Node Polynomials for Curves on Surfaces

We complete the proof of a theorem we announced and partly proved in [Math. Nachr. 271 (2004), 69-90, math.AG/0111299]. The theorem concerns a family of curves on a family of surfaces. It has two parts. The first was proved in that paper. It describes a natural cycle that enumerates the curves in the family with precisely $r$ ordinary nodes. The second part is proved here. It asserts that, for $r\le 8$, the class of this cycle is given by a computable universal polynomial in the pushdowns to the parameter space of products of the Chern classes of the family.

Göttsche [9] had already conjectured the special case where the pairs consist of a fixed surface and of the divisors in a linear system.This celebrated conjecture was proved independently by Tzeng [26] and Kool-Shende-Thomas [18].For the history of the case of plane curves, see [16,Remark 3.7,p. 78] and the more recent [4].
A part of our conjecture has now been proved by Laarakker [19,Theorem A,p. 4921].He defined a cycle γ(r) which, under suitable genericity assumptions on the family, is supported on U (r), and its class [γ(r)] is given by a universal polynomial in the y(a, b, c).Although he did not prove that the polynomial is Bell (see his footnote [19, p. 4918]), he did prove that [γ(r)] is "multiplicative" when the family of surfaces is a direct sum of families over the same base (see [19,Lemma 5.5 and Remark 5.6,p. 4936]).When the family is trivial, Göttsche had observed that this multiplicative property implies the polynomial is Bell.However, when the family is nontrivial, the multiplicative property is insufficient.
In [16] we applied our theorem in several enumerations involving nontrivial families of surfaces, including the family of all planes in P 4 .In [19,Theorem B,p.4922] Laarakker proved that the number of r-nodal plane curves of degree d in P 3 meeting the appropriate number of general lines, is given by a universal polynomial in d of degree ≤ 9 + 2r.Moreover, he explicitly computed the polynomial for r ≤ 12.In [21] Mukherjee, Paul, and Singh did the same; they obtained a recursive formula, and verified that their results agree with Laarakker's.In [5] Das and Mukherjee treated the case where the curves may have one additional nonnodal singularity.In [22] Mukherjee and Singh did the same for rational curves.
In [16,Remark 2.7,p.74] we conjectured that universal polynomials also enumerate curves with any given equisingularity type.In [13,Theorem 10.1,p. 713] Kazaryan gave a "topological justification" of our conjecture, but gave no algebraic proof.He worked with a linear system on a fixed surface, and found several explicit formulas for curves with singularities of codimension ≤ 7. A few of these formulas had been given in [15,Theorem 1.2,p. 210].In [2] Basu and Mukherjee gave recursive formulas for the number of curves in a linear system on a fixed surface that have r nodes and one additional singularity of codimension ≤ 8 − r.In particular, their formula for 8-nodal curves recovers ours in this case; see [15,Theorem 1.1,p. 210].
In [20] Li and Tzeng and, independently in [23], Rennemo proved the existence of universal polynomials enumerating divisors with isolated singularities of given topological or analytical types in a trivial family of varieties of arbitrary dimension.
In short, we work here over an algebraically closed field of characteristic 0 with pairs (F/Y, D), where Y is a Cohen-Macaulay algebraic scheme, F/Y is a smooth projective family of surfaces, and D is a relative, or Y -flat, effective divisor on F .We let π : F → Y denote the structure map.
In Section 2, given a pair (F/Y, D), we recall from [15, pp. 226-227] the construction and elementary properties of its induced pairs (F i /X i , D i ).Then we prove some further properties.Intuitively, (F i /X i , D i ) represents a family of curves that sit on blowups of the surfaces of F/Y and that have one less i-fold point.
In Section 3, the main results are Lemmas 3.2 and 3.3, which concern properties of certain subschemes of the relative Hilbert scheme Hilb 3r D/Y .In Section 4, we develop some results of bivariant intersection theory for use in the subsequent sections.Our treatment here generalizes and improves our shorter one in [15].
In Section 5, we state the main theorem, Theorem 5.4.Then we prove a key recursion relation; we prove the theorem for r ≤ 7; and we explain what more is needed for r = 8.The difficulty is that the induced pair (F 2 /X 2 , D 2 ) does not satisfy the hypotheses of the theorem, as D 2 /X 2 has nonreduced fibers in codimension 7 above the relative quadruple-point locus X 4 of D/Y .Therefore, the recursion that works for r ≤ 7 must be corrected accordingly.In Section 6 we find an expression for the correction term, and in Section 7 we prove that the correction term is equal to C[X 4 ] for some integer C that is independent of the given (F/Y, D).Our proof illustrates the advantage of developing intersection theory over any universally catenary Noetherian base.Thus, to complete the proof of the theorem, it suffices to compute the integer C in a particular case, such as that of 8-nodal quintic plane curves, which we did in [16,Example 3.8,p. 80].
However, our proof requires an additional genericity hypothesis: the analytic type of a fiber of D at an ordinary quadruple point must not remain constant along any irreducible component of X 4 .This hypothesis comes into play at just one spot in the proof of Lemma 7.4 to ensure a certain map is flat.We believe that Lemma 7.4 and Theorem 5.4 hold without this hypothesis.At any rate, the hypothesis is usually fulfilled in practice.

