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1.1 Log geometry In this paper we propose a solution to some of these problems, extending an idea in section 9 of [HPS17] and applying it to the construction of [Her19]. The spaces these authors study have a naturally defined log structure. The theory of log geometry was introduced by Kato, based on ideas of Deligne and Illusie, in [Igu89]. It studies schemes X equipped with a log structure consisting of a sheaf of monoids MX , together with a morphism αX : MX → OX which we require ∗ to be an isomorphism on the submonoid OX . Associated to this is another sheaf, the −1 ∗ characteristic sheaf, the quotient MX /α OX . There are notions of log flatness, log étaleness and log smoothness satisfying similar properties to the classical notion. We focus on the category of fs log schemes, whose definition can be found in [Igu89]. In the introduction we will include the phrase fs to emphasise the difference from the classical construction, in the body however we will leave it implicit. We will conflate Spec k with ∼ ∗ the trivial log point, the log scheme with X = Spec k, MX = k and αX the inclusion. In nice cases, for instance on the of stable curves, this structure spec- ifies a collection of normal crossings divisors. Then section9of[HPS17], and earlier work of Holmes in [Hol], suggest looking at only log blow-ups, weighted blow ups of

2 intersections of these divisors. In nice situations (when X is log smooth over the triv- ial log point) this presents a much smaller category of blow-ups but still allows the singularities to be resolved and the small b-Chow construction to be applied. This construction however still only applies for log smooth schemes. In this paper we extend the above construction to all log schemes by constructing Gysin maps even when the underlying schemes are not smooth but so long as the log structure is locally ∗ Nr free, locally isomorphic to OX ⊕ . This produces the following theorem, a translation of the above list, whose proofs are unfortunately necessarily scattered throughout this paper:

Theorem 1.1 (Constructions 2.15 and 2.16, Example 2.26 and Corollary 2.18 respec- † tively). There is a A∗ : (fs) Log Sch → Gr Ab from the category of (fs) log schemes of finite type over Spec k to the category of graded Abelian groups.

(i) If f : X → Y is a proper map then there is a well-defined pushforward morphism † † f∗ : Ak(X) → Ak(Y ). (ii) If f : X → Y is a log flat map of relative dimension d then there is a well-defined ∗ † † pullback morphism f : Ak(Y ) → Ak+d(X). (iii) If f : X → Y is a morphism of relative dimension d = dim X − dim Y with X and Y both log smooth over Spec k then there is a well-defined Gysin pullback ! † † morphism f : Ak(Y ) → Ak+d(X). (iv) If f : X → Y is a strict morphism of log schemes with X → Y a of rank d then f ∗ is an isomorphism.

(v) If i : D → X is the strict inclusion of a closed subscheme and j : U → X the strict inclusion of the open complement then the following is an , called the excision sequence:

∗ † i∗ † j † Ak(D) −→ Ak(X) −→ Ak(U) → 0

These groups agree with the idea given in section 9 of [HPS17] for schemes log smooth over the trivial log point. As far as the definition exists one could view this as a Chow theory for the valuativisation of [Kat89]. From these one can construct a theory of bivariant classes, following [Ful98]. Let f : X → Y be a log morphism, then the k f group of k dimensional bivariant classes A† (X −→ Y ) consists of compatible collections † † of maps A∗(Z) → A∗−k(Z ×Y X) for all morphisms Z → Y , with Z ×Y X the (fs) log fibre product. We will show the following Poincaré duality statements, firstly one can identify the theory relative to a point with cycle classes:

∗ Theorem 1.2 (Theorem 2.24). Let X be a log scheme. Then the group A† (X → Spec k) † is canonically isomorphic to A∗(X). and secondly, one has a duality pairing:

3 Theorem 1.3 (Corollary 2.23). Let g : Y → Z be a log smooth morphism of relative ∗ dimension k and f : X → Y any morphism. Then the composition map A† (X → Y ) → ∗−k A† (X → Z) is an isomorphism. One important application of this is to describe the bivariant theory relative to the

Artin fan of a log scheme X. The Artin fan AX of a log scheme X was introduced in [ACMW14] Section 3 and has the property that it is log étale over the trivial log point, and has a strict morphism X → AX . Since it is log étale over a point we can apply the above theorem to deduce the following corollary:

Corollary 1.4 (Example 2.27). There are canonical isomorphisms

† ∼ ∗ ∗ ∼ ∗ A∗(X) = A† (X →AX ) and A†(AX → Spec k) = A† (AX →AX )

† This gives the group A∗(X) the structure of a module over the Chow cohomology ring ∗ A†(AX →AX ).

1.2 Log normal cone constructions

∗ The ring A†(AX → AX ) contains combinatorial data about the log structure on X, and correctly described should be a generalisation of the polytope algebra of McMullen, following [FS97]. Elements of A∗(X → Y ) are constructed via finding an embedding of the nor- mal cone CX/Y into a vector bundle stack EX using tools developed by Manolache ℓ in [Man12a] and [Man12b]. The log normal cone C•/• was constructed by Herr in [Her19] and one can view it as a log scheme over X by insisting that map to X be strict. This is the natural extension of the normal cone to log geometry constructed from the log cotangent complex L•/•, described by [Ols03]. Unfortunately the log cotangent com- plex is not as well behaved as the cotangent complex, this presents itself here as the fact that if π : X˜ → X is a log étale morphism and X → Y is any morphism then π∗(Cℓ ) 6∼ Cℓ , it is only a substack. X/Y = X/Y˜ One can view this as the fact that π is not itself flat and so poorly suited to taking fibre products. Correcting this on the level of log schemes requires a foundation of derived log schemes and Chow groups for these, a formidable challenge. Instead one could correct it on the level of cycles. We do this using the Gysin maps constructed above, combined with [Her19] Remark 2.14.

