Intersection Theory of Virtual Classes

In Gromov–Witten theory, the need to intersect virtual classes arises repeatedly. When two moduli spaces \(M_1,M_2\) have maps to a common space \(S\), the moduli space of objects satisfying both conditions simultaneously is the fiber product \(M_1\times_SM_2\), and we wish to obtain a class on this fiber product from \([M_1]^\vir\) and \([M_2]^\vir\). (§Perfect Obstruction Theory, ⁋Proposition 3)

In [Algebraic Varieties] §Intersection Product, we saw how to form the intersection product of two classes, but in our setting this is not well-defined because most spaces are not smooth. If \(S\) were smooth and the two maps were proper, as an alternative we could push forward the two classes to \(S\) and then intersect them, but the resulting class would live on \(S\), which is not what we want. For this reason, we must rigorously define the refined Gysin pullback, which we accepted without proof in [Stacks] §Integration on Deligne–Mumford Stacks, §§Product Spaces and Diagonal Intersections.

On schemes, the Chow group refers to that of [Algebraic Varieties] §Chow Groups, ⁋Definition 5, and on stacks, to the \(\mathbb{Q}\)-coefficient version of [Stacks] §Integration on Deligne–Mumford Stacks; we omit subscripts and write it simply as \(A_k\).

Gysin morphism

To define the Gysin morphism, we need the normal cone. Since we already examined this in [Algebraic Varieties] §Intersection Product, §§Deformation to Normal Cone, here we briefly review as much as will be needed later.

For a closed embedding \(i:W\hookrightarrow X\) and the ideal sheaf \(\mathcal{I}\) of \(W\), the normal cone of \(i\) is defined by

\[C_{W/X}=\Spec_W\Bigl(\bigoplus_{n\geq0}\mathcal{I}^n/\mathcal{I}^{n+1}\Bigr)\]

Here, \(\mathcal{I}/\mathcal{I}^2\) is the first-order part of the functions vanishing on \(W\), namely the linear coordinates in the directions normal to \(W\), which intuitively collects the directions pointing away from \(W\) at each \(w\in W\). Meanwhile, since \(\mathcal{I}^n/\mathcal{I}^{n+1}\) is generated by products of elements of \(\mathcal{I}/\mathcal{I}^2\), there exists a surjection of graded algebras

\[\Sym(\mathcal{I}/\mathcal{I}^2)\rightarrow\bigoplus_{n\geq0}\mathcal{I}^n/\mathcal{I}^{n+1}\]

In this setting, the ambient space \(\Spec_W\Sym(\mathcal{I}/\mathcal{I}^2)\) is the vector space \((\mathcal{I}/\mathcal{I}^2\otimes k(w))^\vee\) formed by the directions pointing away from \(W\) at each point \(w\in W\); if \(\mathcal{I}/\mathcal{I}^2\) is locally free, the collection of these fibers forms a vector bundle, which is called the normal bundle.

As emphasized above, the difference between the normal bundle and the normal cone is that the normal bundle is a vector space formed by the directions pointing away from \(W\) at each point \(w \in W\), so linear combinations of vectors must fall back into itself. In contrast, since the normal cone collects only the directions pointing away from \(W\) at the point \(w\), linear combinations of these directions do not necessarily fall back into itself. Intuitively, the normal cone is a subset of the normal bundle collecting precisely the directions we wish to observe, and by definition the normal cone is closed under scalar multiplication, so it forms a cone in this vector space. To see how this cone is given, let us examine its closed subscheme structure. For the surjection

\[\Sym(\mathcal{I}/\mathcal{I}^2)\rightarrow\bigoplus_{n\geq0}\mathcal{I}^n/\mathcal{I}^{n+1}\]

the degree-\(n\) kernel consists of those degree-\(n\) homogeneous polynomials in the linear coordinates that fall into \(\mathcal{I}^{n+1}\), namely the lowest-degree terms of the equations satisfied by \(X\) near \(W\), and these equations cut out, in each fiber of the ambient space, the directions along which \(X\) does not extend in the lowest-degree approximation. For instance, if \(X=\{\x\y=0\}\subset\mathbb{A}^2\) and \(W\) is the origin, then the fiber of the normal bundle is a \(2\)-dimensional vector space with coordinates \(\x,\y\), but since \(\x\y\) falls into \(\mathcal{I}^3\), we have \(\bigoplus_n\mathcal{I}^n/\mathcal{I}^{n+1}\cong\mathbb{C}[\x,\y]/(\x\y)\) and the normal cone is the two lines \(\{\x\y=0\}\).

