Degree of subvarieties
3.1 Compactification of Drinfeld modular varieties
In [30] Pink constructs the Satake compactification SrF,K of a Drinfeld modular variety SF,rK with K ⊂GLr( ˆA). It is a normal projective variety which contains SF,rK as an open dense subvariety.
IfKis amply small,SrF,Kis endowed with a natural ample invertible sheafLrF,K. In [30], the space of global sections of its k-th power is defined to be the space of algebraic modular forms of weight k onSF,rK.
If K ⊂ GLr(AfF) is arbitrary (not necessarily contained in GLr( ˆA)) and g ∈GLr(AfF) is chosen such that gKg−1 ⊂GLr( ˆA), we can define
SrF,K:=SrF,gKg−1 (3.1.1) and, if K is amply small,
LrF,K :=LrF,gKg−1. (3.1.2)
As in Step (v) of the proof of Theorem 1.1.2, one can show, using part (i) of the following proposition for K ⊂ GLr( ˆA), that this defines SrF,K and LrF,K up to isomorphism.
Proposition 3.1.1. (i) Forg ∈GLr(AfF)and a compact open subgroupK′ ⊂ g−1Kg the morphism πg : SF,rK′ → SF,rK defined in Section 2.1 extends uniquely to a finite morphism πg : SrF,K′ → SrF,K defined over F. If K is amply small, then there is a canonical isomorphism
LrF,K′ ∼=πg∗LrF,K.
(ii) Any inclusion ιFF, b′ : SFr′′,K′ → SF,rK of Drinfeld modular varieties extends uniquely to a finite morphism ιFF, b′ : SrF′′,K′ → SrF,K defined over F′. If K is amply small, then there is a canonical isomorphism
LrF′′,K′ ∼=ιFF, b′ ∗LrF,K.
Proof. This follows from Proposition 4.11 and 4.12 and Lemma 5.1 in [30]. Note that these statements automatically hold for arbitrary levelsKand K′ (not nec-essarily contained in GLr( ˆA) respectively GLr′(Ab′)) because the equations (3.1.1) and (3.1.2) define the Satake compactification of a general Drinfeld modular va-riety as the Satake compactification of a Drinfeld modular vava-riety with level contained in GLr( ˆA) resp. GLr′(Ab′).
3.2 Degree of subvarieties
In this section, SF,rK always denotes a Drinfeld modular variety with K amply small.
Definition 3.2.1. The degree of an irreducible subvariety X ⊂SF,rK is defined to be the degree of its Zariski closure X in SrF,K with respect to LrF,K, i.e., the integer
degX := degLr
F,KX =
∫
SrF,K
c1(LrF,K)dimX ∩[X],
where c1(LrF,K) ∈ A1SrF,K denotes the first Chern class of LrF,K, the cycle class ofX inAdimXSrF,Kis denoted by[X]and∩is the cap-product betweenAdimXSrF,K and AdimXSrF,K.
The degree of a reducible subvariety X ⊂ SF,rK is the sum of the degrees of all irreducible components of X.
Remark: Note that our definition of degree for reducible subvarieties differs from the one used in many textbooks where only the sum over the irreducible components of maximal dimension is taken.
Lemma 3.2.2. The degree of a subvariety X ⊂ SF,rK is at least the number of irreducible components of X.
Proof. This follows by our definition of degree because LrF,K is ample and the degree of an irreducible subvariety of a projective variety with respect to an ample invertible sheaf is a positive integer (see, e.g., Lemma 12.1 in [15]).
Proposition 3.2.3. (i) Let πg : SF,rK′ → SF,rK be the morphism defined in Section 2.1 for g ∈GLr(AfF) and K′ ⊂g−1Kg. Then
degπg−1(X) = [g−1Kg :K′]·degX (3.2.1) for subvarieties X ⊂SF,rK and
degπg(X′)≤degX′ (3.2.2) for subvarieties X′ ⊂SF,rK′. In particular, we have
degTg(X)≤[K:K ∩g−1Kg]·degX (3.2.3) for subvarieties X ⊂SF,rK.
(ii) For any inclusion ιFF, b′ : SFr′′,K′ → SF,rK of Drinfeld modular varieties and for any subvariety X ⊂SFr′′,K′, we have
degX = degιFF, b′ (X). (3.2.4) Proof. We use the projection formula for Chern classes (see, e.g., Proposition 2.5 (c) in [15]):
If f :X →Y is a proper morphism of varieties and L is an invertible sheaf on Y, then, for all k-cycles α ∈Ak(X), we have the equality
f∗(c1(f∗L)∩α) =c1(L)∩f∗(α) (3.2.5) of (k−1)-cycles in Ak−1(Y).
For the proof of (3.2.1) and (3.2.2), we first assume that X ⊂ SF,rK and X′ ⊂ SF,rK′ are irreducible. For this, note that πg : SF,rK′ → SF,rK is finite of degree [g−1Kg : K′] by Theorem 2.1.1 and ´etale by Proposition 2.1.3 because K is amply small. The latter implies that the scheme-theoretic preimage of X under πg is a reduced closed subscheme of SF,rK′ (by Theorem III.10.2 in [22]
the fibers of the generic points of the irreducible components of X are regular).