Theorem 1.5 (Theorem 3.2). Let π : X˜ → X be a log étale map with X˜ and X locally free, and f : X → Y a log morphism with Y log smooth over the trivial log point. Then ℓ ∗ ℓ ℓ ! ℓ let i : CX/Y˜ → π CX/Y be the inclusion. There is an equality i∗[CX/Y˜ ]= π [CX/Y ]. This produces the following corollary:

Corollary 1.6 (Corollary 3.3). The virtual fundamental class construction applied to † † CX/Y produces a well defined class in A∗(X).

4 We use this work finally to both recast Proposition 3.6.2 of [Ran19a] in terms of these Gysin maps and to extend the construction of log Gysin maps found in Section 3 of [Her19] substantially.

1.3 Acknowledgements The initial part of this work was carried out as a PhD student in Cambridge under Mark Gross, and continued both as a Postdoctoral Researcher at NCTS Taipei and a Visiting Assistant Proffesor in Boston College. I must thank Mark Gross for introduc- ing me to log geometry and teaching me the details of the subject. The theory was streamlined through discussion between myself and Leo Herr, and I am greatly indebted to him. I also owe a debt of gratitude to David Holmes for discussing these ideas with me in Leiden, and Michel van Garrel for organising a wonderful conference at MFO Oberwolfach where I discussed some of these ideas with him, Dhruv Ranganathan and Jonathan Wise. Navid Nabijou has been a constant source of inspiration. Finally many intimate details of log structures were unveiled to me by Dan Abramovich.

2 The cycle group

Given a log scheme X we can at a first attempt just take the Chow group of X. Given a modification X˜ → X with both X˜ and X smooth we can follow Shokurov in [Sho96] and Holmes-Pixton-Schmidt in [HPS17] and construct compatible Gysin maps by taking Gysin pullbacks. We begin with a motivating lemma from [NIZ06]:

Lemma 2.1. Let X be a log scheme log smooth over the trivial log point. Then X is smooth if and only if the stalks of the characteristic sheaf are isomorphic to Nr.

Proof. Suppose that X is smooth. Then we have a chart for the log structure over an étale open subset U ⊂ X with at least one point mapping to the origin of Spec k[P ]:

U ch ch

Spec k[P ] Spec k[P ]

Spec k Spec k by assumption U is smooth and the map ch : U → Spec k[P ] is smooth. Taking the cotangent exact sequence:

∗ 0 → ch ΩSpec k[P ]/ Spec k → ΩU/ Spec k → ΩU/ Spec k[P ] → 0 we see that ΩSpec k[P ]/ Spec k is free in a neighbourhood of the origin. But this implies that P ∼= Nr is free.

5 Conversely if the stalks of the characteristic sheaf of X are free then we have an étale cover of X by smooth schemes. Therefore X is smooth.

This will be enough to construct the desired Gysin maps, having locally free stalks as above acts to specify compatible maps to smooth schemes. Let us notate such schemes.

Definition 2.2. Let X be a log scheme such that at every point the stalk of the characteristic sheaf is isomorphic to Nr, where r may vary from point to point. We call such an X a locally free log scheme.

As a word of warning, in the sense of cones such schemes would be called “smooth”, but the phrase “smooth log scheme” is evidently indicative of something else, while free cones also includes simplicial cones. I decided to stick to this direct description of the monoids. Our first step is to construct Gysin maps for certain morphisms of log schemes, locally free blow-ups morphisms of [Kat99] exposited in the introduction.

Definition 2.3. A log blow-up π : X˜ → X is the morphism induced by a sheaf of log ideals on X. When it is finite it is degree one, in particular if X is log smooth over the trivial log point π is birational. We call a log blow-up π : X˜ → X with X˜ and X locally free a locally free log blow-up. A log modification π : X˜ → X is a log morphism for which there exists a log blow-up Z → X˜ such that the morphism Z → X is also a log blow-up.

By Theorem 3.16 of [Kat99] any morphism f : X → Y may be lifted to an integral morphism f˜ : X˜ → Y˜ after passing to log modifications X˜ → X and Y˜ → Y . In fact an inspection of the proof shows that this can be done with Y˜ → Y a log blow-up. Studying only log blow-ups is sufficient for constructing colimits over all log mod- ifications, they are by definition cofinal within the system of log modifications. Since they are degree one when finite they are good analogues of the birational maps used in the b-Chow definition.

Construction 2.4. Let π : X˜ → X be a locally free log blow-up. We claim that this has a canonical perfect obstruction theory of relative dimension 0. Let U → X be an étale open set around x ∈ X on which we have a chart for the log structure. After potentially restricting U we have that this chart is of the form U → An, where n is the n n rank of MX at x. The morphism π is the fibre over a blow-up πAn : BlI (A ) → A , n with BlI (A ) also smooth. Therefore πAn is lci and has a canonical rank zero perfect obstruction theory. We now glue these local models to an étale atlas for the obstruction theory. To be precise suppose that U is an étale open set with a chart An and V is a smaller étale open set with a chart Am. There is an induced generisation map on charts of the form φ : An → Am, given by projection to a subspace, together with an induced map on n m the blow ups, φBl : BlI (A ) → BlI (A ). But over the image of V the map φBl is the pullback of φ. Since φ is smooth the two obstruction theories glue canonically.