Now if \(i\) is a regular embedding of codimension \(d\) ([Scheme Theory] §Complete Intersections, ⁋Definition 1), then \(\mathcal{I}/\mathcal{I}^2\) is a locally free sheaf of rank \(d\) on \(W\) ([Scheme Theory] §Complete Intersections, ⁋Proposition 5), and since the surjection above is an isomorphism for an ideal generated by a regular sequence, there are no extra equations, so \(C_{W/X}\) coincides with the entire rank \(d\) normal bundle \(N_{W/X}=\Spec_W\Sym(\mathcal{I}/\mathcal{I}^2)\). More generally, for a morphism \(g:X'\rightarrow X\) and \(W'=X'\times_XW\), the ideal sheaf of \(W'\) is the ideal generated by \(\mathcal{I}\) on \(X'\), so by the same surjection \(C_{W'/X'}\) defines a closed subcone of \(g^\ast N_{W/X}\). In particular, for a closed subvariety \(V\subseteq X\), \(C_{W\cap V/V}\) is a closed subcone of \(C_{W/X}\).

For any \(k\)-dimensional subvariety \(V\subseteq X\), one can verify that \(C_{W\cap V/V}\) has pure dimension \(k\), so in particular there exists a fundamental class \([C_{W\cap V/V}]\in A_k(C_{W/X})\). Since \(A_k(X)\) is generated by the classes \([V]\) of \(k\)-dimensional subvarieties, the assignment \([V]\mapsto[C_{W\cap V/V}]\) defines a group homomorphism

\[\sigma:A_k(X)\rightarrow A_k(C_{W/X})\]

which we call the specialization with respect to \(i\). On the other hand, since \(N_{W/X}\rightarrow W\) is a vector bundle, there is a flat pullback ([Algebraic Varieties] §Chow Groups, ⁋Proposition 7)

\[p^\ast:A_{k-d}(W)\rightarrow A_k(N_{W/X})\]

which is an isomorphism by homotopy invariance. ([Algebraic Varieties] §Chow Groups, ⁋Example 9) Composing these two morphisms yields the operation we desire.

Definition 1 For a codimension \(d\) regular embedding \(i:W\hookrightarrow X\) and \(N=N_{W/X}\), the Gysin morphism of \(i\)

\[i^!:A_k(X)\rightarrow A_{k-d}(W)\]

is defined as the composition \(i^!:=(p^\ast)^{-1}\circ\sigma\) of the specialization and the inverse of the flat pullback.

Geometrically, \(i^!\alpha\) is obtained by intersecting \(\alpha\in A_k(X)\) with \(W\) and recording the result as a class on \(W\). Specialization allows us to view \(\alpha\) as a class in the normal cone (in this case, the normal bundle) along \(W\), and taking the inverse of the flat pullback (that is, intersecting this cycle with the zero section) yields our desired result.

However, since \(\sigma\) was defined by choosing a representative \(V\) of a generator \([V]\), it must preserve rational equivalence in order to be well-defined on \(A_k(X)\). This is not obvious merely from the formulas, since it is not evident from the definition why the normal cones of two rationally equivalent cycles should be rationally equivalent. Moreover, we have not yet shown that \(C_{W\cap V/V}\) has pure dimension \(k\). What ensures this is the deformation to the normal cone from [Algebraic Varieties] §Intersection Product, ⁋Proposition 9 (Deformation to Normal Cone).

Proposition 2 For a closed embedding \(i:W\hookrightarrow X\), the following hold.

1. For every \(k\)-dimensional subvariety \(V\subseteq X\), \(C_{W\cap V/V}\) has pure dimension \(k\).
2. The specialization \(\sigma\) preserves rational equivalence.

Therefore, \(\sigma:A_k(X)\rightarrow A_k(C_{W/X})\) is a well-defined group homomorphism, and in particular, \(i^!\) of Definition 1 is well-defined.

Proof

First, we construct the deformation to the normal cone on schemes. When \(X\) is affine and \(\mathcal{I}\) corresponds to an ideal \(I\subseteq A\), consider the graded algebra in the variable \(t\)

\[\widetilde{A}=\bigoplus_{n\in\mathbb{Z}}I^nt^{-n}\subseteq A[t,t^{-1}],\qquad I^n=A\quad(n\leq0)\]

This is an algebra over \(A[t]\) and a subring of \(A[t,t^{-1}]\), so it is torsion-free with respect to non-zero elements of \(\mathbb{C}[t]\); since a torsion-free module over a PID is flat, \(M=\Spec\widetilde{A}\) is flat over \(\mathbb{A}^1=\Spec\mathbb{C}[t]\). Moreover, \(\widetilde{A}[t^{-1}]=A[t,t^{-1}]\) and \(\widetilde{A}/(t)\cong\bigoplus_{n\geq0}I^n/I^{n+1}\), and the surjection from \(\widetilde{A}\) to \((A/I)[t]\) sends components with \(n>0\) to \(0\) and components \(At^{-n}\) with \(n\leq0\) to \((A/I)t^{-n}\). Since this construction glues over affine opens, it is well-defined over a general \(X\) as well.

