The Chow Ring of the Stack of Rational Curves with at most 3 Nodes

0.17 MB·English

by  Damiano Fulghesu

THE CHOW RING OF THE STACK OF RATIONAL
CURVES WITH AT MOST 3 NODES
9
DAMIANO FULGHESU
0
0
2 Abstract. In this paper we explicit the rational Chow ring of
n the stack M≤3 consisting of nodal curves of genus 0 with at most
0
a 3 nodes: it is a Q-algebra with 10 generators and 11 relations.
J

9

]
G
1. Introduction
A
h. In this paper we explicit the ring A∗(M≤3)⊗Q (Theorem 6.3). Our
0
at technique is to compute A∗(M≤n)⊗Q for n ≤ 3 by induction on n. For
0
m n = 0 we have M0 = BPGL , and this case is well understood [Pan].
0 2
[ The inductive step is based on the following fact: if n ≤ 4, then

1 the top Chern class of the normal bundle of Mn into M≤n is not a 0-
0 0
v divisor in A∗(Mn)⊗Q. As a consequence, by an elementary algebraic
9 0
0 Lemma (2.1) we can reconstruct the ring A∗(M≤n)⊗Q from the rings
0
2 A∗(M≤n−1)⊗QandA∗(Mn)⊗Q, provided thatwehaveanexplicit way
1 0 0
of extending each class in A∗(M≤n−1)⊗Q to a class in A∗(M≤n)⊗Q
1. 0 0
(as we have seen in [Fulg2]), and then computing the restriction of
0
9 this extension to A∗(Mn)⊗Q.
0
0
We find 10 generators: the classes γ for all trees Γ with at most
: Γ
v three nodes (they are 5), plus the Mumford class k . The remaining 4
2
i
X generators are somewhat new with respect to the tautological classes
r introduced for stable curves.
a
The ideal of relations is determined essentially through two useful
technical algebraic Lemmas (2.1 and 2.3).
2. Algebraic lemmas
Here we state and prove two algebraic lemmas for future reference.
The first one is Lemma 4.4 [Ve-Vi]:
Lemma 2.1. Let A,B and C be rings, f : B → A and g : B → C ring
homomorphism. Let us suppose that there exists an homomorphism of
abelian groups φ : A → B such that the sequence
φ g
A −→ B −→ C −→ 0
1
2 DAMIANOFULGHESU
is exact; the composition f ◦φ : A → A is multiplication by a central
element a ∈ A which is not a 0-divisor.
Then f and g induce an isomorphism of ring
(f,g) : B → A× C,
A/(a)
p q
where the homomorphism A −→ A/(a) is the projection, while C −→
A/(a) is induced by the isomorphism C ≃ B/kerg and the homomor-
phism of rings f : B → A.
Proof. Owing to the fact that a is not a 0-divisor we immediately
have that φ is injective. Let us observe that the map (f,g) : B →
A× C is well defined for universal property and for commutation
A/(a)
of the diagram:
B g // C
f q
(cid:15)(cid:15) (cid:15)(cid:15)
A p // A/(a)
ρ
Now let us exhibit the inverse function A× C −→ B. Given (α,γ) ∈
A/(a)
A × C, let us chose an element β ∈ B such that g(β) = γ. By
A/(a)
definition of q we have that f(β)−α lives in the ideal (a) and so (it is
an hypothesis on f ◦φ) there exists in A an element α˜ such that:
f(β)−α = f(φ(α˜))
from which:
f(β −φ(α˜)) = α
we define then ρ(α,γ) := β −φ(α˜). In order to verify that it is a good
definition, let us suppose that there exist an element β ∈ B such that
0
(f,g)(β ) = 0, to be precise there exists an element α ∈ A such that:
0 0
φ(α ) = 0 and furthermore
0
0 = f(φ(α )) = aα
0 0
but we have that a is not a divisor by zero, so necessary we have α = 0
0
and β = 0. we conclude by noting that from the definition of ρ we
0
(cid:3)
have immediately that it is an isomorphism.