6 This therefore defines a vector bundle stack on the étale topology of X. I suspect that this descends to the Zariski topology, but in any case by [CL11] Theorem/Definition 3.7 there is a well defined theory of étale perfect obstruction theories. We write the associated Gysin map π! which is in degree zero, it preserves dimension. Remark 2.5. When X˜ and X are locally free and log smooth over the trivial log point then X˜ and X are also smooth. Therefore π induces a Gysin map agreeing with the above construction. Thus these are an extension of the groups constructed in [HPS17], restricted only to log blow-ups. We need to check that this construction is functorial under further blow-ups. Lemma 2.6. Let π′ : X′′ → X′ and π : X′ → X be locally free log blow-ups. Then (π′ · π)! = π′! · π!. Proof. After passing to étale open sets we may assume that we have charts for the log n n n structures, BlJ A → BlI A → A . Our claim is equivalent to the compatibility of the obstruction theories for this sequence of maps, and they are evidently compatible.

We can then define the Chow groups analogously to [HPS17] as being a colimit over all log blow-ups.

Definition 2.7. Let X be a log scheme. The category Fr BlX is the (non-empty) category whose objects are log blow-ups π : X˜ → X with X˜ locally free and whose morphisms are further log blow-ups over X. The above construction gives a contravari- † ant functor A∗(−): Fr BlX → Gr Ab. The log Chow groups of X, A∗(X), are the colimit over A∗(−). Remark 2.8. As in [HPS17] the category of log blow-ups is filtered, and these groups may be described in terms of cycles on some blow-up, modulo identification under (Gysin) pullback. Using 2.12 and 2.13 there is a canonical map from this to the inverse limit over all log blow-ups, with transition maps given by pushforward. This is the analogue of the embedding of the small b-Chow ring into the large b-Chow group. This inverse limit in the context of logarithmic geometry was suggested as an object of study by [Ran19a] in Proposition 3.6.2. We will see that relations in our groups imply the relations there. Calculating these colimits is at least as hard as calculating Chow groups. There are two examples where we can directly describe the groups, for toric varieties described in [FS97] Theorem 4.2 and the discussion before it, applying Poincaré duality and cofinality of smooth blow-ups to move from the Chow cohomology groups used there to the Chow groups used here. The authors describe these groups as the polytope algebra of McMullen, [McM93]. The second case is when the log structure on X is Deligne–

Faltings rank one, in the notation of [Che10]. Here the category Fr BlX is the singleton X, with only the identity map. Let us calculate a non-trivial example of these Gysin maps and see that there are some subtleties:

7 2 Example 2.9. Let Spec kN2 be the origin in A with its induced log structure. The 1 1 blow up at the origin induces a locally free log blow-up P → Spec kN2 where P has globally non-trivial log structure. The only possible class in dimension zero is again a multiple of a point. Calculating the normal cone we see that we are intersecting two copies of the zero section inside O(1) on P1, so the multiplicity is 1.

There is a problem with this, unless X is log flat over the trivial log point formation of the fundamental class does not commute with this pullback. Under sufficiently nice conditions however this does commute. Our first step is to have some notation for when all the log blow-ups of a given log scheme X have the same dimension.

Definition 2.10. Let X be a log scheme such that for the generic point of every log strata ζ the sum rank Mζ − codim ζ is constant on each irreducible component. This condition is stable under log flat morphisms and we call such a log scheme of pure log dimension. The negation of this is not stable under log flat morphisms. If in addition every generic point of a component of X has log structure 0 or N then we say that X is maximal pure log dimension dim X. For every further log blow-up of X, X˜ → X we have that dim X˜ = dim X. We say that a locally free log blow-up π : X˜ → X is compatible with the fundamental class (of X) if π![X] = [X˜]. We say that X has compatible fundamental class if X is locally free and every locally free blow-up is compatible with the fundamental class.

The idea is that if X has maximal pure log dimension then the Gysin pullback of [X] to any log blow-up is again top dimensional and so possibly equal to the fundamental class. The following lemma proves that this intuition is indeed correct.

Lemma 2.11. Let X be a locally free log scheme of maximal pure log dimension. Then X has compatible fundamental class.

Proof. Let π : X˜ → X be a locally free blow-up. Then as stated above dim X˜ = dim X, and so π![X] is a top dimensional class on X˜. We are free to suppose that X˜ is the ! union X˜1,... X˜n, so that π [X] = P ni[X˜i] and the claim is that ni = mult X˜i where mult denotes the length at the generic point. Let us fix one of the components of X˜,

X˜1 Since we are dealing with top dimensional classes this is local on X, so we may move to étale open sets U˜ and U of X˜ and X such that U˜ contains the generic point of X˜1 and where we have charts fitting into the following diagram:

U˜ U

n n BlI (A ) A

∗ There is an associated embedding of normal cones Cπ → chU˜ NπI , which we wish to An show is an isomorphism. There is an affine bundle U → U with a relatively smooth An An blow up π : BlI ( U ) → U which is smooth over U given by pulling back charts. Say

8 ∗ ∗ that U is cut out as the zero section by x1,...xn, then π x1,...π xn cut out U˜. These ˜ An remain regular since U is n inside BlI ( U ). In particular the normal cone of U˜ → U is a bundle. ∗ We now have a sub-bundle of the same dimension as chU˜ NπI , hence these are equal and the desired class is just multi[Ui].

