The Kleiman–Piene conjecture and node polynomials for plane curves in P3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {P}^3$$\end{document}

For a relative effective divisor C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {C}$$\end{document} on a smooth projective family of surfaces q:S→B\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q:\mathcal {S}\rightarrow B$$\end{document}, we consider the locus in B over which the fibres of C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {C}$$\end{document} are δ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta $$\end{document}-nodal curves. We prove a conjecture by Kleiman and Piene on the universality of an enumerating cycle on this locus. We propose a bivariant class γ(C)∈A∗(B)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma (\mathcal {C})\in A^*(B)$$\end{document} motivated by the BPS calculus of Pandharipande and Thomas, and show that it can be expressed universally as a polynomial in classes of the form q∗(c1(O(C))ac1(TS/B)bc2(TS/B)c)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_*(c_1(\mathcal {O}(\mathcal {C}))^a c_1(T_{\mathcal {S}/B})^b c_2(T_{\mathcal {S}/B})^c)$$\end{document}. Under an ampleness assumption, we show that γ(C)∩[B]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma (\mathcal {C})\cap [B]$$\end{document} is the class of a natural effective cycle with support equal to the closure of the locus of δ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta $$\end{document}-nodal curves. Finally, we apply our method to calculate node polynomials for plane curves intersecting general lines in P3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {P}^3$$\end{document}. We verify our results using nineteenth century geometry of Schubert.


The Kleiman-Piene conjecture
All schemes we consider are separated and of finite type over C. Let B be a base scheme, and let q : S → B be a smooth family of surfaces, i.e. a smooth projective morphism of relative dimension 2. By a curve, we mean a proper 1-dimensional scheme, not necessary irreducible or reduced. Let C be a relative effective (Cartier) divisor on the family S → B, i.e. an effective Cartier divisor on S, such that the morphism C → B is flat. Fix a non-negative integer δ. We call a curve δ-nodal if it is reduced, has δ nodes and no other singularities. Consider the following counting problem: Assume that B is of pure dimension n. In [19], Kleiman and Piene construct a natural effective cycle U (δ) with support equal to the closure of the locus of δ-nodal curves. For δ ≤ 8 they prove that the class [U (δ)] is given in a rather specific form as a polynomial in the classes Kleiman and Piene work with certain assumptions on the dimensions of equisingular strata in the family. For a curve C, let D be its equisingularity type. It can be represented by an Enriques diagram, which encodes the numerical invariants of the singularities of C [18]. Conversely, for an equisingularity type (or Enriques diagram) D, we write B(D) ⊂ B for the locus of curves in the family C → B, with equisingularity type D.
One of the invariants of an equisingularity type is the codimension cod(D). It is the 'expected codimension' in which curves with equisingularity type D appear in a family. More precisely, in [18], it is characterized as the codimension of the locus of curves with equisingularity type D in the universal family C → |L| of any sufficiently ample complete linear system. The hypotheses on the family C → B under which the class U (δ) is constructed and which we will denote by DIM KP are the following: • The locus of non-reduced curves B(∞) has codimension > δ; • For each equisingularity type D, the locus B(D) has at least the expected codimension cod(D), or codimension > δ.
Here we use the convention codim(∅) = ∞. In [19] and [17], the authors prove the following theorem: Moreover, in [19] the following conjecture is made. In this paper we propose a class γ (C) ∈ A δ (B), enumerating the δ-nodal curves, inspired by the BPS calculus of Pandharipande and Thomas [26]. We will show that if B is complete, the class γ (C) ∩ [B] is the rational equivalence class of a natural cycle with support B(δ). For this we work with hypotheses DIM, similar to but slightly weaker than DIM KP , and an additional ampleness assumption AMP. By means of a family version of an algorithm by Ellingsrud, Göttsche and Lehn [8], we show that without assumptions, the class γ (C) is a universal polynomial in the classes (a, b, c). This will be the content of Theorem A below.

