X,rat
• η ∈Γ(U,Ωm )
X,adj
• η ∈Γ(U \supp(S),Ωm ) and η is regular at all ϕ∈S with center(ϕ)∈U
X,adj
Proof. The firstimplicationistrivial,soassumethatthe secondconditionis true. Letπ :Y →X
be a desingularization that is described by S and set V :=π−1(U).
ByCorollary3.3andLemma3.6wehaveπ#(Γ(U,Ωm ))=Γ(V,Ωm )=Γ(V,Ωm ). Since
X,adj Y,adj Y,reg
π inducesanisomorphismV \supp(π∗(S))∼=U\supp(S)itremainstocheckthatπ#(η)isregular
in supp(π∗(S))∩V. Since Y is regular the locus of non-regularityof π#(η) has pure codimension
1. Hence, by Lemma 3.5, it is sufficient to check regularity of π#(η) at the formal prime divisors
in π∗(S) with center in V. Equivalently, working on X, we have to check regularity of η at the
corresponding formal prime divisors in S. (cid:3)
Inthe followingdefinitionweassumethatwe havechosenfreegeneratorsω ∈Ωm (e.g.,
ϕ,m FϕJtK|E
ω =(ds ∧···∧ds ∧dt)⊗m) for each formal prime divisor ϕ:SpecF JtK→X.
ϕ,m ϕ,1 ϕ,l−1 ϕ
e
Definition 4.4 (Valuations Associated to Formal Prime Divisors). Let ϕ:SpecF JtK →X be a
ϕ
formal prime divisor. Define κ := ord ◦ϕ# : E(X) → Z, which is a divisorial valuation. We
ϕ t
“extend the valuation” to Ωm as follows: If η ∈ Ωm and ϕ#(η) = fω , then κ (η) :=
X,rat X,rat ϕ,m ϕ
κ (f). Finally we can also define “valuations” κ : Γ(X,O (k)) → Z for k ∈ Z by setting
ϕ ϕ X
κ (f):=κ (f/xk) for any index i s.t. center(ϕ)∈U .
ϕ ϕ i i
The map fromΩm is obviously well-defined because two free generatorscan differ only by a
X,rat
unit in F JtK which has order 0. We should also make sure that the definition for the map from
ϕ
Γ(X,O (k))doesnotdependonthechoiceoftheindexi. Assumethatj 6=iisanotherindexwith
X
center(ϕ)∈U . Thenf/xk =(x /x )kf/xk andhenceκ (f/xk)=kκ (x /x )+κ (f/xk). Since
j i j i j ϕ i ϕ j i ϕ j
center(ϕ) ∈ U ∩U we have that x /x ∈ O is invertible and so is ϕ#(x /x ) ∈ F JtK.
i j j i X,center(ϕ) j i ϕ
But then again κ (x /x )=ord (ϕ#(x /x ))=0.
ϕ j i t j i
Definition 4.5 (Adjoint Order). Let ϕ:SpecF JtK→X be a formal prime divisor and 0≤i≤
ϕ
l+1 an index s.t. center(ϕ)∈U . We define the adjoint order at ϕ as α :=−κ (γ ).
i ϕ ϕ i
ADJOINT COMPUTATION FOR HYPERSURFACES USING FORMAL DESINGULARIZATIONS 7
This definition is again independent of the index i by an analogous reasoning as above.
Theorem 4.6 (Global Sections of Twisted Pluricanonical Sheaves). Let S be a formal desingu-
larization of X. Then
Γ(X,OX(n)⊗OX ΩmX,adj)∼={f ∈Γ(X,OX(n+m(d−l−2)))|κϕ(f)≥mαϕ for all ϕ∈S}.
Proof. By Lemma 4.2 we can view OX(n)⊗OX ΩmX,adj as a subsheaf of OX(n)⊗OX (ωX0 )⊗m ∼=
O (n+m(d−l−2)). Let f ∈Γ(X,O (n+m(d−l−2))) be a globalsection andη its preimage
X X
in Γ(X,O (n)⊗ (ω0 )⊗m) via this isomorphism. Projecting η to the sections over U we find
X OX X i
f
η 7→xn⊗ γ⊗m.
i xn+m(d−l−2) i
i
We have to check whether f/xn+m(d−l−2)γ⊗m ∈Γ(U ,Ωm ) for all i. Again by Lemma 4.2 this
i i i X,adj
form is adjoint at all regular points. Applying now Lemma 4.3 it is equivalent to check that
κ (f/xn+m(d−l−2)γ⊗m)≥0
ϕ i i
for all i and any formal prime divisor ϕ∈S with center(ϕ)∈U . This again is equivalent to
i
κ (f/xn+m(d−l−2))≥−mκ (γ )=mα
ϕ i ϕ i ϕ
for all ϕ∈S. (cid:3)
5. The Algorithm
Concerningthe actualcomputationweclosewithafew remarksandgiveanexplicitalgorithm.
First since a hypersurface X is in particular a complete intersection we know that
Γ(X,OX(n+m(d−l−2)))∼=(E[x0,...,xl+1]/hFi)n+m(d−l−2)
(see, for example, [5, Exer. III.5.5.(a)]). If 4 is some well-ordering on exponents compatible with
the group structure and µ is the leading exponent of F w.r.t. 4 then we can write
0
(E[x ,...,x ]/hFi) =hxµ ||µ|=n+m(d−l−2) and µ4µ i .
0 l+1 n+m(d−l−2) 0 E
Secondwewanttocommentonthecomputationofadjointorders(seeDefinition4.5). Therefore
we have to determine κ (γ ). More generally let η ∈Ω1 be arbitrary. Further let u ,...,u ∈
ϕ i X,rat 1 l
E(X) be a transcendence basis over E. As a generator of Ω1 we choose as before ω =
FϕJtK ϕ,1
ds ∧···∧ds ∧dt and we can write η =fdu ∧···∧du . Then by the rules of calculus
ϕ,1 ϕ,l−1 1 l
e
∂(ϕ#(u ),...,ϕ#(u ))
ϕ#(η)=ϕ#(f) 1 l ds ∧···∧ds ∧dt
ϕ,1 ϕ,l−1
∂(s ,...,s ,t)
(cid:12) ϕ,1 ϕ,l−1 (cid:12)
(cid:12) (cid:12)
and hence (cid:12) (cid:12)
(cid:12) (cid:12)
∂(ϕ#(u ),...,ϕ#(u ))
1 l
κ (η)=κ (f)+ord .
ϕ ϕ t
∂(s ,...,s ,t)
(cid:12) ϕ,1 ϕ,l−1 (cid:12)
(cid:12) (cid:12)
In order to compute the order of the Jacobian(cid:12)one can use approximat(cid:12)ive methods, i.e., compute
(cid:12) (cid:12)
with truncations of the involvedpower series of sufficiently high precision. In any case one has to
beabletocomputeintheuniversalmoduleofdifferentialsofthefieldextensionF |E. Thisproves
ϕ
still a little difficult in current computer algebra systems. It is therefore preferable to compute
the adjointorders,whenpossible,simultaneouslywiththeformaldesingularization. This involves