The crucial fact here is that \(M\rightarrow\mathbb{A}^1\) is a flat family, whose fiber at each point \(t\in \mathbb{A}^1\) is given by

\[M_t\cong X\quad(t\neq0),\qquad M_0\cong C_{W/X}\]

That is, \(M\) can be thought of as a flat family in which \(X\) degenerates to the normal cone \(C_{W/X}\) as \(t\rightarrow0\). More concretely, the locus \(t\neq0\) in \(M\) is precisely given by \(X\times(\mathbb{A}^1\setminus\{0\})\); if we embed \(W\) into each fiber as the closed subscheme \(W\times\mathbb{A}^1\subseteq M\), this becomes \(W\) itself inside \(X\) in the fibers with \(t\neq 0\), and becomes the zero section of \(C_{W/X}\) at \(t=0\).

Now we must check where a \(k\)-dimensional subvariety \(V\subseteq X\) goes under this process. Let \(M_V\rightarrow\mathbb{A}^1\) be the family obtained by constructing the deformation to the normal cone as above for the closed embedding \(W\cap V\hookrightarrow V\). If \(V\) is affine and its coordinate ring is a quotient \(B\) of \(A\), then the ideal of \(W\cap V\) is \(IB\), and \(M_V\) is given by

\[M_V=\Spec\Bigl(\bigoplus_{n\in\mathbb{Z}}(IB)^nt^{-n}\Bigr)\]

Since \(I^n\rightarrow(IB)^n\) is surjective, this algebra is a quotient of \(\widetilde{A}\). That is, \(M_V\) is a closed subscheme of \(M\). Furthermore, since the above algebra is a subring of the integral domain \(B[t,t^{-1}]\), \(M_V\) is also integral. Now, the locus \(t\neq0\) in \(M_V\) is \(V\times(\mathbb{A}^1\setminus\{0\})\), so the entire \(M_V\) can be obtained by taking the closure of this locus in \(M\); hence \(M_V\) has dimension \(k+1\). In other words, \(M_V\) is the trace left by \(V\) as it moves along with \(X\), and its \(t=0\) fiber \(C_{W\cap V/V}\) is the limit of \(V\). Finally, this limit is the zero locus of the function \(t\), which is not identically zero on the \((k+1)\)-dimensional variety \(M_V\), so by [Commutative Algebra] §Krull Dimension, ⁋Theorem 6 (Codimension one Principal Ideal Theorem), all of its irreducible components have dimension \(k\). Therefore, \(C_{W\cap V/V}\) has pure dimension \(k\), which proves the first assertion.

Now we prove the second assertion, namely that \(\sigma\) preserves rational equivalence. To this end, we express \(\sigma\) as the composition of three operations, each already well-defined on Chow groups. The first operation is the flat pullback by the projection \(X\times(\mathbb{A}^1\setminus\{0\})\rightarrow X\):

\[A_k(X)\rightarrow A_{k+1}(X\times(\mathbb{A}^1\setminus\{0\}))\]

which sends \([V]\) to \([V\times(\mathbb{A}^1\setminus\{0\})]\). The second operation goes backwards through the restriction from \(M\) to the open subset \(X\times(\mathbb{A}^1\setminus\{0\})\). By the localization exact sequence of [Algebraic Varieties] §Chow Groups, ⁋Proposition 8 (Localization Exact Sequence):

\[A_{k+1}(M_0)\rightarrow A_{k+1}(M)\rightarrow A_{k+1}(X\times(\mathbb{A}^1\setminus\{0\}))\rightarrow0\]

this restriction is surjective, and its kernel consists of classes coming from \(M_0\). Thus, a class on \(X\times(\mathbb{A}^1\setminus\{0\})\) can be lifted to a class on \(M\), and the ambiguity of this lift lies in classes coming from \(M_0\). A straightforward calculation shows that we may choose \([M_V]\) as a lift of \([V\times(\mathbb{A}^1\setminus\{0\})]\). The final operation is the intersection with the principal Cartier divisor \(M_0=\{t=0\}\), \(A_{k+1}(M)\rightarrow A_k(M_0)\). This operation preserves rational equivalence and eliminates the ambiguity of the second operation, since intersecting with \(M_0\) a class supported on \(M_0\) amounts to capping with the first Chern class of the line bundle \(\mathcal{O}_M(M_0)\), and this line bundle is trivial because \(M_0\) is the zero locus of the function \(t\).