These log blow-ups are almost birational, there may be some components that are contracted but away from these the map is birational. A corollary is:

Corollary 2.12. Suppose that X has compatible fundamental class and π : X˜ → X a ! locally free blow-up, then π∗π ([X]) = [X].

From this lemma we can deduce another corollary (the proof is obvious in the quasi- projective case using the moving lemma and the above result, but it remains true in the general case):

Corollary 2.13. Let Y be a log scheme. Then there is a blow-up π : Y˜ → Y and a collection of closed embeddings ij : Zj → Y˜ and nj such that each Zj has ! compatible fundamental class and π [Y ]= P nji∗[Zj].

Proof. We prove this by Noetherian induction on the dimension of the images of various strata in Y . Let Y1 → Y be the barycentric subdivision of Y . This is of maximal pure log dimension and every irreducible component has generic log structure N. If Y is quasi-projective then we can apply the moving lemma [Sta20] to π!([Y ]) to get a transverse representative. We apply this idea, together with Noetherian induction, to prove the claim for all Y .

Choose an open affine of Y1, U1, with complement D1. Since U1 is affine it is ! (i) quasi-projective and we represent π [U1] by a collection of classes P niZ1 which are transverse to all the log strata. The closure of these need not be transverse, but we can refine Y1 further until they are, changing U1 by log blow-ups. Since the representatives have compatible fundamental class on U1, they retain this after a further blow-up of ! ¯(i) U1. The difference between π ([Y ]) and P niZ1 is supported over D1. Apply the same argument to these classes over an open affine U2 of D1, the fibre over U2 of any log blow-up is again quasi-projective, with complement D2 and extend the blow-ups as needed over U1. Therefore by Noetherian induction we reduce to the case where Dn is a point inside Y . Then we have a collection of classes on Pm which we want to write as being transverse to the log structure. But these are represented by a collection of hyperplanes.

2.1 Actions of morphisms We now must construct the usual paraphenalia of Chow theory, a log flat pullback, a proper pushforward, Gysin maps etc. We first demonstrate that a little care must be taken with log flat pullback and proper pushforward.

9 Example 2.14. Consider the square:

1 π P Spec kN2

π

Spec kN2 Spec kN2

1 where P → Spec kN2 is the morphism in the previous example and the other two morphisms are the identity. We compare π∗ and π! applied to the class of a point to see that in this case they must agree. In particular here the log flat pullback is not given by taking inverse image, even though all of the maps are also flat. Now consider the square:

P1 π P1

π 1

1 1 P Spec kN2 this square is Cartesian in the category of fs log schemes. We compare pushforward of the fundamental class of P1, working both ways around this square and starting in the ! 1 1 bottom left. The composite 1∗1 ([P ]) is of course [P ]. Unfortunately the composite ! 1 1 π π∗([P ]) is zero, since π∗([P ]) is already zero. In both cases the problem stems from the fact that neither class is represented by a cycle with compatible fundamental class.

Construction 2.15. Let f : X → Y be a with X and Y locally free. Suppose that we have f˜ : X˜ → Y˜ an integral lift of f, with log blow-up πY and modification πX respectively, Y˜ locally free and X¯ → X˜ a log blow-up with X¯ locally free. In diagrams we have the following picture:

ψ ˜ f˜ X¯ X X˜ Y˜ φX πX πY f XY

˜ ˜ ! There is a canonical map from A∗(X) to A∗(Y ) defined by f∗ψX˜ ∗φX , which we Y˜ denote f# . This notation already supresses various of the dependencies of this map, for instance on the choice of X¯. We first show independence from the choice of integral lift after passing to a common refinement using the Gysin maps. Any two choices of integral lift are dominated by a third, so let us assume that we have fˆ : Xˆ → Yˆ a integral lift of f˜ (which is already itself integral!), with log blow-ups ˆ ˇ ˆ πY˜ and log modifications πX˜ respectively, Y locally free, and ψXˆ : X → X a further ˇ ˇ ¯ log blow-up such that X is locally free and admits a log blow-up map φX¯ : X → X. In short we find ourselves with this diagram:

10 fˆ

ψ ˆ f˜| ˆ ˇ X ˆ ˜ ˆ Y ˆ X X X ×Y˜ Y Y ¯ φX π ¯ ˜ ˜ π ˜ X πY |X Y ψ ˜ f˜ X¯ X X˜ Y˜ φX πX πY f XY

˜ ˜ ˆ where since f is integral the underlying scheme of X ×Y˜ Y is the product of the underlying schemes. Suppose that [V ] is a class on X representing a strict embed- ding V → X where V has compatible fundamental class and we wish to show that ! Y˜ Yˆ πY˜ f# ([V ]) = f# ([V ]). After replacing X by V we may assume that X has compatible ! Y˜ Yˆ fundamental class. Now we wish to prove that πY˜ f# ([X]) = f# ([X]). First note that ˜ ˜ ˆ ˜ since f is integral the morphism πY˜ |X˜ : X ×Y˜ Y → X has an induced perfect obstruc- ! ˜ ˜ ˆ tion theory and πY˜ |X˜ ([X]) = [X ×Y˜ Y ] by the same arguments as in 2.11. Then by definition:

! Y˜ ! ˜ ! πY˜ f# ([X]) = πY˜ f∗ψX˜ ∗φX ([X]) ! ˜ ˜ = πY˜ f∗([X]) ˜ ! ˜ =(f|Yˆ )∗(πY˜ |X˜ ) ([X]) ˜ ˜ ˆ =(f|Yˆ )∗([X ×Y˜ Y ]) ˆ ˇ = f∗ψXˆ ∗([X]) Yˆ = f# ([X]) therefore this induces a well-defined map on the cycle classes on X which are represented by cycles with compatible fundamental class. ˆ ˆ ˆ ˜ Now suppose that f : X → Y is a further integral lift of fψX˜ with blow-ups πX¯ ˇ ˆ ˇ and πY˜ respectively and ψXˆ : X → X is a blow-up with X locally free and admitting ˇ ¯ a blow-up map φX¯ : X → X. In short we have the following picture:

ψ ˆ fˆ Xˇ X Xˆ Yˆ φ ¯ X ¯ ˜ πX πY ψ ˜ f˜ X¯ X X˜ Y˜ φX πX πY f XY

˜ Yˆ ! Yˆ and we wish to compare (fψX˜ )#φX ([V ]) and f# ([V ]) for some cycle class [V ] where V has compatible fundamental class. Then as before we may replace X by V . By direct ˇ ˜ Yˆ ! Yˆ comparison of classes on X we have an equality (fψX˜ )#φX ([X]) = f# ([X]). † Therefore we get a well defined map on the level of log Chow groups f∗ : A∗(X) →

11 † A∗(Y ) by representing a cycle class by a sum of cycles with compatible fundamental class. If X and Y are not locally free then we may pass to a blow-up of X and Y which are.

We can now construct a log flat pullback map.

Construction 2.16. Let f : X → Y be a log flat morphism with X and Y locally ˜ free. Let f : X˜ → Y˜ be a log flat and integral lift of f with log blow-ups πY and log ¯ ˜ ¯ modifications πX respectively, and ψX˜ : X → X a blow-up with X locally free and write φX : X¯ → X for the induced blow-up of X. As before we obtain a diagram looking as follows:

f˜ X˜ Y˜

πX πY f XY

The map f˜ is log flat and integral, hence has flat underlying morphism. Therefore there ¯ ˜∗ ! # is a canonical map A∗(Y ) → A∗(X) given by πX ∗f πY and we write this fX¯ . We note that if [V ] is a class on Y induced by a strict embedding V → Y such that both V and −1 # f V have compatible fundamental classes then f ([V ]) = [X ×Y V ]. As before we prove that this is independent of the choice of f˜, and compatible with the Gysin maps. First suppose that we had a further choice of integral lift of f˜ (which is of course already integral). Then we have a diagram like the following:

f¯ X¯ Y¯

˜ ˜ πX πY f˜ X˜ Y˜

πX πY f XY

So long as [V ] has compatible fundamental class the above remark shows that f #([V ]) is independent of the choice of integralisation. Now suppose that we have an integralisation f˜ : X˜ → Y˜ with Y˜ locally free and a ˇ ˇ ˜ ¯ ¯ ¯ further blow-up πX : X → X admitting a blow-up map φX˜ : X → X. Let f : X → Y ˜ ¯ be an integralisation of fφX˜ with Y locally free. Then we have the following diagram:

X¯ Y¯

˜ πY

φ ˜ f˜ Xˇ X X˜ Y˜ πX πY f XY

12 ! # ˜ # ! where we wish to show the equalities πX f (α)=(φX˜ f) πY (α). If α is the class of a cycle V with compatible fundamental class these hold by the above formula. ∗ † † Therefore this extends to a morphism f : A∗(Y ) → A∗(X) since every cycle is equivalent to one which has compatible fundamental class after aprropriate blow-ups and we may further refine the source to ensure that the inverse image has compatible fundamental class as well. If X and Y are not locally free then we may pass to a blow-up which is. This contruction is compatible with fibre products. Given f : X → Y a locally free ′ log flat morphism and g : V → Y any morphism then f : X ×Y V → V is also log flat, however X ×Y V need not also be locally free. Nonetheless we can choose a refinement of X ×Y V to a log scheme which is locally free, and with the induced map to V still log flat. Given a cycle on V with compatible fundamental class the pullback is still given by taking the inverse image.

Fulton constructs one more tool, Gysin maps.

Construction 2.17. Suppose that D → X is the strict inclusion of a regularly embedded subscheme. If D has compatible fundamental class then for any blow-up

πX : X˜ → X the embedding D˜ := D ×X X˜ → X˜ remains a regularly embedded sub- scheme and the induced Gysin maps are compatible. Therefore they induce a morphism † † A∗(X) → A∗(D). We call such a map a strict Gysin map. For strict closed and open embeddings these constructions simplify considerably and give us an excision sequence. Similarly for strict affine or projective bundles we can apply the same proofs as in [Ful98].

Corollary 2.18. Let D be a strict closed subscheme of X and U the complement with strict log structure. Then there is an exact sequence:

† † † A∗(D) → A∗(X) → A∗(U) → 0 If X → B is a strict rank k affine bundle then log flat pullback gives an isomorphism † † A∗(B) → A∗+k(X). Similarly if X → B is a strict rank k there is an † ∼ † i isomorphism A∗(X) = L A∗+i(B)ξ . Of course there are various relations between the different maps, functoriality etc which guarantee that we have canonical orientations. These are not immediate since both definitions mix pushforward, pullback and Gysin maps in non-trivial ways.