Remark 2.2. The Lemma 2.1 will be used for computing the Chow
ring of M≤k+1 when the rings A∗[M≤k] ⊗ Q and A∗[Mk+1] ⊗ Q are
0 0 0
known. As a matter of fact, given an Artin stack X and a closed Artin
substack Y −→i X of positive codimension, we have the exact sequence
of groups (see [Kre] Section 4)
A∗(Y)⊗Q −→i∗ A∗(X)⊗Q −j→∗ A∗(U) −→ 0
CHOW RING OF RATIONAL CURVES 3
By using the Self-intersection Formula, it follows that:
i∗i 1 = i∗[Y] = c (N ),
∗ top Y/X
when c (N ) is not 0-divisor we can apply the Lemma.
top Y/X
We will also use the following algebraic Lemma:
Lemma 2.3. Given the morphisms
p1 p2
A −→ A −→ B ←− A ←− A
1 1 2 2
in the category of rings, where the maps p and p are quotient respec-
1 2
tively for ideals I and I . Then it defines an isomorphism
1 2
A × A
∼ 1 B 2
A × A = .
1 B 2
(I ,I )
1 2
Proof. Let us consider the map
(p ,p ) : A × A → A × A ,
1 2 1 B 2 1 B 2
by surjectivity of p e p this map is surjective, while the kernel is the
1 2
ideal (I ,I ). (cid:3)
1 2
3. The open substack M0
0
Γ := •
As we have seen in [Fulg1]
MΓ ≃ BAut(C). (1)
0
we have that the stack M0 is the classifying space of PGl . Owing
0 2
to the fact that PGl ∼= SO and following [Pan] we have
2 3
A∗(M0)⊗Q ∼= Q[c (sl )].
0 2 2
Since (as shown in [Fulg2]) c (sl ) = (1/2)k , we can write
2 2 2
Proposition 3.1.
A∗(M0)⊗Q = Q[k ].
0 2
4 DAMIANOFULGHESU
4. The first stratum
Γ :=• •
Weorderthetwo components. The automorphismgroupisC ⋉(E×
2
E), (where C is the order two multiplicative group) and the action of
2
its generator τ over E ×E exchanges the components.
Then the induced action of τ on the ring A∗ ⊗ Q ∼= Q[t ,t ]
E×E 1 2
exchanges the first Chern classes t and t . The invariant polynomials
1 2
are the symmetric ones which are algebrically generated by: {(t +
1
t )/2,(t2 + t2)/2}. By recalling the description of Mumford classes
2 1 2
given in [Fulg2], we have A∗(M1)⊗Q = Q[k ,k ]. Let us consider the
0 1 2
two inclusions i and j (respectively closed and open immersions) and
the ´etale covering φ
M1 φ // M1 (2)
0 0C
C
C
f CCiC
C
C!!
M0 // M≤1
0 j 0
we obtain the following exact sequence
A∗(M1)⊗Q −→i∗ A∗(M≤1)⊗Q −j→∗ A∗(M0)⊗Q −→ 0
0 0 0
for what we have seen we have:
Q[k ,k ] −→i∗ A∗(M≤1)⊗Q −j→∗ Q[k ] −→ 0.
1 2 0 2
Nowwehave thatthefirst Chern classofthenormalbundleNM10(M≤01)
is
1
i∗i [M1] = φ (t +t ) = −k
∗ 0 ∗2 1 2 1
SinceA∗M1 isanintegraldomainwecanapplyLemma(2.1)andobtain
0
the ring isomomorphism
A∗(M≤1)⊗Q ∼= Q[k ,k ]× Q[k ] ∼= Q[k ,k ].
0 1 2 Q[k2] 2 1 2
where the map q : Q[k ] → Q[k ,k ]/(k ) = Q[k ] tautologically sends
2 1 2 1 2
k into k .
2 2
So we have
Proposition 4.1.
A∗(M≤1)⊗Q ∼= Q[k ,k ].
0 1 2
CHOW RING OF RATIONAL CURVES 5
5. The second stratum
Γ :=• • • We order the two components with one node.
In this case the group of automorphism of the fiber is
Aut(CΓ) ∼= C2 ⋉(Gm ×E ×E) =: C2 ⋉H,
where the action of τ sends an element g ∈ Gm into g−1 and exchange
the components isomorphic to E.
WecanidentifyA∗(B(Gm)3)withA∗(MΓ)andA∗(BAut(Γ)⋉(Gm)3)
0
with A∗(MΓ). Set
0 f
t = ψ1(∞,1) t = ψ1(∞,2) r = ψ2(∞)
1 2
the action induced by τ on these classes is τ(r,t ,t ) = (−r,t ,t ).
1 2 2 1
With reference to the map
B(Gm)3 −→φ BAut(Γ)⋉(Gm)3
we recall that φ∗ is a ring isomorphism between A∗(M1) ⊗ Q and
0
A∗(B(Gm)3)C2.