Therefore, the composition of the three operations

\[A_k(X)\rightarrow A_k(M_0)=A_k(C_{W/X})\]

is independent of the choice of lift and is well-defined as a group homomorphism between Chow groups. To show that this coincides with \(\sigma\), it suffices to see where \([M_V]\) goes in the final step. Since we know that \(M_V\) is integral and \(t\) is not identically zero on it, the intersection of \(M_0\) and \([M_V]\) is the zero scheme of \(t\) on \(M_V\), which is the class of the \(t=0\) fiber \(C_{W\cap V/V}\) of \(M_V\).

The operation in Definition 1 cuts a class on \(X\) itself by \(W\). The issue is that the class we actually wish to intersect is usually not on \(X\); for instance, in the example discussed in the introduction, what we wanted was a class on the fiber product \(M_1\times_SM_2\) when two moduli spaces \(M_1,M_2\) map to a common space \(S\). This fiber product can be thought of as cutting \(M_1\times M_2\) by the preimage of the diagonal in \(S\times S\):

\[(M_1\times M_2)\times_{S\times S}S=\{(m_1,m_2,s)\mid(f_1(m_1),f_2(m_2))=(s,s)\}=\{(m_1,m_2)\mid f_1(m_1)=f_2(m_2)\}=M_1\times_SM_2\]

but while the class we need to cut lies on \(M_1\times M_2\), the space cutting it is the diagonal inside \(S\times S\), so Definition 1 above does not apply directly. Instead, since we have the morphism \(M_1\times M_2\rightarrow S\times S\), we can have the above construction factor through this morphism.

To formulate this in general, let \(i:W\hookrightarrow X\) be a regular embedding of codimension \(d\), let \(N=N_{W/X}\) be its normal bundle, and for an arbitrary morphism \(g:X'\rightarrow X\), consider the fiber product

\[W'=X'\times_XW=g^{-1}(W)\]

Here, \(W'\hookrightarrow X'\) is a closed embedding, but it does not need to be a regular embedding, so \(N_{W'/X'}\) is not necessarily well-defined as a bundle in the above construction; however, as seen before, the normal cone \(C_{W'/X'}\) is a closed subcone of the rank \(d\) vector bundle \(g^\ast N\) over \(W'\), so it suffices to use this. That is, we send a class on \(X'\) to a class on \(C_{W'/X'}\) via the specialization of Proposition 2, regard it as a class on \(g^\ast N\), and then pull it down to \(W'\) via the inverse of the flat pullback isomorphism

\[p^\ast:A_{k-d}(W')\rightarrow A_k(g^\ast N)\]

of \(p:g^\ast N\rightarrow W'\) (that is, by intersecting with the zero section of \(g^\ast N\)) to obtain a class on \(W'\).

Definition 3 In the situation above, we define the refined Gysin morphism of \(i\),

\[i^!:A_k(X')\rightarrow A_{k-d}(W')\]

as the composition of the specialization to \(W'\hookrightarrow X'\) and the inverse of the flat pullback along \(g^\ast N\rightarrow W'\).

The essence of the definition is that the resulting class lives on the actual preimage \(W'\), while the dimension still maintains the expected value \(k-d\). In particular, even when \(W'\) is larger than the expected dimension, this operation yields a well-defined \((k-d)\)-dimensional class.

As we have seen in §Perfect Obstruction Theory, §§Zero Locus of a Vector Bundle, the prototype of a virtual class is as follows. Suppose we are given a rank \(r\) vector bundle \(E\rightarrow V\) over a smooth space, and a section \(s:V\rightarrow E\). Let \(Z(s)\) be the zero locus of this section, and let \(0_E: V\hookrightarrow E\) be the zero section of \(E\). Then the class we expect is the class formed by embedding the normal cone \(C_{Z/V}\) into \(E\vert_Z\) and then cutting it by the zero section. That is, this construction is precisely the case where, in Definition 3, we take

\[X=E,\quad X'=W=V,\quad g=s: V\rightarrow E,\quad i=0_E: V\rightarrow E\]

as the input. In this setting, in the total space of \(E\), \(\Spec_V\Sym(E^\vee)\), the zero section \(0_E\) is given precisely by the condition that all linear coordinates in the fiber direction vanish, so it is given by sending all parts of degree \(1\) or higher to \(0\) via the map \(\Sym(E^\vee)\rightarrow\mathcal{O}_V\). That is, \(0_E\) is a regular embedding of codimension \(r\), its conormal sheaf is \(E^\vee\), and its normal bundle is \((E^\vee)^\vee=E\), so that \(N=E\) and \(d=r\).