Hence the scheme-theoretic preimage is equal to the subvariety πg−1(X). Note that the degree of a finite, flat surjective morphism of varieties f : V → W is preserved under base extension by our characterisation of the degree as rank of the locally free OW-module f∗(OV) in Section 0.1. Therefore the restriction of πg to the subvarietyπ−g1(X) is also finite of degree [g−1Kg :K′] and we have the equality
πg∗[π−g1(X)] = [g−1Kg :K′]·[X]
of cycles on SrF,K. For d := dimX, with Proposition 3.1.1 (i) and the above
For the proof of (3.2.2), we note that
πg∗[X′] = deg(πg|X′)·[πg(X′)]
as cycles onSrF,K. Again with the projection formula and Proposition 3.1.1 (i), we get
because the set of irreducible components of π−1g (X) is exactly equal to
∪n i=1
{irreducible components of π−g1(Xi)}
and this union is disjoint. Therefore, the formula (3.2.1) follows from the irre-ducible case.
IfX′ ⊂SF,rK′ is reducible with irreducible components X1′, . . . , Xk′, then the set of irreducible components of πg(X′) is a subset of {πg(X1′), . . . , πg(Xk′)}, hence we have
degπg(X′)≤
∑k i=1
degπg(Xi′) and the inequality (3.2.2) follows from the irreducible case.
The inequality (3.2.3) immediately follows from (3.2.1) and (3.2.2) because Tg(X) = πg(π−11(X))
where π1 and πg are projection morphisms SF,rKg →SF,rK with Kg :=K ∩g−1Kg and
degπ1 = [K:Kg] = [K:K ∩g−1Kg].
Finally, for the proof of (3.2.4) we use that ιFF, b′ : SFr′′,K′ → SF,rK is a closed immersion by Proposition 2.2.2 because K is amply small. For an irreducible subvariety X ⊂SFr′′,K′, we therefore have the equality
ιFF, b′
∗[X] = [ιFF, b′ (X)]
of cycles onSrF,K. The same calculation as in the proof of (3.2.2) withd := dimX therefore gives
degιFF, b′ (X) =
∫
SrF,K
c1(LrF,K)d∩[ιFF, b′ (X)]
=
∫
Sr′F′,K′
c1(ιFF, b′ ∗LrF,K)d∩[X] = degX
because ιFF, b′ ∗LrF,K ∼=LrF′′,K′ by Proposition 3.1.1 (ii).
If X ⊂ SFr′′,K′ is reducible with irreducible components X1, . . . , Xl, then ιFF, b′ (X) has exactly the irreducible components ιFF, b′ (X1), . . . , ιFF, b′ (Xl) because ιFF, b′ is a closed immersion. Therefore, the formula (3.2.4) forX reducible follows from the irreducible case.
We will use the following two consequences of B´ezout’s theorem to get an upper bound for the degree of the intersection of two subvarieties of SF,rK:
Lemma 3.2.4. For subvarieties V, W of a projective variety U and an ample invertible sheafL on U, we have
degV ∩W ≤degV ·degW, where deg denotes the degree with respect to L.
Proof. See Example 8.4.6 in [15] in the case thatV and W are irreducible.
IfV =V1∪· · ·∪VkandW =W1∪· · ·∪Wlare decompositions into irreducible components, then
V ∩W =∪
i,j
Vi∩Wj.
Therefore, each irreducible component of V ∩W is an irreducible component of some Vi∩Wj. By our definition of degree for reducible varieties this implies
degV ∩W ≤∑
i,j
deg(Vi∩Wj).
Hence by the case that V and W are irredubible, we get degV ∩W ≤∑
Proof. Recall that we defined the degree of a subvariety of SF,rK as the de-gree of its Zariski closure in the compactification SrF,K with respect to the line bundleLrF,K. In view of the previous lemma, it is therefore enough to show the following inequality of degrees of Zariski closures inSrF,K with respect to LrF,K:
degV ∩W ≤degV ∩W . For this, note thatV ∩W ⊂V ∩W and
V ∩W ∩SF,rK=V ∩W = (V ∩SF,rK)∩(W ∩SF,rK) = (V ∩W)∩SF,rK because SF,rK is Zariski open inSrF,K. Therefore
V ∩W =V ∩W ∪ (Y ∩(V ∩W)) (3.2.6)
where Y := SrF,K \SF,rK denotes the boundary of the compactification. Since the irreducible components of V ∩W are the Zariski closures of the irreducible components of V ∩W, they all have non-empty intersection with SF,rK. Hence, every irreducible component of V ∩W is not contained in Y ∩(V ∩ W) and therefore by (3.2.6) an irreducible component of V ∩W. The desired inequality
degV ∩W ≤degV ∩W follows.