We can describe A∗(BH)⊗Q = Q[r,t ,t ] as the polynomial ring in
1 2
r with coefficients in Q[t ,t ], so we write a polynomial P(r,t ,t ) as
1 2 1 2
k riP (t ,t ).
i=0 i 1 2
PThe polynomial P is invariant for the action of τ if and only if the
coefficients of the powers of r in P(r,t ,t ) are equal to those of the
1 2
polynomial P(−r,t ,t ).
2 1
That is to say that P with even index are invariant for the ex-
i
change of t and t , while those with odd index are anti-invariant. An
1 2
anti-invariant polynomial Q is such that Q(t ,t ) + Q(t ,t ) = 0 and
1 2 2 1
consequently it is the product of (t −t ) by an invariant polynomial.
1 2
It is furthermore straightforward verifying that any such polynomial is
invariant for the action of τ.
So an algebraic system of generators for (A∗ ⊗Q)C2 is given by
(Gm)3
u := t +t u := t2 +t2, u := r(t −t ), u := r2.
1 1 2 2 1 2 3 1 2 4
We know that
φ∗k = −u
1 1
φ∗k = −u
2 2
1
φ∗(γ ) = (t −r)(t +r) = (u2 −u )+u −u
2 1 2 2 1 2 3 4
where γ = c (N ), and there exists a class x ∈ A2M2 ⊗Q such
2 2 M20/M≤02 0
that u = π∗x.
3
6 DAMIANOFULGHESU
Claim 5.1. The ideal of relations is generated on Q[k ,k ,γ ,x] by the
1 2 2
polynomial
(2x+(2k +k 2))2 −(2k +k 2)(4γ −k 2) = 0. (3)
2 1 2 1 2 1
Proof. From direct computation we have that relation (3) holds and
the polynomial is irreducible. On the other hand let us consider the
map f : A3 → A4 defined as (t ,t ,r) 7→ (u ,u ,u ,u ). If the generic
Q Q 1 2 1 2 3 4
fiber of f is finite then f(A3) is an hypersurface in A4 and we have
Q Q
done. Now for semicontinuity it is sufficient to show that a fiber is
finite. Let us consider the fiber on 0. We have that u ,u ,u ,u are
1 2 3 4
simultaneously zero iff t = t = r = 0. (cid:3)
1 2
NOTE: In the following we do not explicit the argument above.
Now set η := 2x+(2k +k2), we have that A∗(M2)⊗Q is isomorphic
2 1 0
to the graded ring Q[k ,k ,γ ,η]/I where the ideal I is generated by
1 2 2
the polynomial η2−(2k +k 2)(4γ −k 2). Since φ∗φ is multiplication
2 1 2 1 ∗
by two, we also have the following relation
1
η = φ (t −t )(2r −t +t ) .
∗ 1 2 1 2
(cid:18)2 (cid:19)
Let us consider the cartesian diagram
A∗(M≤2)⊗Q j∗ // Q[k ,k ]
0 1 2
i∗ 2 q
(cid:15)(cid:15) (cid:15)(cid:15)
Q[k ,k ,γ ,η]/I p // Q[k ,k ,η]/I
1 2 2 1 2
where I is the ideal generated by η2 +(2k +k 2)k 2.
2 1 1
The map q is injective so i∗ is injective too.
We set inA∗(M≤2)⊗Q the classes γ := i 1 and q := i η, the ring we
0 2 ∗ ∗
want (from injectivity of i∗) is isomorphic to the subring ofA∗(M2)⊗Q
0
generated by k ,k ,γ ,γ η so we have
1 2 2 2
Proposition 5.2.
Q[k ,k ,γ ,q]
A∗(M≤2)⊗Q = 1 2 2
0 (q2 +γ2(2k +k 2)(k 2 −4γ ))
2 2 1 1 2
CHOW RING OF RATIONAL CURVES 7
6. The third stratum
The third stratum splits into two components.
The first component
•
Γ′ :=
3 •nnnnnnn•PPPPPPP•
We order components in ∆ . Let us note that the component cor-
1
responding to the central vertex has three points fixed by the other
three components, consequently, given a permutation of the external
vertices, there is an unique automorphism related to the central vertex.