The induced pairs
The induced pairs (F i /X i , D i ) of a given pair (F/Y, D) play a central role in the present work.So, in this section, we recall the theory and develop it further.Here F and Y need only be Noetherian, and F/Y need only be of finite type.
2.1.The induced pairs.From [15, pp. 226-227], let's recall the construction and elementary properties of the induced pairs, but make a few minor changes appropriate for the present work.
Denote by p j : F × Y F → F the jth projection, by ∆ ⊂ F × Y F the diagonal subscheme, and by I ∆ its ideal.Say D is defined by the global section σ of the invertible sheaf O F (D). Then σ induces a section σ i of the sheaf of relative twisted principal parts, Take the scheme of zeros of σ i to be X i , and set X 0 := F .Then X 1 = D. Further, a geometric point of X i , that is, a map ξ : Spec(K) → X i , where K is an algebraically closed field, is just a geometric point ξ of F at which the fiber D π(ξ) has multiplicity at least i.Also, as i varies, the X i form a descending chain of closed subschemes.
The sheaf P i−1 F/Y (D) fits into the exact sequence, where the first term is the symmetric power of the sheaf of relative differentials, twisted by O F (D). Hence P i−1 F/Y (D) is locally free of rank i+1 2 by induction on i.Therefore, at each scheme point x ∈ X i , we have where, as usual, cod x (X i , F ) stands for the minimum min(dim O F,η ) as η ranges over the generizations of x in Denote by β : F → F × Y F the blowup along ∆, and by E the exceptional divisor.Set ϕ := p 1 β and π := p 2 β.Then π : F → F is again a smooth family of surfaces, and projective if π is; in fact, over a point ξ of F , the fiber F ξ := π −1 (ξ) is just the blowup (via p 1 ) of the fiber F π(ξ) := π −1 π(ξ) at ξ.For each i, set F i := π −1 (X i ), and denote by π i : F i → X i the restriction of π .In sum, we have this diagram: In addition, given r ≥ 1, set As F has no associated points on E, the subscheme ϕ −1 D is an effective divisor; so D i is a divisor on F .If i ≥ 1, then In [15, p. 227], we proved the second assertion of the next lemma.Taking a little more care, we now prove the first too.Later, in Lemma 2.8, we relate X i and r i .
Lemma 2.2.For each i ≥ 1, the subscheme X i of F is the largest subscheme over which D i is effective.Furthermore, Let Γ ⊂ F × Y T be the graph subscheme of t, and since Γ is a local complete intersection; see [7, display (6), p. 601]; so the projection formula yields In particular, on every fiber of π i , the restriction of D i is effective.Furthermore, π i is flat.Hence, D i is relative effective.Thus the lemma holds.Proof .It follows from [17,Proposition 3.4,p. 422] that the formation of F (1) and E (1) commutes with base change.Set g : F × Y Y → F .By [11,Proposition 16.4.5,p. 19], we have , and the section σ i pulls back to the corresponding section σ i .Hence the zero scheme of σ i is equal to 2.5.Arbitrarily near points.Recall the following notions, notation, and results.First, as in [17, Definition 3.1, p. 421], for j ≥ 0, iterate the construction of π : F → F from π : F → Y to obtain π (j) : F (j) → F (j−1) with π (0) := π, with π (1) := π , and so forth.By [17,Proposition 3.4,p. 422], the Y -schemes F (j) represent the functors of arbitrarily near points of F/Y ; the latter are defined in [17,Definition 3.3,p. 422].As in [17, Definition 3.1, p. 421], we denote by ϕ (j) : F (j) → F (j−1) the map equal to the composition of the blowup and the first projection, and by E (j) ⊂ F (j) the exceptional divisor.
Given a minimal Enriques diagram D on j + 1 vertices, fix an ordering θ of these vertices.Also, let U be the unweighted diagram underlying D. By [17, Theorem 3.10, p. 425], the functor of arbitrarily near points with (U, θ) as associated diagram is representable by a Ysmooth subscheme F (U, θ) of F (j) .
By [ in fact, Ψ is a an embedding in characteristic 0. The construction and study of Ψ is based on the modern theory of complete ideals.Finally, set The definitions yield q(G(D)) ⊃ Y (D).Further, take any y ∈ q(G(D)) Y (∞), and let D be the diagram of D K , where K is the algebraic closure of k(y).Then the definitions yield a natural injection α : D → D such that each V ∈ D has weight at most that of α(V ).So deg But G(D) and the G(D ) are locally closed.Thus Y (D) is constructible.
To prove the second bound in (2.5), note that G(D) has a unique point, z say, lying over the given y.Now, Q(D)/Y is smooth of relative dimension dim D by [17,Theorem 3.10,p. 425].Thus, as desired, (2.6) We say that (F/Y, D) is strongly 8-generic if it is 8-generic and if the analytic type of D π(x) at an ordinary quadruple point x ∈ X 4 is not constant along any irreducible component Z of X 4 ; that is, the cross ratio of the four tangents at x is not the same for all x ∈ Z. Proposition 2.8.Fix r.Assume that Y is universally catenary and that (F/Y, D) is r-generic.Then, for each i ≥ 2, the induced pair Proof .Fix i.Let D be a minimal Enriques diagram.Let x be a generic point of the closure of X i (D ).Then x ∈ X i (D ) as X i (D ) is constructible by Lemma 2.6 applied with (F i /X i , D i ) and D for (F/Y, D) and D. Set y := π(x).Let K be an algebraically closed field containing k(x); then K contains k(y) too.
Consider the curves D K and (D i ) K .Note x ∈ X i (D ).So the curve (D i ) K is reduced, and is obtained from D K as follows: blow up F K , at the K-point, x K say, defined by x; take the preimage of D K ; and subtract i times the exceptional divisor.Hence D K is reduced and of multiplicity either i or i + 1 at x K .In the latter case, (D i ) K contains the exceptional divisor; in the former, it doesn't.In either case, let D be the diagram of However, x is the generic point of a component, X say, of the closure of Combine the last two displays; then (2.6) yields Therefore, (2.3) and (2.7) yield the desired lower bound: , and set y := π(x).Then (2.12) Proof .Plainly we may assume x is the generic point of X.Let K be an algebraically closed field containing k(x), so k(y).Then D K is reduced as x / ∈ Y (∞).Let D be the diagram of D K , and D that of D i .Then X is a component of the closure of X i (D ).So we may appeal to the proof of Proposition 2.8.Note that equation (2.10) is trivial here, and we do not need Y to be universally catenary.
Since x ∈ X i , at the corresponding K-point, D K is of multiplicity at least i.Hence D has a root of weight at least i.
But the opposite inequality is (2.2), which always holds.So equality holds.Thus, in (2.11), the first equation holds.The second follows from it and (2.8). Suppose Hence (2.9) and (2.8) yield (2.12).Thus the corollary is proved.