Then in this situation, since

\[W'=s^{-1}(0_E(V))=Z,\qquad g^\ast N=E\vert_Z\]

as we observed before Definition 3, \(C_{Z/V}\) embeds into \(E\vert_Z\). Concretely, just as the zero section corresponds to the algebra map sending the degree \(1\) part to \(0\), the section \(s\) corresponds, with its degree \(1\) part being \(s^\vee:E^\vee\rightarrow\mathcal{O}_V\), to an algebra map \(\Sym(E^\vee)\rightarrow\mathcal{O}_V\). Locally, if \(s=(s_1,\ldots,s_r)\), then \(s^\vee\) is \(e_i\mapsto s_i\). Since \(Z\) is the fiber product \(V\times_{s,E,0_E}V\) of the two sections, its structure sheaf is the quotient of \(\mathcal{O}_V\) by the elements \(s^\vee(e)-0=s^\vee(e)\) \((e\in E^\vee)\), and therefore the ideal sheaf of \(Z\) is \(\mathcal{I}=s^\vee(E^\vee)\), locally \((s_1,\ldots,s_r)\). From this we obtain a surjection \(E^\vee\vert_Z\rightarrow\mathcal{I}/\mathcal{I}^2\), which gives a closed embedding \(C_{Z/V}\hookrightarrow E\vert_Z\). Therefore, applying Definition 3 to a section of a vector bundle yields an operation that acts by the localized Euler class on general cycles, and allows us to treat virtual classes in the same manner. We write the refined Gysin morphism obtained in this way as

\[0^!_{E,s}:A_k(V)\rightarrow A_{k-r}(Z)\]

Since \(0_E\) is a regular embedding regardless of the smoothness of \(V\), this is defined for an arbitrary \(V\), and for \(j:Z\hookrightarrow V\),

\[j_\ast 0^!_{E,s}\gamma=c_r(E)\cap\gamma\]

holds. That is, the top Chern class formula holds after pushing forward to \(V\), whereas \(0^!_{E,s}\gamma\) itself is a class that lives more precisely on \(Z\).

Also, the following is almost immediate from the definition.

Proposition 4 For a regular embedding \(i:W\hookrightarrow X\) of codimension \(d\) and \(N=N_{W/X}\), the following hold:

1. If \(X\) is a smooth variety of dimension \(n\), then \(i^![X]=[W]\in A_{n-d}(W)\).
2. For all \(\beta\in A_\ast(W)\), we have \(i^!i_\ast\beta=c_d(N)\cap\beta\).

Excess intersection and functoriality

Since \(i^!\) in Definition 3 goes from \(A_k(X')\) to \(A_{k-d}(W')\), the result, regardless of the actual dimension of \(W'\), is always a class of dimension \(k-d\). When \(W'\) is larger than this, especially when \(W'\hookrightarrow X'\) is a regular embedding of codimension strictly less than \(d\), the refined Gysin morphism can be computed concretely as follows.

Proposition 5 (Excess intersection formula) In the setting of Definition 3, suppose that \(i':W'\hookrightarrow X'\) is also a regular embedding of codimension \(d'\) with normal bundle \(N'\). Then \(N'\) becomes a subbundle of \(g^\ast N\), and the quotient

\[E:=g^\ast N/N'\]

is a vector bundle on \(W'\) of rank \(e=d-d'\). Then for all \(\alpha\in A_k(X')\),

\[i^!(\alpha)=c_e(E)\cap(i')^!(\alpha)\]

holds.

The rank \(e\) bundle \(E\) is called the excess bundle. Thus, the refined Gysin morphism automatically performs the correction of multiplying the actual intersection \((i')^!\alpha\) by the Euler class of the excess bundle, and if \(W'\) has exactly the expected dimension, then \(d'=d\) and \(e=0\), so without any correction, \(i^!\alpha=(i')^!\alpha\). In the extreme case where \(g=i\), we have \(i'=\id_W\), so \(E=N\), which reduces to the self-intersection formula of Proposition 4.

Example 6 If we take \(X=\mathbb{P}^2\) and a line \(W=L\), then \(i:L\hookrightarrow\mathbb{P}^2\) is a codimension \(1\) regular embedding, and since the ideal sheaf of \(L\) is \(\mathcal{O}_{\mathbb{P}^2}(-1)\), the conormal sheaf is \(\mathcal{O}_L(-1)\) and \(N_{L/\mathbb{P}^2}=\mathcal{O}_L(1)\). Two phenomena appear in this setting.