Theorem 2.19. These constructions are functorial and commute with one another in Cartesian squares in the fine log category restricting to proper pushforward along saturated morphisms.

Proof. Functoriality of log flat pull back follows from the description as a Cartesian product for classes with compatible fundamental class. Functoriality of proper pushfor- ward follows by functoriality of pushforward for the classical pushforward. Functoriality

13 for strict Gysin maps follows directly from the classical statement, since strict fs fibre products have underlying scheme the product of underlying schemes. To prove commutativity we suppose that we have a Cartesian diagram in the fs category:

f ′ W V

g′ g f XY where X, Y and Z are locally free and let Wˆ be a locally free resolution of W . Sup- pose that f is a strict regular embedding and g is proper or log flat. Since exactness and integrality are preserved under fs base change, strict fibre products commute with passing to underlying schemes and Gysin maps we can find compatible diagrams for ′∗ ! ′! ∗ ! ′ ′! the pullback or pushforward along g and the equality g f = f g or f g∗ = g∗f holds. ∗ Now suppose that f is proper and g log flat and we wish to show that g f∗[Z] = ′∗ ′ g f∗[Z] where Z ⊂ X has compatible fundamental class. By taking the base change along Z → X we may assume that Z = X, and after refining further we may assume that X, W and the image of X have compatible fundamental class. ˜ To calculate f∗[X] we choose an integralisation f : X˜ → Y˜ , and a locally free blow- up X˚. We would end with a class on Y˜ which we may assume is locally free. Let V˜ be a locally free blow-up of V admitting a map, g˜, to Y˜ . To calculate g˜∗ we choose an integralisation g¯ : V¯ → Y¯ . Let X¯ be the induced blow-up of X˜, and note that we are free to assume that X˚ maps to this. Write W¯ for the induced fibre product. In short we have the following diagram with Cartesian squares:

V¯ ˜ ψV

V˜ g¯

V

g˜ f¯ X¯ Y¯ ¯ ˜ πX ψY

π ˜ f˜ X˚ X X˜ Y˜ πX ψY

f XY

The composite we wish to calculate is

∗ ! ˜ ! πV˜ ∗g¯ ψY˜ f∗πX˜ ∗πX ([X])

14 By the above this is equal to

∗ ¯ ! πV˜ ∗g¯ f∗πX¯ ∗πX ([X]) and since X has compatible fundamental class this is

∗ ¯ ¯ πV˜ ∗g¯ f∗([X])

Meanwhile to calculate g′∗ we must choose an integralisation of the induced map Wˆ → X, which we may assume lives over X˚, so W˚ → X˚. This produces a class on Wˆ and we wish to push this forward to a class on V . To do so we integralise this morphism, which we may assume occurs over V˜ , so we have W˜ → V˜ and a locally free blow-up of W˜ , Wˇ . In short we have the following diagram:

ψ ˜ Wˇ W W˜ ˆ ψW

f¯′ ˜′ W¯ f V¯ ˆ ψW ¯ ψV

π ˆ W˚ W Wˆ V˜ g¯′ g¯ W V

˚g′ f¯ X¯ Y¯ ¯ πX

X˚ X˜ Y˜ πX

X

and we wish to calculate ˜′ ! ′∗ ! f∗ψW˜ ψWˆ πWˆ ∗˚g πX [X] This is equal to ′ ! ′∗ ! ¯ ¯ ψV˜ ∗f∗ψW ψWˆ πWˆ ∗˚g πX [X] ¯′ ¯ Since X and W have compatible fundamental class this is equal to ψV˜ ∗f∗[W ], or ex- panding this via functoriality: ¯′ ¯′∗ ¯ ψV˜ ∗f∗g [X] Finally the square with bars is Cartesian in the category of schemes and g¯ is flat and ∗ so we can commute f¯∗ and g¯ . Therefore the two expressions are equal.

15 We recall the Artin fan construction of [ACMW14] and [AW18]. This associates to a log scheme X a canonical log Artin stack AX such that there is a strict map X →AX and AX is log étale over the trivial log point. A scheme X is log flat over Spec k if and only if it is flat over AX , and similarly for log étale or log smooth. So now suppose that X is log flat over the trivial log point. Then we can pullback classes from AX via flat pullback. Given a log strata of X, XM , of codimension n there is a canonical subscheme in AX of codimension n, and flat pullback maps this class with [XM ]. When X is a toric variety with its toric log structure the Chow groups are generated by the closure of torus orbits and hence this pullback is in fact an isomorphism.

2.2 Bivariant classes and Poincaré duality The most useful part of the theory is the study of bivariant classes, which we are now able to introduce:

Definition 2.20. We define the log bivariant theory of ψ : X → Y of relative dimension k k to be the group A† (f : X → Y ) generated by collections of morphisms fφ for every † † map φ : Z → Y mapping A∗(Z) → A∗−k(X ×Y Z) and commuting with saturated proper pushforward, log flat pullback and all strict Gysin maps.

For instance any Cartier divisor D with compatible fundamental class on any blow- up of X induces an element in the log Chow cohomology A1(D → X). Similarly a strict map between schemes log smooth over the trivial log point can be refined to one between log schemes with smooth underlying scheme, and the induced Gysin map commutes with further refinements. Once we have a version of Poincaré duality we will be able to drop the strictness assumption.