Virtual double points
The minimal Enriques diagram rA 1 consists of r roots of weight 2 and no other vertices.The corresponding scheme G(rA 1 ) is particularly important, as it is equal to the subscheme of the Hilbert scheme Hilb 3r D/Y associated to the geometric fibers of D/Y with at least r distinct singular points.Moreover, we need to consider it for various (F/Y, D) and r.So, for clarity, we set G(F/Y, D; r) := G(rA 1 ).
In this section, we first recall the basic properties of G(F/Y, D; r), which were treated in [17, Proposition 5.9, p. 439].Then we fix r ≥ 1, and assume (F/Y, D) is r-generic.For each i ≥ 1, we find a natural large open subscheme of such that the associated geometric fibers of D i /X i have exactly r i nodes.None lies on the exceptional divisor of a fiber of F i .Further, adding the exceptional divisor to the fiber of D i /X i yields a fiber of D i−1 /X i−1 , and thus establishes an isomorphism from the preceding open subscheme to a natural open subscheme of H i−1 , which is dense in the preimage of X i .These results are treated in Lemmas 3.2 and 3.3 below for later use.F/Y over which the universal family is smooth; in other words, H(r) parameterizes the unions of r distinct reduced points in the geometric fibers of F/Y .By convention, if r = 0, then H(r) and Hilb r F/Y are both equal to Y ; if r ≤ −1, then both are empty.If r ≥ 1, then Proposition 5.9 on p. 439 in [17] asserts that H(r) = Q(rA 1 ) and that the map Ψ : H(r) → Hilb 3r F/Y is given on T -points, where T is a Y -scheme, by sending a subscheme W of F T , say with ideal I, to the subscheme W with ideal I 2 (note that W is flat, because the standard sequence Finally, for an arbitrary fixed r and for i ≥ 0, set and (2) for every i ≥ 1 with r i ≥ 0, if we set then U i is dense in X i , and there is a natural isomorphism of F -schemes Proof Let z ∈ H i × F U i ; let x be its image in U i , and set y := π(x).Let K be an algebraically closed field containing k(z), so k(x) and k(y) too.Then D K is reduced since Y (∞) is empty, and D K has multiplicity exactly i at the K-point x K defined by x since x ∈ U i , so x / ∈ X i+1 .Hence (D i ) K is reduced, and does not contain the exceptional divisor E K .
Let D be the diagram of D K , and D that of (D i ) K .By [17, Proposition 2.8, p. 420], we have with equality if and only if D K has an ordinary i-fold point at x K .Now, (D i ) K has at least r i singular points since z ∈ H i ; hence, formula (2.6.2) in [17, p. 419] yields cod(D ) ≥ r i since D has at least r i roots, each root has multiplicity at least 2, and the summands in that formula corresponding to the other vertices of D are nonnegative.So the right-hand side of (3.2) is at least r.However, r ≥ cod(D) since y ∈ U .So equality obtains everywhere.Hence D K has an ordinary i-fold point at x K .Furthermore, (D i ) K has exactly r i singular points, each is an ordinary double point, and none lies on E K ; also, (D i ) K and E K meet transversally in i points.
We define γ i as follows.A T -point of its source H i × F U i is given by a map T → U i and a T -smooth subscheme W ⊂ F T of relative length r i whose squared ideal defines a subscheme W ⊂ F T contained in (D i ) T .Owing to the discussion above, in every geometric fiber of F T /T , the fibers of (D i ) T and E T meet transversally in i points.Hence, since (D i ) T and E T are relative effective divisors, their intersection is a T -smooth subscheme Z ⊂ F T of relative length i.
Let Z be the subscheme of F T defined by the squared ideal of Z. Then Z is contained in the sum and the latter scheme is to be the target of γ i .We define γ i by sending W to W ∪ Z. Plainly, γ i is injective on T -points since W is determined by W ∪ Z as the part off E T .
To prove It is given by a map T → U i and a T -smooth subscheme S ⊂ F T of relative length r i−1 such that its squared ideal defines a subscheme S ⊂ F T contained in (D i−1 ) T .Then (D i−1 ) T = (D i ) T + E T by (2.3), and (D i ) T is relative effective by Lemma 2.2 since U i ⊂ X i .Let W be the part of S off E T .Plainly, W is a T -smooth subscheme of F T , and its squared ideal defines a subscheme contained in (D i ) T .