essentially repeated application of the chain rule of differential calculus.
With these remarks and the above notation it is now obvious how to derive an algorithm.
Correctness of the following is immediate by Theorem 4.6, and termination is trivial because
formal desingularizations can be computed and consist of finitely many formal prime divisors.
8 TOBIASBECKANDJOSEFSCHICHO
Algorithm 1 Adjoints(F :E[x0,...,xl+1],m:N,n:N):2E[x0,...,xl+1]
Require: an irreducible,homogeneouspolynomialF ofdegree d(not equalto x for any 0≤i≤
i
l+1), defining X ⊂Pl+1
E
Ensure: a basis for the space of global sections of O (n)⊗ π (ω⊗m) represented by homoge-
X OX ∗ Y
neous polynomials of degree n+m(d−l−2) where π :Y →X is any desingularization
1: Let B ⊂E[x0,...,xl+1] be a set representing a basis of (E[x0,...,xl+1]/hFi)n+m(d−l−2);
2: Compute a formal desingularization S of X and adjoint orders αϕ for all ϕ∈S;
3: C :=0∈E∞×|B|; {a matrix with an undetermined number of rows}
4: for ϕ∈S do
5: A:= b∈BcbTrunc(ϕ#(b),mαϕ)= 0≤j<mαϕajtj; {with aj linear in Fϕ[cb |b∈B]}
6: C :=PAddConstraints(C,{aj}0≤j<mαPϕ);
7: Let K ⊂E#B be a basis of ker(C);
8: return { b∈Bcbb|(cb)b∈B ∈K};
P
Here Trunc(ϕ#(b),mα ) means the truncation of ϕ#(b) at order mα . It remains to explain
ϕ ϕ
the functionAddConstraints. It is meantto stacknew rowsontopofthe matrix C, representing
the linear constraints imposed by the formal prime divisor ϕ.
Assume E ⊂ F′ ⊂ F is a tower of field extensions where F over F′ is simple (algebraic or
transcendental). Let a:= y c ∈F[c |b∈B]. We want to find values c ∈E s.t. the linear
b∈B b b b b
constraint a=0 is fulfilled. We are done if we know how to translate the constraint equivalently
to a finite number of lineaPr constraints over the smaller field F′. Using this step recursively and
considering the fact that F over E is finitely generated, we finally get a set of constraints with
coefficients in E. If y c =0 is sucha constraint,the function AddConstraints wouldstack
b∈B b b
the row vector (y ) on top of the matrix C. We distinguish two cases:
b b∈B
P
• IfF overF′ is algebraic,say,ofdegreee+1,chooseabasis {f } ofF as anF′-vector
r 0≤r≤e
space. Then
0= y c = c y f = y c f
b b b b,r r b,r b r
!
b∈B b∈B 0≤r≤d 0≤r≤e b∈B
X X X X X
holds if and only if y c =0 for all 0≤r ≤e.
b∈B b,r b
• NowassumeF=F′(s)istranscendental. W.l.o.g. wemayassumethatthey areactually
b
polynomials in F′[s]P, otherwise multiply the equation by the common denominator. Let e
be the maximal degree of all the y . Then
b
0= y c = c y sr = y c sr
b b b b,r b,r b
!
b∈B b∈B 0≤r≤e 0≤r≤e b∈B
X X X X X
holds again if and only if y c =0 for all 0≤r ≤e.
b∈B b,r b
P
6. Example
Let E := Q and write x := x ,y := x ,z := x ,w := x . The homogeneous polynomial
0 1 2 3
F := w3y2z +(xz +w2)3 ∈ Q[x,y,z,w] of degree d = 6 defines a hypersurface X ⊂ P3, i.e.,
Q
l = dim(X) = 2. We compute a formal desingularization S using Algorithm 1 of [1]. Amongst
others,wegetaformalprimedivisorϕ:SpecF JtK→X definedbytheQ-algebrahomomorphism
ϕ
x7→1,
y 7→−8t3,
ϕ# :Q[x,y,z,w]/hFi→F JtK: s
ϕ z 7→ 6s4t6,
w7→−8αt3− 8t4+ 4αt5+ 1αt7+ 1 αt9+O(t11)
s s s2 s3 2s4
where Fϕ =Q(s)[α] and α has minimal polynomial α2+s.
ADJOINT COMPUTATION FOR HYPERSURFACES USING FORMAL DESINGULARIZATIONS 9
Firstwe wantto compute the adjointorderofthis formalprime divisor. Thereforewe consider
the rational differential form
xd−1 x5
dy ∧dz = dy ∧d z.
∂F/∂w x x 6x2z2w+12xzw3+3y2zw2+6w5 x x
Now we apply the map induced by ϕ# and find the differential form
1
d−8t3∧d64t6.
786432αt17+ 1572864t18− 1966080αt19+O(t20) s s
s4 s4 s5
According to Definition 4.4 we generate by ds∧dt and rewrite this form again to
1
ds∧dt.
512αt9+ 1024t10− 1280αt11+O(t12)
s s s2
The coefficient has order −9, so κ =9 as of Definition 4.5.
ϕ
Assume now, we wantto compute the globalsections of π (ω )⊗ O (1) (where π :Y →X
∗ Y OX X
is any desingularization, not necessarily the one described by S), i.e., we have m =1 and n=1.
We compute n+m(d−l−2)=1+1(6−2−2)=3. Therefore we first need a set B projecting
bijectively to the component of Q[x,y,z,w]/hFi of homogeneous degree 3. Since the defining
equation is of degree 6 we can choose the set of all monomials of degree 3:
B :={x3,x2y,xy2,y3,x2z,xyz,y2z,xz2,yz2,z3,x2w,xyw,y2w,xzw,yzw,z2w,xw2,yw2,zw2,w3}
Applying ϕ# to the generic form of degree 3 we find:
ϕ#( c b)=(c )t0+(−8c − 8αc )t3+(−8c )t4+( 4αc )t5+
b∈B b x3 s x2y s x2w s x2w s2 x2w
P (6s24cxy2 + 6s4cx2z+ 6s24αcxyw− 6s4cxw2)t6+(s13αcx2w+ 6s24cxyw+ 1s228αcxw2)t7+
(−32αc + 128c )t8+O(t9)
s3 xyw s2 xw2
A form is adjoint iff its ϕ#-image vanishes with order greateror equal to mκ =1·9=9, i.e.,
ϕ
the coefficients of t0,...,t8 have to vanish. Viewing B as an ordered basis we can write this as a
matrix in Q(s)[α]9×20:
1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
 