The group Aut(CΓ′3) is therefore isomorphic to S ⋉(E3). We have
3
A∗ ⊗Q ∼= Q[w ,w ,w ],
E3 1 2 3
on which S acts by permuting the three classes
3
w := ψ1 w := ψ1 w := ψ1
1 ∞,1 2 ∞,2 3 ∞,3
So we have
φ∗k = −(w +w +w ),
1 1 2 3
φ∗k := −(w2 +w2 +w2),
2 1 2 3
φ∗k := −(w3 +w3 +w3);
3 1 2 3
conesequently
A∗(MΓ′3)⊗Q ∼= Q[k ,k ,k ].
0 1 2 3
We fix the following notation
MΓ′3 f // Ψ(Γ ,Γ′) pr2 // MΓ2 ≤2
0 2 3 0
(cid:16) (cid:17)
f g
pr1 Π
φ
(cid:15)(cid:15) (cid:15)(cid:15)
++ MΓ′3 in // M≤3
0 0
where f is the union of all f .
α
First of all let us notice that the class φ∗γ′ := π∗c (N ) = w w w
3 3 i 1 2 3
depends on Mumford classes in the following way 6γ′ = −(k 3−3k k +
3 1 1 2
2k ) so we can write
3
A∗(MΓ′3)⊗Q = Q[k ,k ,γ′].
0 1 2 3
8 DAMIANOFULGHESU
The restriction of γ to A∗(MΓ′3)is
2 0
1 1
γ := pr c (pr∗(N )/N ) = φ f∗ c (pr∗(N )/N )
2 1∗2 2 1 in pr2 ∗ 2 2 1 in pr2
1
= π (w w )
∗ 1 3
2
from which φ∗γ = w w +w w +w w consequently, by writing 2γ =
2 1 3 1 2 2 3 2
k 2 −k we have
1 2
A∗(MΓ′3)⊗Q = Q[k ,γ ,γ′].
0 1 2 3
In order to restrict the class q let us notice that we can write
f∗r = 0 f∗t = w f∗t = w
1 1 2 3
from which we have
1 1
f∗ (t −t )(2t−t +t ) = − (w −w )2
1 2 1 2 1 3
(cid:18)2 (cid:19) 2
and so
1
φ∗q = φ∗φ − (w −w )2w w
∗ 1 3 1 3
(cid:18) 2 (cid:19)
= −((w −w )2w w +(w −w )2w w +(w −w )2w w )
1 3 1 3 1 2 1 2 2 3 2 3
we can therefore write q = −γ (k 2 −4γ ) + 3γ′k . With reference to
2 1 2 3 1
the inclusions
MΓ′3 −→i M≤2 ∪MΓ′3 ←−j M≤2
0 0 0 0
we have the fiber square:
A∗(M≤2 ∪MΓ′3)⊗Q j∗ // Q[k1,k2,γ2,q]/I
0 0
i∗ 2 ϕ
(cid:15)(cid:15) (cid:15)(cid:15)
Q[k ,γ ,γ′] p // Q[k ,γ ]
1 2 3 1 2
where I is the ideal generated by the polynomial q2−γ2(2k −k 2)(k 2−
2 2 1 1
4γ ) and the map ϕ is surjective and such that
2
kerϕ = (q +γ (k 2 −4γ ), k +2γ −k 2).
2 1 2 2 2 1
Now let us observe that from Lemma (2.3) the ring in question is
isomorphic to
Q[k ,γ ,γ′]× Q[k ,k ,γ ,q]
A/(0,I) := 1 2 3 Q[k1,γ2] 1 2 2 .
(0,I)
Set, with abuse of notation
k := (k ,k ) γ := (γ ,γ ) γ′ := (γ′,0)
1 1 1 2 2 2 3 3
k := (k 2 −2γ ,k ) q := (−γ (k 2 −4γ ),q)
2 1 2 2 2 1 2
CHOW RING OF RATIONAL CURVES 9
Straightforward arguments lead us to state the following
Lemma 6.1. The classes k ,γ ,γ′,k ,q generate the ring A.
1 2 3 2
Let us compute the ideal of relations. Let T(k ,γ ,γ′,k ,q) be a
1 2 3 2
polynomial in Q[k ,γ ,γ′,k ,q]. It is zero in A iff
1 2 3 2
T(k ,γ ,γ′,k 2 −2γ ,−γ (k 2 −4γ )) = 0 in Q[k ,γ ,γ′]
1 2 3 1 2 2 1 2 1 2 3
T(k ,γ ,0,k ,q) = 0 in Q[k ,γ ,k ,q]
1 2 2 1 2 2
in particular this implies that T is in the ideal of γ′. Consequently
3
the polynomial T =: γ′T is zero in A iff T(k ,γ ,γ′,k 2−2γ ,−γ (k 2−
3 1 2 3 1 2 2 1
4γ )) is zero in Q[k ,γ ,γ′]. The ideal of relations in A is so generated
2 1 2b3 b
by γ′(−k +2γ −k 2) and γ′(q +γ (k 2 −4γ )).