1. Suppose a smooth conic \(Q\) is tangent to \(L\) at a point \(p\), and let \(g:Q\hookrightarrow\mathbb{P}^2\) be its embedding. Then \(W'=g^{-1}(L)\) is, as a scheme, the length \(2\) divisor \(2p\) on \(Q\). Since \(W'\) is a Cartier divisor in \(Q\), that is, a codimension \(1\) regular embedding, we have \(d'=d=1\) and \(e=0\), and by Proposition 5 (Excess intersection formula) and Proposition 4,

   \[i^![Q]=(i')^![Q]=[W']=2[p]\]

   holds. That is, the result of Definition 3 is the class of the actual intersection \(W'\) itself, and its degree

   \[\deg i^![Q]=L\cdot Q=2\]

   coincides with the Bézout number. ([Algebraic Varieties] §Bézout’s Theorem) Even though the two curves are not transversal at \(p\), the refined Gysin morphism yields a well-defined class concentrating the intersection number \(2\) at the single tangent point.

2. This time, let \(X'=L\) itself with \(g=i\), intersecting \(L\) with itself. Then \(W'=g^{-1}(L)=L\) in its entirety, which has dimension \(1\), strictly larger than the expected dimension \(0\). Since \(i'=\id_L\), we have \(d'=0\), \(N'=0\), and the excess bundle becomes

   \[E=g^\ast N_{L/\mathbb{P}^2}=\mathcal{O}_L(1),\qquad e=1\]

   . Applying Proposition 5 (Excess intersection formula),

   \[i^![L]=c_1(\mathcal{O}_L(1))\cap[L]=[\mathrm{pt}]\]

   yields the self-intersection number \(L\cdot L=1\). The excess dimension of \(1\) is precisely absorbed into the Euler class of the excess bundle.

In differential topology, in both cases one could slightly perturb one of them to make them transversal and then count the intersection points. That is, in the first case one slightly pushes \(L\) so that it meets \(Q\) at two distinct points, and in the second case one moves \(L\) to another nearby line. In algebraic geometry, we do not have the freedom to move things around so freely; the intuition provided by Example 6 is that the refined Gysin morphism yields the same answer on the actual intersection without moving anything at all.

Meanwhile, for this operation to be useful, it must also interact well with morphisms. Compatibility with proper pushforward was already used when we derived fiber integration in [Stacks] §Integration on Deligne–Mumford Stacks, §§Integration along fibers, and what remains to be added is the fact that different Gysin morphisms commute regardless of order.

Proposition 7 For a regular embedding \(i:W\hookrightarrow X\), the following hold:

1. If \(g:X'\rightarrow X\) is flat, then \(i^!\) commutes with flat pullback.
2. If \(j:V\hookrightarrow Y\) is another regular embedding, then \(i^!j^!=j^!i^!\) on the appropriate fiber product.

The second commutativity ensures that when intersecting several conditions in sequence, the order does not affect the result; from this comes the associativity of the operation intersecting two classes via the diagonal below.

Relative Intersection of Virtual Classes

Thus far, we have defined Gysin morphisms on schemes, but what we are actually interested in is the Gysin morphism on the moduli of stable maps, namely on Deligne–Mumford stacks. However, not much new is particularly required for this, because we have already established in [Stacks] §Integration on Deligne–Mumford Stacks that Chow groups with \(\mathbb{Q}\)-coefficients and proper pushforwards exist, and that refined Gysin pullbacks are constructed on them and are compatible with base change. Therefore, all formulas remain valid even if we substitute moduli stacks into the places of \(X,X',W,W'\) in the preceding sections. Furthermore, since the above construction uses the fact that \(i\) is an embedding only étale-locally, refined Gysin pullbacks are defined in the exact same manner for morphisms that are unramified and étale-locally regular embeddings, that is, regular local immersions. The reason this generalization is strictly necessary is that if a stack \(S\) has nontrivial automorphisms, the diagonal \(\Delta_S:S\rightarrow S\times S\) is not an embedding, yet it is always a regular local immersion as long as \(S\) is smooth.

Now let us return to the setting of the introduction. Let \(S\) be a smooth Deligne–Mumford stack of pure dimension \(s\), let \(M_1,M_2\) be Deligne–Mumford stacks, and let \(f_i:M_i\rightarrow S\) be morphisms. As seen before Definition 3,

\[M_1\times_SM_2=(M_1\times M_2)\times_{S\times S}S\]

and since \(\Delta_S\) is a codimension \(s\) regular local immersion, we can apply Definition 3 with \(i=\Delta_S\), \(X'=M_1\times M_2\), and \(g=f_1\times f_2\). Here, \(W'=M_1\times_SM_2\).