Consider a geometric point of T , say with (algebraically closed) field K. Then D K is reduced since Y (∞) is empty, and D K has multiplicity exactly i at the center of K since T maps into U i .Hence (D i ) K is reduced, and (D i ) K ∩ E K is a scheme of length i.Now, S K is K-smooth of length r i−1 .Hence S K consists of r i−1 distinct reduced points, of which at most i lie on E K .So W K consists of at least r i−1 − i, or r i , distinct reduced points.By choosing any r i of them, we obtain a K-point of H i × F U i .But then, by the discussion of such points right after (3.2), there was no choice: (D i ) K has exactly r i singular points, and all are ordinary nodes.Hence W K consists exactly of r i distinct reduced points.Thus W is of relative length r i .
Therefore, W defines a T -point of H i × F U i .According to the discussion above, this T -point is carried by γ i to the T -point of H i−1 × F U i that is given by R, where R := W ∪ Z and Z := (D i ) T ∩ E T .To prove that γ i is surjective on T -points, so bijective on T -points, so an isomorphism, it remains to prove that R = S.
The  Given t ∈ T , to check if I t vanishes, we may replace T by Spec(O T,t ).Thus we may assume that T is of the form Spec(A), where A is local and that T is nonempty.Then it suffices to prove the ideal I ⊂ A of T vanishes, or equivalently, T = T .
There exists a flat local homomorphism A → B such that B is complete and its residue class field is algebraically closed.Then A → B is faithfully flat.So I vanishes if I ⊗ A B does.Thus we may replace A by B, and so assume that A is complete and its residue class field is algebraically closed.
Consider the composition Via it, T is the preimage of U i .Hence T = T if and only if (D i ) T is effective, owing to Lemma 2.2.By the same token, (D i−1 ) T is effective.
Consider the local ring C of F T at the closed point of the center of the blowing up As F/Y is a smooth family of surfaces and as C is complete with algebraically closed residue class field, C is a power series ring; say C = A [[u, v]].Say that the section T → F T is defined by mapping u, v to a, b ∈ A. Replacing u, v by u − a, v − b, we may assume that T → F is defined by mapping u, v to 0, 0. Let , where f j is homogeneous of degree j in u and v. Then To prove that f i−1 = 0, denote the maximal ideal of A by m, and write Then it suffices to prove that a j ∈ m n for all j and n ≥ 0. Since T maps into H i−1 , there is a T -smooth subscheme S ⊂ F T of relative length r i−1 whose squared ideal defines a subscheme S ⊂ F T contained in (D i−1 ) T .Since A has an algebraically closed residue class field K, the fiber S K consists of r i−1 distinct points.Of them, exactly i lie on E K according to our discussion above.Further, the fiber S K is the singular locus of (D i−1 ) K , which consists of r i−1 ordinary double points, and i of them constitute (D i ) K ∩ E K , which is a transverse intersection.
By replacing u and v with suitable linear combinations of themselves, we may assume that the u-axis is not tangent to D K at the center of the blowing up.Now, A is complete; so by Hensel's lemma, S decomposes into the disjoint sum of r i−1 sections.Of them, i sections meet E T .Hence, they correspond to A-algebra maps Set b j := s j (v) and c j := s j (w).Then, for all j, let's check that The first relation holds as the closed points of the sections lie on E K .The second holds because these same points lie on (D i ) K ∩ E K .Further, the points are distinct; so mod m, the c j are distinct elements of K.
Proceeding by induction on n ≥ 1, suppose that b j ∈ m n for all j.Set Then f defines the pullback of (D i−1 ) T .Now, S ⊂ (D i−1 ) T .Hence, for each j, From (3.3), we obtain the following linear system of equations for the a j : The coefficient matrix is Vandermonde.Its determinant is invertible in A, as the c j are distinct mod m.As f i−1 (c j , 1) ∈ m n+1 for all j, solving yields a j ∈ m n+1 .
To complete the proof, we must show b j ∈ m n+1 for each j.Set I j := Ker(s j ).Then f ∈ I 2 j as S ⊂ (D i−1 ) T .Hence ∂ f /∂w ∈ I j .Therefore, for some d j ∈ A. Now, a j ∈ m n+1 ; so (3.3) yields But b j ∈ m n .So (3.4) yields b j (∂f i /∂u)(c j , 1) ∈ m n+1 .But (c j , 1) is, mod m, a simple root of f i ; so (∂f i /∂u)(c j , 1) / ∈ m.Thus b j ∈ m n+1 , as desired.