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
 0 −8 0 0 0 0 0 0 0 0 −8α 0 0 0 0 0 0 0 0 0 
 s s 
 0 0 0 0 0 0 0 0 0 0 −8 0 0 0 0 0 0 0 0 0 
 s 
 0 0 0 0 0 0 0 0 0 0 4 α 0 0 0 0 0 0 0 0 0 
 s2 
 0 0 64 0 64 0 0 0 0 0 0 64α 0 0 0 0 −64 0 0 0 
 s2 s s2 s 
 0 0 0 0 0 0 0 0 0 0 1 α 64 0 0 0 0 128α 0 0 0 
 s3 s2 s2 
 0 0 0 0 0 0 0 0 0 0 0 −32α 0 0 0 0 128 0 0 0 
 s3 s2 
 
The third row from the bottom corresponds to the constraint
64c + 64c + 64αc − 64c =0.
s2 xy2 s x2z s2 xyw s xw2
Reordering this in terms of the basis α0,α1 of Q(s)[α] over Q(s) we find
(64c + 64c − 64c )α0+(64c )α1 =0
s2 xy2 s x2z s xw2 s2 xyw
and hence the two equivalent constraints
64c + 64c − 64c =0 and 64c =0.
s2 xy2 s x2z s xw2 s2 xyw
The second one is clearly equivalent to 64c = 0 which is already a constraint over Q. The
xyw
first one can be multiplied by s2 and rewritten in terms of s0 and s1:
(64c )s0+(64c −64c )s1 =0
xy2 x2z xw2
This yields another two constraints
64c =0 and 64c −64c =0
xy2 x2z xw2
10 TOBIASBECKANDJOSEFSCHICHO
over Q. Altogether the third row from the bottom corresponds to the following matrix in Q3×20:
0 0 64 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 64 0 0 0 0 0 0 0 0 0 0 0 −64 0 0 0
 