Definition 2.21. The final piece of data that we need is a product map for cycles. Given cycles α and β on X and Y respectively we may lift these to actual closed subschemes of X and Y with X and Y locally free. Then the product X ×Spec k Y is again locally free, and we have an associated cycle α × β. These depend on α and β up to rational equivalence and is stable under Gysin pullback once α and β have compatible fundamental class.

Remark 2.22. In [HPS17] and [Ran19b] much of the hard work deals with the fact that intersection products do not commute with pushforward. Given a refinement π : X˜ → X and cycles α and β on X˜ there is not an equality π∗(α ∩ β)= π∗α ∩ π∗β.

This does not occur here, the map π∗ sends the class α ∩ β on X˜ × X˜ to the class α ∩ β on X × X, α to α on X and β to β, as colimits the groups remember the log schemes on which the cycles live, at least up to further blow-ups.

A certain amount of algebra copies across from the classical case. In particular we get the following version of Poincaré Duality for these groups.

16 Corollary 2.23. Let f : X → Y be an arbitrary log morphism and g : Y → Z be a log ∗ smooth morphism of relative dimension d. Then the composition map A† (f : X → Y ) ∗−d ∗ to A† (gf : X → Z) is an isomorphism with inverse Γf · g (−). The distinguishing feature of classical Chow groups is the presence of an isomorphism between the bivariant theory of the structure map to a point and the cycle groups. We will prove that, although not all log schemes are log flat over the trivial log point, this result continues to hold by embedding X into the enveloping vector bundle E.

Theorem 2.24. Let X be a log scheme. Then there is a canonical isomorphism −∗ † A† (X → Spec k) → A∗(X). Proof. For any locally free log scheme we may embed V strictly as the zero section of a vector bundle E defined by the N faces of the log structure, together with sections ∼ defined up to an action of the étale fundamental group. From the local charts E|U = U × An → An where the identification is the one induced by the sections we see that E is in fact log flat over the trivial log point. ∗ Let f be a bivariant class in A† (X → Spec k) and let α be f(Spec k), the corre- sponding evaluation against a fundamental class. Then let g : V → Spec k be a log scheme with V locally free, we have the following diagram:

V × X E × X X

V0 E Spec k

Since the horizontal rows are strict regular embeddings and flat morphisms the whole square has commuting Gysin maps. In particular we see that the value of α([V ]) is 0!([E] × α). Therefore this evaluation map was an isomorphism.

Remark 2.25. There is no such analogous construction over Spec k†, and in general −∗ † −∗ there is not a natural map A∗(X) → A (X → Spec k ) splitting the one A (X → † Spec k ) → A∗(X). The fundamental reason is that if X and Y are locally free there is no reason that X ×Spec k† Y need be locally free. By [BGS95] and applied in [BGv19b] we can construct the Chow cohomology group of the central fibre of a strict semi-stable degeneration. This allows one to intersect with classes occuring as limits of families from the general fibre. In our setting this can be interpreted by showing that after some log blow-ups the limits of such families can be represented by cycles on the central fibre which are log flat over the standard log point. These have a well defined log flat pullback and pulling back along a strict resolution of the diagonal gives the desired product.

We can now apply this corollary to obtain several useful examples.

Example 2.26. Let X → Y be a morphism with Y log smooth over the trivial log point, suppose that X is of pure log dimension dX , Y is of pure log dimension dY , and

17 d write d = dY − dX . Then there is a Gysin pullback map in A† (X → Y ) corresponding to the fundamental class in A† (X). dX

Example 2.27. We apply the above Poincaré duality result to the map X →AX for ∗ ∼ ∗ a log scheme X to obtain canonical isomorphism A†(X → Spec k) = A† (X →AX ). † ∗ This makes A∗(X) into a module over the ring A†(AX ), which is in turn isomorphic † to A−∗(AX ). This is equivalent to the statement that one can intersect cycles with the closure of locally free log strata, since these strata are intersections of Cartier divisors. Furthermore when X is a toric variety with the toric log structure the toric divisors † † † on blow-ups of X span both A∗(X) and A∗(AX ). Therefore the ring A∗(AX ) is also isomorphic to the McMullen polytope algebra Π. I do not know of a construction of a polytope algebra for a general Artin fan.

Extending this to the relative setting we obtain:

Example 2.28. Let f : X → Y be a morphism of log stacks. The relative Artin fan AX/Y of [ACMW14] admits a strict log étale map from Y , and hence there is an ∗ ∼ ∗ isomorphism A†(X → Y ) = A† (X →AX/Y ).

3 The log normal cone

Given a morphism of log schemes f : X → Y Leo Herr in [Her19] defines a log normal ℓ cone, CX/Y , a cone strict over X of dimension equal to the dimension of LogY . If f is strict this is simply the usual normal cone. In general one factors f locally as a strict closed embedding into a smooth cover of Y . Equivalently this is the restriction of the cone of LogX over LogY to X. Let us give the standard example, due to W. Brauer, of how this can fail to be well behaved under log blow-ups, and how to fix it:

2 2 Example 3.1. Let i : Spec kN2 → A be the origin inside A with its toric log structure. 2 2 Let φ : Bl0(A ) → A be the blow up at the origin. The fibre product of these two 1 1 2 morphisms defines a log structure on P . We write j : P → Bl0(A ) for the induced inclusion. The normal cone of i is the trivial rank two bundle on Spec k, while the normal cone P1 ∗ ℓ ∼ ℓ for j is O(−1) on . It is of course clear for rank reasons that φ Ci 6= Cj , even though the log blow-ups are log flat. We can however form the following diagram:

∗ ℓ ℓ φ Ci Ci

1 P Spec kN2

2 φ 2 Bl0(A ) A

18 Where the bottom row has a canonical perfect obstruction theory. This induces a ℓ ! ℓ relative dimension zero Gysin map for the top row, so we may compare [Cj ] and φ ([Ci ]) ∗ ℓ inside φ Ci . In this case we are simply pulling back and capping with the top of the excess obstruction bundle, which gives [O(−1)] embedded inside the trivial ℓ rank two bundle, exactly equal to the pushforward of the cone Cj .