Intersection theory
For use in the remaining sections, we extend the intersection theory of bivariant classes developed in [8, Chapter 17] and generalized over any universally catenary base in [14, Sections 2 and 3], in [25], and in [24,Chapter 42].However, only [8] is cited below.
4.1.Push down.Assume that f : X → Y is a map of schemes such that its orientation class [f ] is defined [8,Section 17.4,p. 326].If f is also proper, define an additive map where f * : A * (f ) → A * (Y ) is the proper push-forward operation discussed in (P 2 ) on p. 322 of [8].
Assume that a is a polynomial in Chern classes of vector bundles on X.
Proof .By (A 12 ) on p. 323 of [8], product and push-forward commute; so Apply the projection formula of (A 123 ) of [8, p. 323] with f := f , with g := f , with h := 1 Y , with c := [f ] and with The definitions of f * , of [f ], and of ). Apply this projection formula again, but now with f := 1 X , with g := p 1 , with h := f , with c := a and with d The formula just before Proposition 17.4.1 on p. 327 of [8] . So the functoriality of pushforwards, stated in (A 2 ) on p. 323 of [8], yields Putting it all together yields By hypothesis, a is a polynomial in Chern classes of vector bundles on F .But product and pullback commute by property (A 13 ) on p. 323 of [8].So p * 1 (a) is the same polynomial in the same Chern classes of the pullbacks under p 1 of those vector bundles.But, as stated just before Proposition 17.3.2on p. 325 of [8], Chern classes commute with all bivariant classes.Thus But a and b are arbitrary in (4.1); moreover, p 1 and p 2 may be interchanged.So Assume g : Y → Y , and consider f : This property results from property (A 23 ) on p. 323 of [8] as follows: where s ν ∨ = c(ν) −1 denotes the Segre class of ν ∨ .