BPS numbers
Let C be a locally planar, reduced curve of arithmetic genus g. In [26] the authors consider the following transformation of the generating series of topological Euler characteristics of Hilbert schemes C [i] of i points on C, which defines the BPS numbers n r ,C of C. They prove the following: Theorem 1.3 (Pandharipande-Thomas) The numbers n r ,C are zero, unless g − δ ≤ r ≤ g, where δ is the δ-invariant of C, i.e., g − δ is the geometric genus of C.
Shende proves in [30] that the number n g−i,C equals the degree of the subvariety of i-nodal curves in the versal deformation space of C. In particular it is positive for 0 ≤ i ≤ δ.
Let B be a scheme and let p : C → B be a family of (not necessarily reduced) curves, i.e., a projective flat morphism of relative dimension 1. Assume that the fibres are locally planar curves of genus g. Let be the relative Hilbert scheme of i points on the fibres of C → B. We define constructible functions n r = n r (C) on B by In other words, n r is the function that assigns the number n r ,C red b to a point b ∈ B. By Theorem 1.3, the function n g−δ is supported on the locus of curves that have δinvariant ≥ δ or are non-reduced. In the same paper it is shown that n g−δ,C = 1 for a δ-nodal curve C.
Let q : S → B be a smooth family of surfaces and let C ⊂ S be a relative effective divisor. We will use the embedding C → S to analogously define classes n cl r ∈ A * B.
In fact, C [i] B is a subscheme of S [i] B = Hilb i (S/B), cut out regularly (in particular, in the expected codimension i) by the tautological bundle O(C) [i] B (see Lemma 2.7). The scheme S [i] B is smooth over B [1] and the virtual tangent bundle T C [i] B /B , as defined in [10, B.7.6], is given by the class Here the homomorphism denotes the Gysin push-forward as defined in [10,Chapter 17]. We define to be equal to the degree-δ part of n cl g−δ (C). We will show that it reflects some of the properties of n g−δ (C). In fact, in Proposition 3.1, we will compare n r and n cl r by means of the Chern-Schwartz-MacPherson class. [12] also mention the CSM-class of the constructible function n r as an invariant counting nodal curves. Moreover they consider an analogous class, using the virtual tangent bundle of Hilbert schemes of points of the curve. However, the use of the relative tangent bundle, which is natural from the point of view of [19], is essential for our results.

Results
Recall that a line bundle L on a smooth projective surface S is called δ-very ample if for any finite subscheme Z ⊂ S of length δ + 1, the map H 0 (S, L) → H 0 (Z , L| Z ) is surjective [5]. For a line bundle L on a smooth family of surfaces S → B, consider the following ampleness hypotheses, which we denote by AMP: Now let C and q : S → B be given as above, and let L = O(C). Without making any assumptions on the dimensions of the equisingular strata, AMP guarantees that the class γ (C) ∩ [B] is supported on the locus of curves with δ-invariant ≥ δ. If B is equidimensional, it will follow (Proposition 4.4) that γ (C) ∩ [B] is the class of a natural effective cycle with support B(δ) if we assume the following hypotheses, which we denote by DIM: • The locus of δ-nodal curves, if non-empty, has codimension δ.
As explained in [18], for an equisingularity type D we have cod(D) ≥ δ(D), with equality only for δ-nodal curves. It follows that DIM is slightly weaker than DIM KP . To summarize, we will prove the following theorem: Theorem A Let B be a scheme and fix an integer δ. Let C be a relative effective divisor on a flat family of smooth surfaces S → B. Then the class γ (C) can be expressed universally as a polynomial of degree δ in classes of the form The conjecture of Kleiman and Piene is a family version of the Göttsche conjecture [11]. For a sufficiently ample line bundle L on a smooth surface S, the latter asserts that the degree of the Severi locus of δ-nodal curves in the complete linear system |L|, is given by a universal polynomial in the numbers L 2 , (L.K ), K 2 and c 2 (S), for K the canonical divisor on S. Equivalently, the number of δ-nodal curves in a general linear system P δ ⊂ |L| is given by such a polynomial.
The Göttsche conjecture was first proved using algebraic methods by Tzeng in [31]. Other proofs were given in [16,24] and [21]. In [23] and [27] the result is generalized to other singularity types.
Our theorem implies the Göttsche conjecture for δ-very ample L, but it is not independent from existing results. In fact, our method can be seen as a family version of the proof in [21]. Moreover, sharper results are known in terms of the required ampleness [20].