The solution therefore seems to be that this cone defines a log class on itself different from the fundamental class. Indeed for a large class of targets it does:

Theorem 3.2. Let X → Y be a log morphism such that Y has compatible fundamental class and X locally free and suppose that π : X˜ → X is a locally free log blow-up. Then ℓ ℓ ˜ ℓ ! ℓ CX/Y˜ is a subscheme of CX/Y ×X X with inclusion i and i∗([CX/Y˜ ]) = π ([CX/Y ]). Proof. We apply Vistoli’s rational equivalence from [Vis89]. Since X is locally free there is a strict embedding of X into a log scheme U log smooth over Y which is therefore free in a neighbourhood of the image of X, say U. The scheme U has compatible fundamental class and the blow-up of X lifts to a blow-up of U, πU : U˜ → U. Then πU ℓ is an lci morphism, so the normal cone to it is a vector bundle stack, NU/U˜ . We take the following Cartesian diagram:

X˜ U˜

X U

In [Vis89] Vistoli produces an explicit rational equivalence:

ℓ ℓ [C ℓ ℓ ] = [C ℓ ℓ ] C × U/˜ C N ˜ × X/N ˜ X/U U X/U U/U U U/U

ℓ ℓ ℓ inside A∗(CX/U ×U NU/U˜ ). Pulling back this relation along the zero section of NU/U˜ ! ℓ ∼ ℓ ˜ ℓ gives an equality [π CX/U ] = i∗[CX/˜ U˜ ] inside A∗(X ×X CX/U ). The log cotangent bundle of U over Y pulls back to that on U˜ since the map U → Y is log smooth and U˜ → U log étale. Therefore the above equivalence descends to one in the quotient cone.

As a corollary one obtains:

Corollary 3.3. Let f : X → Y be a log morphism with X locally free and Y having ℓ compatible fundamental class. Let CX/Y be the log normal cone with log structure pulled ℓ back from X. Then there is a canonical class in CX/Y given by the fundamental class. Another corollary answers a question of Herr in [Her19] on the existence of Gysin maps.

19 Construction 3.4. We extend the construction of [Her19] Section 3 to produce Gysin maps in the classical sense defined on the level of cycle classes. Let f : X → Y be a log morphism. The data of a rank d log perfect obstruction theory is the same as the data of a rank d perfect obstruction theory for X → AX/Y . The map X → AX/Y is d strict, and so the construction of [Man12a] produces a Gysin map in A† (X → AX/Y ) which we call the Herr-Gysin pullback. Composing with the log étale map AX/Y → Y d we obtain a Gysin map in A† (X → Y ). When Y is equidimensional and has maximal pure log dimension the stack AX/Y is also equidimensional of the same dimension, it admits an étale cover by log blow-ups of Y . By the above corollary the Gysin map is stable under further log blow-ups of Y or X. This pure dimensionality is automatically the case when Y is connected and log smooth over the trivial log point.

When X → Y is strict the log normal cone is given by the normal cone, and this strictness often holds for many log moduli problems. We can give a refinement of the phenomena seen in [Ran19a] using the following corollary of 3.2.

Corollary 3.5. Let M be a locally free finite-type log stack with a map M → M to a log smooth locally finite-type moduli stack M and a perfect obstruction theory E. Let π : M˜ → M be a locally free log étale morphism and M˜ the fibre product. Then there is an equality π![M]vir = [M˜ ]vir.

M Mlog Mlog Example 3.6. Suppose that is the moduli stack of curves g,n and Mg,n → g,n is the strict classifying map for a space of stable maps. Then we may choose a log Mlog Mlog Mlog blow-up of an finite-type open substack π : fg,n → g,n such that fg,n is locally free. We write Mfg,n for the fibre product which is locally free and we may arrange to have compatible fundamental class. This stack inherits a virtual fundamental class. The Mlog Mlog pushforward of the fundamental class of fg,n is the fundamental class of g,n and so vir vir π∗([Mfg,n] ) = [Mg,n] . Nlog Mlog Now suppose that g,n → g,n is another log blow-up such that the pullback of Mg,n, which we write Ng,n. Choose a further blow-up of an open finite-type substack Nlog Nlog Nlog Mlog ψ : e g,n → g,n with e g,n locally free and admitting a map to fg,n. Then we have a commuting cube:

Neg,n Ng,n φ

Mfg,n Mg,n

Nlog Nlog e g,n g,n

Mlog Mlog fg,n g,n

20 vir vir vir ! vir satisfying π∗([Neg,n] ) = [Ng,n] and by the above [Neg,n] = φ [Mfg,n] . This im- vir vir plies φ∗[Neg,n] = [Mfg,n] . Pushing forward this relation to A∗(Mg,n) we obtain the compatibility of [Ran19a] Proposition 3.6.2.

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