Derived classes. Consider the setup of Section 2.1. We have the Chern classes
, where E is the exceptional divisor on F .Let β : E → ∆ ∼ = F also denote the restriction of β : F → F × Y F .Recall that β i : X i → F and β i : F i → F denote the inclusions.Let v, w j , e ∈ A * (F ) also denote their own pullbacks via ϕ = p 1 β.As in [15,16], we have We also have the following relations:  c) is commutative.)The corresponding classes for the map π i : F i → X i (defined in Section 2.1) are We have π = p 2 β, and Since p 1 ι = p 2 ι, we have is the class that acts as identy on A * (∆) and since p 2 ι is an isomorphism, we get Thus we have 5 The main theorem Proof .This follows from the second part of Lemma 2.6, with D = rA 1 : Let z be a generic point of G(F/Y, D; r).Then cod z (G(F/Y, D; r), H(r)) ≤ 3r because Hilb 3r D/Y is the zero scheme of a section of a locally free sheaf of rank 3r on Hilb 3r F/Y by [1, Proposition 4, p. 5].
Proposition 5.3.Fix r ≥ 1.Then the following formula holds: where β i : X i → F denote the inclusions.
Proof .Notice that the sum in (5.1) is finite, as 3) and as U (D, s) = 0 for s < 0 by Definition 5.1.
Let's first explain set-theoretically why the formula should hold.Consider a closed point y ∈ Y (r).The curve D y has precisely r nodes, and if we blow up one of the nodes, x ∈ X 2 say, the strict transform (D 2 ) x has r −1 nodes.Hence, above y ∈ Y (r), we get r points x ∈ X 2 (r −1).But not all r − 1-nodal curves of D 2 /X 2 arise in this way: if x ∈ X 3 is an (ordinary) triple point of D π(x) , then (D 2 ) x = (D 3 ) x + E x , hence it has the 3 nodes (D 3 ) x ∩ E x .Therefore, settheoretically, X 2 (r − 1) consists of two parts, one mapping r : 1 to Y (r), the other mapping 1 : 1 to Y (D 4 + (r − 4)A 1 ).The second part is equal to the part of X 3 (r − 4) not contained in X 4 .The part contained in X 4 is equal to X 4 (r − 8) minus the part contained in X 5 , and so on.
The fact that this reasoning is valid on the cycle level is precisely what Lemma 3.3 shows.It therefore only remains to show that there is a natural map e., a family of r − 1 double points in D 2 ⊂ F 2 over T , none of which lie on E), we send it to the image ϕ (Z ∪ 2E) in F to get a family over T of r double points in D. This map induces an r : 1 map from the components of the cycle U (D 2 , r − 1) that are supported on X 2 X 3 to U (D, r).
Our main result is Theorem 5.4, the first part of which was proved in [16].We now prove the last part of the theorem, namely the part concerning the expression for u(D, r) = [U (D, r)], where U (D, r) is the cycle introduced in Definition 5.1.
Recall from Section 2 that for a given pair (F/Y, D), the subscheme X i ⊂ F denotes the scheme of zeros of the natural section σ i of P i−1 F/Y (D).After making some simplifications, we prove the theorem when X 4 = ∅.This proof is easy, and it yields the case r ≤ 7. We then consider the case r = 8, which is more difficult due to the presence of nonreduced fibers in codimension r 2 = 7 in the family of curves of the induced pair (F 2 /X 2 , D 2 ).Theorem 5.4 (Main).Let π : F → Y be a smooth projective family of surfaces, and D a relative effective divisor.Assume Y is Cohen-Macaulay and equidimensional.Fix an integer r ≥ 0, and assume Then either Y (rA 1 ) is empty, or it has pure codimension r; in either case, its closure Y (rA 1 ) is the support of a natural nonnegative cycle U (D, r).
Let b s (D) be the polynomial in v, w 1 , w 2 output by Algorithm 2. for i = 2, 3, 4 by Corollary 2.9.Therefore, X i is a local complete intersection in F , and F is smooth over the Cohen-Macaulay scheme Y , hence is Cohen-Macaulay, and so X i is too.Since Y is equidimensional, so is F , and hence so is X i .
By [16,Lemma 2.4,p. 73]  Proof .The proof is by induction on r.For r = 1, we have Assume next that r ≥ 2 and that the theorem holds for all families verifying the hypotheses of the theorem with r replaced by r < r.In particular, the statement then holds for the induced pairs (F i /X i , D i ), for i = 2, 3, defined in Section 2.1.Indeed, X i (∞) = ∅, and (F i /X i , D i ) is r i -generic by Proposition 2.8; that is, (ii) of the theorem holds with r replaced by r i .
To simplify the notation, let us write where P m is the mth Bell polynomial and z 1 , . . ., z m are variables.Then we get By definition, a s (D i ) = π i# b s (D i ).By applying (4.4) to the polynomials b s (D) (cf.[16, Algorithm 2.3]), By the binomial property of the Bell polynomials [3, equation (4.9), p. 265], we have Plugging this in and using the definition of b s (D) and a s (D), we get where The induced pairs (F i /X i , D i ), i = 3, 4 satisfy the conditions of the theorem, with r replaced by r i , hence, by the case r ≤ 7 of the theorem: Note that, since F is Cohen-Macaulay, [X i ] = x i ∩ [F ] with x i as in Algorithm 2.3 in [16].The induced pair (F 2 /X 2 , D 2 ) does not satisfy the conditions for r replaced by r 2 .Indeed, note that and that D 4 is relative effective on F 4 /X 4 by Lemma 2.2; hence, D 2 has nonreduced fibers above X 4 .So X 2 (∞) = X 4 , and hence has codimension r 2 = 7 in X 2 .
However, from what we have seen above, if we restrict the family F 2 → X 2 to X 2 X 4 , then is the class of the 7-nodal curves of that family.So the difference is the correction term we are looking for.It is the class of a cycle of codimension 7, supported on the codimension 7 subscheme X 4 of X 2 .As Theorem 7.5 shows, (5.In the case of a trivial family, Göttsche observed that this multiplicativity implies that the universal polynomials are Bell polynomials.However, as observed by Laarakker, this conclusion does not follow in the case of a nontrivial family. Let a j (D), a j (D ), a j (D ) be the classes, introduced in Theorem 5.4, for the three pairs.Clearly, a j (D) = a j (D ) + a j (D ).So (for r ≤ 8), in the notation of (5.2), By the binomial property of the Bell polynomials, the right-hand side is equal to Hence the Bell polynomial shape of the universal polynomials is in agreement with the multiplicative property of the generating series of (F/Y, D).