Proposition 8 In the situation above, for classes \(\gamma_1\in A_{k_1}(M_1)\) and \(\gamma_2\in A_{k_2}(M_2)\), the class

\[\Delta_S^!(\gamma_1\times\gamma_2)\in A_{k_1+k_2-s}(M_1\times_SM_2)\]

is well-defined. Here \(\gamma_1\times\gamma_2\in A_{k_1+k_2}(M_1\times M_2)\) is the exterior product.

This proposition requires the base \(S\) to be smooth, but the moduli of stable maps is not smooth in general. Smooth bases used in practice include the target \(X\) itself when gluing two stable maps at marked points, and the genus \(0\) moduli for a convex target \(\mathbb{P}^N\). (§Moduli Space of Stable Maps, ⁋Proposition 5)

The exterior product \(\gamma_1\times\gamma_2\) is a class of dimension \(k_1+k_2\), and intersecting it along the Gysin morphism defined by the codimension \(s\) regular local immersion \(\Delta_S\) lowers the dimension by \(s\), resulting in dimension \(k_1+k_2-s\). Intuitively, \(\Delta_S^!(\gamma_1\times\gamma_2)\) intersects the two classes along the locus where \(M_1\) and \(M_2\) take the same value in \(S\). By the commutativity in Proposition 7, one can show that this operation is also associative for three or more factors. A familiar example to us is when \(M_1,M_2\) are both closed substacks of \(S\) and \(\gamma_i\) are their fundamental classes, in which case \(\Delta_S^!(\gamma_1\times\gamma_2)\) is the refined intersection class in [Stacks] §Integration on Deligne–Mumford Stacks, ⁋Proposition 9 (Diagonal intersection formula).

Now let us write out what \(\Delta_S^!(\gamma_1\times\gamma_2)\) actually computes. Before Definition 3, we verified that the following diagram is cartesian:

That is, this is the case where

\[X=S\times S,\qquad W=S,\qquad X'=M_1\times M_2,\qquad W'=M_1\times_SM_2,\qquad i=\Delta_S, \qquad g=f_1\times f_2\]

in Definition 3. To compute this, if we first consider the ideal sheaf \(\mathcal{I}\) of the diagonal on an étale local model, the conormal sheaf \(\mathcal{I}/\mathcal{I}^2\) is isomorphic to \(\Omega_S\), so the normal bundle of \(\Delta_S\) is the tangent bundle \(T_S\). Furthermore, letting \(f\) be the morphism \(f_1\circ\pr_1=f_2\circ\pr_2\) from \(W'=M_1\times_S M_2\) to \(S\), from this computation we have \(g^\ast N=f^\ast T_S\), and therefore the normal cone \(C_{W'/M_1\times M_2}\) lies inside this vector bundle \(f^\ast T_S\). (Discussion right before Definition 3) Hence \(\Delta_S^!(\gamma_1\times\gamma_2)\) is obtained by sending \(\gamma_1\times\gamma_2\) via specialization to a class in \(A_{k_1+k_2}(C_{W'/M_1\times M_2})\), viewing it as a class inside \(f^\ast T_S\), and intersecting it with the zero section to push it down to \(A_{k_1+k_2-s}(W')\).

Locally, this can be viewed as follows. If we take an étale local model of \(S\) in a neighborhood of a point to be a quotient \([U/G]\) for a smooth scheme \(U\) and a finite group \(G\), then the étale local model of \(S\times S\) is \([U\times U/G\times G]\), and on it the image of \(\Delta_S\) is given by the union of graphs \(\{(u,hu)\}\subseteq U\times U\) for each \(h\in G\). Now, choosing one of these, if we take smooth coordinates of \(U\) to be \(\t_1,\ldots,\t_s\), then the graph \(\Gamma_h=\{(u_1,u_2):u_2=hu_1\}\) is the space defined locally by \(s\) equations \(\t_a(u_2)-\t_a(hu_1)=0\), and since \(\Gamma_h\cong U\) is smooth, this is a regular sequence. Pulling these back to the corresponding étale local model of \(M_1\times M_2\), we obtain \(s\) functions

\[\sigma_a(u,v)=\t_a(f_2(v))-\t_a(hf_1(u)),\qquad 1\leq a\leq s\]