Application to plane curves in P 3
The motivation for the project was the following counting problem. For fixed integers δ ≥ 0 and d > 1 write and consider lines 1 , . . . , n ⊂ P 3 . The space of curves of degree d that lie on a plane in P 3 and that intersect the lines 1 , . . . , n has expected dimension δ. We will show that, if we choose the lines 1 , . . . , n sufficiently general, the subspace of δ-nodal curves is finite (and reduced, as a scheme). For d ≥ δ, we can use our method to calculate the number N δ,d of δ-nodal plane curves of degree d intersecting the lines 1 , . . . , n . Let C → B be the universal plane curve of degree d in P 3 . We will show in Proposition 6.1 that for d ≥ δ, we have N δ,d = γ (C)∩[B 1 ,..., n ], in which B 1 ,..., n ⊂ B is the closed subvariety of curves intersecting the general lines 1 , . . . , n . We use this to prove our second main result: Moreover, for δ ≤ 12 these polynomials are the ones given in Appendix A. Remark 1. 6 Our computations suggest that N δ (d) has degree exactly 9 + 2δ, with leading coefficient 3 δ 162 δ! , but we do not prove this.

Chern-Schwartz-MacPherson classes
To any constructible function f on a complete scheme X , one can assign a class c SM ( f ) in the Chow group of X , called the Chern-Schwartz-MacPherson class of f . The existence of well-behaved Chern classes for constructible functions was conjectured by Deligne and Grothendieck and proved by MacPherson in [25]. Several other constructions are known. See [2] for an overview and a new construction in a more general set-up. For a subset V of a scheme X , write 1 V : X → Z for the function with constant value 1 on its support V . Recall that a constructible function is a map f : X → Z that can be written as a finite sum in which V ⊂ X is closed and e is the topological Euler characteristic. For a scheme X let A * (X ) denote the Chow group of X . In the following theorem, we will view A * as a (covariant) functor on the category of complete schemes with proper morphisms to the category of abelian groups. Let c denote the total Chern class.

Remark 2.2
The uniqueness of such a natural transformation follows from resolution of singularities. MacPherson proved the naturality of the homology class, but in fact his argument gives this stronger result (see [10, 19.1.7]). It follows directly from the definitions that for a complete scheme X , we have with π : X → { * } the morphims to a point. At the other extreme, we have the following lemma.

Lemma 2.4
Let X be a scheme and let V ⊂ X be a locally closed subset of dimension n and let V be its closure (with the reduced scheme structure). Then Proof Let g : V → V be a birational morphism from a nonsingular projective variety V . Let U ⊂ V be a dense open over which g is an isomorphism and let Z = V \U be its complement. Write for the boundary of V . Then the we have with f a constructible function on Z . On the other hand, we have Since the functions f and 1 ∂ V are supported on closed subsets of dimension < n, by naturality the same holds for their Chern-Schwartz-Macpherson classes. It follows that we have

Hilbert schemes of points
In [8], Ellingsrud, Göttsche and Lehn describe a method to calculate certain tautological integrals on S [n] . In fact, they give a constructive proof of the following theorem:  from the universal length n subscheme Z on the fibres of S → B. It restricts to The following is proved in [1] and generalizes a well known result by Fogarty: Now let S → B be given as above, and let C ⊂ S be a relative effective divisor.  [14], the locus in S [i] of subschemes of length i supported at a point has dimension i − 1. From this it follows directly that the locus in C (i) over which the fibres of the morphism