An expression for the correction term
We now find an expression for the correction term (5.3).First, in Section 6.1 we define some useful schemes.Then in Lemma 6.2, we give an expression for u(D 2 , 7), obtained via repeated use of the recursion formula of Proposition 5.3.Then in Section 6.3, we introduce classes e(W i ) on X 2 of cycles on X 4 .Finally, in Proposition 6.4, we express (5.3) as a linear combination of the e(W i ).
6.1.Some important schemes.Let F as the induced pair of Let be the induced pair of . Let X 2 D ⊂ F (j+1) 3 be the zero scheme of the section of P 1 induced by that defining the divisor D .

Independence of the correction term
In this section, we prove Theorem 7.5, which asserts that the correction term (5.3) is equal to C[X 4 ], where C is independent of the strongly 8-generic pair (F/Y, D) with Y (∞) = ∅.
We work locally analytically on F at a general closed point x in X 4 .Section 7.1 describes the local setup.Lemma 7.2 asserts that locally we have the properness we need to pushdown classes.Lemma 7.3 asserts that the key classes e(W i ) pull back to their local counterparts e W i .Lemma 7.4 asserts that the coefficient in e W i of X 4 depends only on the analytic type of the ordinary quadruple point x ∈ D π(x) ; namely, on the cross ratio of the four tangents at x.Its proof requires (F/Y, D) to be strongly 8-generic.Finally, we prove Theorem 7.5 by exhibiting a pair (F/Y, D), where X 4 is irreducible and where any given value of the cross ratio appears at some x ∈ X 4 .
7.1.The local setup.Fix an 8-generic pair (F/Y, D) with Y (∞) = ∅, and a general closed point x ∈ X 4 .By general , we mean that x lies on a single irreducible component Z of X 4 and that x is an ordinary quadruple point of D π(x) .Let us arrange for every x ∈ X 4 to be general as follows.First, if two components Z and Z of X 4 meet, then dim(Z ∩ Z ) < dim X 4 .So cod π(Z ∩ Z ) > cod π(X 4 ).But cod(π(X 4 ), Y ) = 8 by (2.11) as (F/Y, D) is 8-generic.Hence we may discard Z ∩ Z .Second we may discard the locus of y ∈ Y , where D y has a singularity x worse than an ordinary quadruple point, again because (F/Y, D) is 8-generic.
Set F := Spec O F,x and Y := Spec O Y,π(x) and D := Spec O D,x .Denote the induced pair of F / Y , D by F 2 / X 2 , D 2 .The bundles of relative principal parts are compatible not only with the base change Y → Y , but also with the maps F → F and F (j) → F (j) ; cf.[11,Proposition 16.4.14,p. 22].So although the X i for i ≥ 2 are defined in terms of F / Y , D , we have X i = Spec O X i ,x .Similarly, the schemes constructed in Section 6 for (F 2 /X 2 , D 2 ) induce the corresponding schemes for F 2 / X 2 , D 2 .Denote by W (j) i the scheme corresponding to W (j) i ; see (6.2).
Next consider the completions of the local rings, giving us a pair F / Y , D .Replacing the principal parts bundles by their completions, cf.[6, Example 16.14, p. 416], construct the X i , the induced pair F 2 / X 2 , D 2 , and the corresponding schemes of Section 6.Since the complete principal parts bundles are pullbacks, X i = Spec O X i ,x , and all the schemes of Section 6 for (F 2 /X 2 , D 2 ) pull back to the corresponding schemes for F 2 / X 2 , D 2 .Denote by W (j) i the scheme corresponding to W (j) i , so to W (j) i .The classes e(W i ) of Section 6.3 are sums of pushdowns of classes on the W (j) i .By Lemma 7.2 below, the W (j) i are proper over X 2 ; hence, we may form the corresponding classes e( W i ) for the pair F 2 / X 2 , D 2 .Denote them by e( W i ).
Each e(W i ) is the class of a cycle U i on X 4 of dimension dim X 4 .Say that the component Z of 1 as the other cases are similar.Let E j ⊂ F (j) be the union of the strict transforms of the exceptional divisors E (1) , . . ., E (j) ; see [17,Definition 3.5,p. 423].It follows from the description in Section 6 that W (j) 1 is supported in E j .The fibers of the exceptional divisor of F 2 are the same as the corresponding fibers of

Lemma 2 . 3 .
Let (F/Y, D) be a pair.Then forming all of the induced pairs (F i /X i , D i ) commutes with arbitrary base change g : Y → Y .

Definition 2 . 4 .
Let Y (∞) denote the subset of Y whose geometric points are those η of Y whose fiber D η is not reduced.Fix a minimal Enriques diagram D; see [15, Section 2, p. 213].Denote by Y (D) the subset of Y whose geometric points are those η whose fiber D η has diagram D.

FinallyDefinition 2 . 7 .
, suppose G(D) h −1 Y (∞) is nonempty.Then, as we have just seen, there is an injection α : D → D , where D ∈ Σ, and each V ∈ D has weight at most that of α(V ).But there are only finitely many such D, as desired.We say that (F/Y, D) is r-generic if for every minimal Enriques diagram D and for every y ∈ Y (D), we have cod y (Y (D), Y ) ≥ min(r + 1, cod D).