whose common zero locus is the étale local model of \(W'\) corresponding to the chosen \(h\). That is, locally \(W'\) is the zero locus of a section \(\sigma=(\sigma_1,\ldots,\sigma_s)\) of the trivial rank \(s\) vector bundle over \(M_1\times M_2\). The key fact is that restricting this trivial bundle to \(W'\) yields \(f^\ast T_S\); indeed, since the ideal \(\mathcal{I}\) of \(\Gamma_h\) is generated by the regular sequence \(\t_a(u_2)-\t_a(hu_1)\), the conormal sheaf \(\mathcal{I}/\mathcal{I}^2\) is a free module having their classes as a basis, and under the isomorphism \(\mathcal{I}/\mathcal{I}^2\cong\Omega_S\), the class of the coordinate difference \(\t_a(u_2)-\t_a(hu_1)\) of the two points corresponds to its first-order approximation \(\dd{\t_a}\). In other words, if we trivialize the normal bundle \(T_S\) by the basis \(\partial/\partial\t_a\) dual to \(\dd{\t_a}\), the \(a\)-th component of \(\sigma\) in this trivialization is precisely \(\sigma_a\).

From the discussion above, locally \(\Delta_S^!\) is the Gysin map of the zero section with respect to the section \(\sigma\). If the equations \(\sigma_1,\ldots,\sigma_s\) defining the fiber product \(W'=M_1\times_SM_2\) form a regular sequence, then \(W'\) is a codimension \(s\) regular embedding in \(M_1\times M_2\), and the normal cone becomes the vector bundle \(f^\ast T_S\) itself, so that \(\Delta_S^![M_1\times M_2]\) would be the fundamental class of the zero locus \(W'\). The issue is that in general there is no guarantee that the equations pulled back in this way form a regular sequence; when this assumption fails, the normal cone might not be a vector bundle and the dimension of \(W'\) may be larger than expected, in which case the above construction provides a class of the expected dimension instead of the fundamental class of the actual zero locus.

Proposition 9 In the setting of Proposition 8, if \(M_2\) is pure-dimensional and \(f_2:M_2\to S\) is a regular embedding of codimension \(c\), then for every \(\gamma_1\in A_k(M_1)\),

\[\Delta_S^!(\gamma_1\times[M_2])=f_2^!\gamma_1\in A_{k-c}(M_1\times_SM_2)\]

holds. Here, the right-hand side is the refined Gysin morphism along \(f_1:M_1\to S\) of \(f_2\).

Proof

The map \(\id\times f_2:M_1\times M_2\to M_1\times S\) is a regular embedding of codimension \(c\), and since \(S\) is smooth, Proposition 4 gives \(f_2^![S]=[M_2]\). Therefore, \(\gamma_1\times[M_2]=(\id\times f_2)^!(\gamma_1\times[S])\). On the other hand, pulling back \(\Delta_S\) to \(M_1\times S\) yields the graph of \(f_1\), \(M_1\to M_1\times S\); since \(S\) is smooth, this is a regular local immersion of codimension \(s\), and since there is no excess, we have \(\Delta_S^!(\gamma_1\times[S])=\gamma_1\). Now, using Proposition 7 to interchange the order of \(\Delta_S^!\) and \((\id\times f_2)^!\), we obtain

\[\Delta_S^!(\gamma_1\times[M_2])=\Delta_S^!(\id\times f_2)^!(\gamma_1\times[S])=f_2^!\Delta_S^!(\gamma_1\times[S])=f_2^!\gamma_1\]

In general, since \(f_2\) is not a regular embedding, we cannot directly use the defining equations of \(M_2\) to cut \(M_1\); thus, as long as \(S\) is smooth, we can think of cutting instead by the diagonal \(\Delta_S\), which is always a regular local immersion. On the other hand, Proposition 9 states that if \(f_2:M_2\rightarrow S\) is already a regular embedding, there is no need to pass through the diagonal, and using the defining equations of \(M_2\) to directly cut \(M_1\) via \(f_2^!\gamma_1\) yields the same class.

Meanwhile, substituting the virtual classes \(\gamma_i=[M_i]^\vir\) into Proposition 8,

\[\Delta_S^!\bigl([M_1]^\vir\times[M_2]^\vir\bigr)\in A_\ast(M_1\times_SM_2)\]

defines the expected-dimensional class on the fiber product. We may also take a virtual class on one factor and an ordinary fundamental class on the other. However, if the fiber product is itself a moduli space and possesses its own POT and VFC, whether that VFC coincides with this class is a separate issue. For instance, comparing the VFC of the moduli space obtained by gluing two stable maps with such an intersection of two VFCs requires separately verifying the compatibility of the two obstruction theories.

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References

[V] A. Vistoli, Intersection theory on algebraic stacks and on their moduli spaces, Invent. Math. 97 (1989), 613–670.