3 2 2 1 3 2 1 2
Finally let us notice that the ideal (0,I) is generated in A by the
polynomial q2 +γ2(2k +k 2)(k 2 −4γ ). So we can conclude that
2 2 1 1 2
A∗(M≤2 ∪MΓ′3)⊗Q = Q[k ,k ,γ ,γ′,q]/J,
0 0 1 2 2 3
where J is the ideal generated by the polynomials
q2 +γ2(2k +k 2)(k 2 −4γ )
2 2 1 1 2
γ′(−k +2γ −k 2)
3 2 2 1
γ′(q +γ (k 2 −4γ ))
3 2 1 2
The second component
Γ′′ := • • • •
3
The group of Aut(CΓ′3) is C2⋉(E×E×Gm×Gm). The action of τ
on this group exchange simultaneously the components related to Gm
and those related to E.
We have the isomorphism
A∗ ⊗Q ∼= Q[v ,...,v ]
E×E×Gm×Gm 1 4
where
v = ψ1 v = ψ1 v = ψ2 v = ψ2
1 ∞,1 2 ∞,2 3 ∞,1 4 ∞,2
by gluing curves such that the two central components corresponds in
the point at infinity.
It follows that the action induced by τ is
τ(v ,v ,v ,v ) = (v ,v ,v ,v )
1 2 3 4 2 1 4 3
Since C has order 2, the invariant polynomials are algebraically gener-
2
ated by the invariant polynomials of degree at most two (see. Theorem
7.5 [CLO]). It is easy to see that a basis for the linear ones is given
10 DAMIANOFULGHESU
by u := v + v , u := v + v . For the vector subspace of invari-
1 1 2 2 3 4
ant polynomials of degree two, we can compute a linear basis by using
Reynolds’ operator
u := v2 +v2, u := v v +v v ,
3 1 2 6 1 3 2 4
u := v2 +v2, u := v v ,
4 3 4 7 1 2
u := v v +v v , u := v v
5 1 4 2 3 8 3 4
now we note that
u = u u −u , u = (u2 −u )/2, u = (u2 −u )/2
6 1 2 5 7 1 3 8 2 4
Consequently we can write
A∗MΓ′3 ⊗Q ∼= Q[u ,...,u ]/I,
0 1 5
where I is the ideal generated by the polynomial
2u u +2u u u −u2u −u2u −2u2 (4)
3 4 1 2 5 2 3 1 4 5
Withreference to thedegreetwo covering MΓ′3 −→φ MΓ′3 wehave: −u =
0 0 1
φ∗k −u = φ∗k . In order to compute the restriction of the closure
1 3 2
of the classes γ and q of M≤2 let us fix tfhe notation of the following
2 0
diagram
M1 ×M0 ×M0
MΓ′3 kkkkkkf2kkkkfk1k//k(kMkkk00k,k155×M10,2×MTT00T,T1)TITTTpTprT2′r′2′TTTTTTTT//)) MΓ2 ≤2
f0 SSSSSSSSSSSfS3SSSSS0S,1S)) 0,2 jj0j,1jjjjpjr2′j′′jjjjjjjjj55 (cid:16)f0 (cid:17)
M0 ×M0 ×M1
0,1 0,2 0,1
Π
φ pr1
)) M(cid:15)(cid:15)Γ′3′ i // M(cid:15)(cid:15)≤3
0 0
where (M0 ×M1 ×M0 )I is the component where the marked points
0,1 0,2 0,1
of the central curve (which is singular) are on different components.
As φ∗ is an isomorphism to the algebra of polynomials which are
invariants for the action of C , let us choose in A∗(MΓ′3′) ⊗ Q classes
2 0
ρ,λ,µ such that u = φ∗ρ ,u = φ∗λ ,u = φ∗µ.
2 4 5
First of all let us compute the restriction of the closure of γ ∈
2
A∗(M≤2)⊗Q, thepolynomialintheclassesψ is 1,wehavec (φ∗(N )) =
0 2 3 i