Lemma 3 . 3 .
equation R = S may be checked locally over T and locally on F .So we may replace T and F by affine open subsets Spec(A) and Spec(B).Then B is étale over a polynomial subring A[x, y].Let I ⊂ B denote the ideal of S. Shrinking F further if necessary, we can find an f ∈ B that generates the ideal of (D i−1 ) T .Then f ∈ I 2 as R determines a T -point of H i−1 .Hence f , ∂f /∂x, ∂f /∂y ∈ I.But those three elements generate the ideal of Z := (D i ) T ∩ E T on a neighborhood N of E T .Hence Z ⊃ S ∩ N .But both Z and S ∩ N are T -flat of relative length i. Hence Z = S ∩ N .But R and S are equal off E T .Thus R = S, as desired.Under the conditions of Lemma 3.2, the closed subscheme suffices to prove T is an open subscheme, as we may takeT = H i−1 × F V i .Let I ⊂ O T denote the ideal of T .Then it suffices to prove that the stalk I t vanishes for all t ∈ T for the following reason.Since I is coherent, the t ∈ T , whereIt vanishes form an open subset T .By hypothesis, T ⊂ T .But, if t / ∈ T , then I t = O T,t ; whence, T ⊃ T .So T = T .Give T the induced structure as an open subscheme of T .Then T is the closed subscheme of T with ideal I | T .But I | T = 0. Thus T is equal to the open subscheme T .

4. 3 .
Blowups.Let ι : W → V be a closed, regular embedding of codimension d.Then the orientation class [ι] is defined.Let V denote the blowup of V along W , with exceptional divisor E. Then E = P ν ∨ , where ν is the normal bundle of W in V .Set ξ := c 1 (O V (−E)).The map f : E → W is flat, hence has an orientation class [f ].Then, by [8, Corollary 4.2.2, p. 75], for k ≥ 1,

Fix a smooth projective 4 . 5 . 1 .Proposition 5 . 2 .
family of surfaces π : F → Y , and a relative effective divisor D on F/Y .For each r ≥ 1, we introduce a natural cycle U (D, r) on Y that enumerates the fibers D y with r nodes.Our first goal is to prove Proposition 5.3, which gives a recursive relation for the class u(D, r) := [U (D, r)] in terms of the classes u(D i , r i ) of the induced pairs (F i /X i , D i ) introduced in Section 2.1.This relation is the key to the proof of our main theorem, Theorem 5.Definition Fix r ≥ 1. Form the direct image on Y of the fundamental cycle [G(F/Y, D; r)], remove the part supported in Y (∞), and denote the result by U (D, r).In other words, U (D, r) is obtained as follows.For each generic point z of G(F/Y, D; r), let n z be the length of O z over O q(z) provided this length is finite and q(z) / ∈ Y (∞); otherwise, let n z be 0. Let {q(z)} be the closure of {q(z)}.U (D, r) := z n z {q(z)}.In addition, let U (D, 0) denote the fundamental cycle of Y , and set U (D, −r) := 0. Finally, set u(D, r) := [U (D, r)].It's a class on Y .Fix r ≥ 1. Assume that the pair (F/Y, D) is r-generic.Then U (D, r) has pure codimension r, and its support is just the closure of Y (rA 1 ).

3 )Remark 5 . 6 .
is equal to C[X 4 ], where the constant C is an integer, which is independent of the given pair (F/Y, D).Hence it suffices to compute C in any particular case; for example, in [16, Example 3.8, p. 80], we worked out the case of 8-nodal quintic plane curves, and found C = 3280.Thus Theorem 5.4 is proved.Assume (F/Y, D) is the direct sum of two pairs (F /Y, D ) and (F /Y, D ) over the same base.Then the r-nodal curves of D → Y consists of the unions of the (r − i)-nodal curves of D → Y and the i-nodal curves of D → Y for i = 0, . . ., r.Hence, the existence of a universal polynomial for r-nodal curves implies that the generating series for (F/Y, D) is equal to the product of the generating series for (F /Y, D ) and (F /Y, D ).This fact was observed by Göttsche in the case of a trivial family [9, p. 525], and by Laarakker in the general case [19, Section 5.1, p. 4935].

Lemma 7 . 2 .
X 4 containing x appears in U i with coefficient C i and in the fundamental cycle |X 4 | with coefficient C i .Set C i := C i /C i .Then the cycles U i and C i |X 4 | become equal after restriction to a neighborhood of Z, so the classes e(W i ) and C i [X 4 ] do too.Thus e W i = C i X 4 ; (7.1) furthermore, C i is independent of the choice of x in Z.The schemes W (j) i are proper over X 2 .Proof .Let us first show that the schemes W (j) i and W (j) i | X 2 have the same support.It suffices to consider only the W (j) ) Recall that the classes a s (D 2 ) on X 2 are obtained by pushing down the classes b s (D 2 ) on F 2 obtained by applying Algorithm 2.3 of[16, p. 72]to the pair (F 2 /X 2 , D 2 ).In the case that X 4 = ∅, the Algorithm would have produced the formula u(D 2 , 7) =17! P 7 (a• (D 2 )) ∩ [X 2 ].Removing the classes e(W i ), we get