0 0 0 0 0 0 0 0 0 0 0 64 0 0 0 0 0 0 0 0
Treating all rows similarly, stacking the computed matrices on top of each other and skipping
zero rows we obtain:
1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 −8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
 
0 0 0 0 0 0 0 0 0 0 −8 0 0 0 0 0 0 0 0 0
 0 0 0 0 0 0 0 0 0 0 −8 0 0 0 0 0 0 0 0 0 
 
 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 
 
 0 0 64 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 
 
 0 0 0 0 64 0 0 0 0 0 0 0 0 0 0 0 −64 0 0 0 
 
 0 0 0 0 0 0 0 0 0 0 0 64 0 0 0 0 0 0 0 0 
 
 0 0 0 0 0 0 0 0 0 0 0 64 0 0 0 0 0 0 0 0 
 
 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 
 
 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 128 0 0 0 
 
 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 128 0 0 0 
 
 0 0 0 0 0 0 0 0 0 0 0 −32 0 0 0 0 0 0 0 0 
 
Doing the same computation for all the other formal prime divisors in S one computes a huge
matrix which turns out to have a one-dimensional kernel spanned by
0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 .
The entries o(cid:0)f this vector are the coefficients of the form xzw+w3. So we have (cid:1)
Γ(X,π∗(ωY)⊗OX OX(1))∼=hxzw+w3iQ.
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SymbolicComputationGroup,JohannRadonInstituteforComputationalandAppliedMathematics,
Austrian Academy of Sciences, AltenbergerStraße69,A-4020Linz, Austria
E-mail address: [email protected]
E-mail address: [email protected]