The smooth case
Let B be a scheme. Let S → B be smooth projective of relative dimension 2 and let C ⊂ S be a relative effective divisor. For a fixed δ, let be the class defined by Eq. (2) in the introduction. The following situation is the model for our results.
In particular, the class n cl g−δ ∩ [B] is supported on the locus of curves that have δinvariant ≥ δ or are non-reduced, i.e. it is the push forward of a class on this locus. If B is of pure dimension n and the family of curves C → B satisfies DIM, we find with β a sum of cycles of dimension < n − δ.
Proof By the defining Eq. (1), the constructible function n g−δ is a linear combination of the terms ). It is easy to see that only the terms with i = 0, . . . , δ are involved. Similarly, n cl g−δ is a linear combination (with the same coefficients) of the classes Therefore it suffices to verify that relation (3) holds for the first δ + 1 terms in the left hand sides of (1) and (2).
Recall that we have defined the class T For i = 0, . . . , δ, we have by Lemma 2.6 and by Lemma 2.7 the following relations It follows that By the defining properties of Chern-Schwartz-MacPherson classes (Theorem 2.1) we obtain By Theorem 1.3 the support of the constructible function n g−δ lies in the locus in B over which the curves in the family C → B have δ-invariant ≥ δ. By the functoriality of the Chern-Schwartz-MacPherson class, the cycle class is the push-forward of a cycle class on this locus. Since c(T B ) is invertible, the same holds for n cl . By [26,Prop. 3.23], the function n g−δ has constant value 1 on the locus of δ-nodal curves. If C satisfies DIM, it follows that the support of the constructible function n g−δ − 1 B(δ) lies in a closed subset of codimension > δ. Hence the last assertion follows from Lemma 2.4. [21] it is shown that both conditions of the proposition are satisfied by the universal curve C → |L| over the linear system of a δ-very ample line bundle L on a smooth surface S, i.e., C satisfies DIM, and the relative Hilbert schemes C [i] |L| are non-singular for i ≤ δ. By Bertini's theorem it then follows that the same holds for the restriction C P δ of the universal curve to a general linear system P δ ⊂ |L|. In particular the set P δ (δ) is finite, and it follows that the degree

Example 3.2 In
equals the number of δ-nodal curves in the linear system P δ .

Remark 3.3
It should be noted that the integrals differ slightly from the ones in [21]. In fact, in loc. cit. the authors consider a linear combination of the integrals whereas we use the relative (virtual) tangent bundles and consider the integrals Interestingly, (4) does not equal (5) in general, but after taking the BPS linear combination of the integrals for i = 0, . . . , δ, they both calculate the number of δ nodal curves in the linear system P δ .
More generally, let B be a scheme and let p : S → B be smooth projective of relative dimension 2. Let L be a line bundle on S and assume L satisfies AMP. Then p * L is a vector bundle and the fiber of the projective bundle

Functoriality and support
Let B be a scheme. Let C be a relative effective divisor on a smooth family of surfaces q : S → B. Let f : B → B be a morphism and consider the relative effective divisor C B = C × B B on the smooth family of surfaces Then we have the following: Proof It suffices to show the relations for the LHS of equation (2). Note that we have Cartesian squares By flatness of the vertical maps, we have The second part follows from the projection formula.
Let C be given as above. Write L = O(C) and assume L satisfies AMP. As in the previous section, we can form the projective bundle |L/B| := P(q * L) over B, with fibre over b the δ-very ample complete linear system |L b |. Let C ⊂ S |L/B| denote the universal relative effective divisor on the family S |L/B| → |L/B|. The bundle π : |L/B| → B has a canonical section induced by the inclusion O(−C) → O, and the divisor C is obtained by restricting the divisor C .

Lemma 4.2 Assume that B is equidimensional, and let W ⊂ |L/B|(δ) be an irreducible component. Then π(W ) is an irreducible component of B.
Proof Clearly π(W ) is closed and irreducible, so it suffices to show that we have dim(π(W )) = dim(B

Definition 4.3 Define a cycle
in which the sum is taken over the irreducible components of |L/B|(δ). Here l(O π(W ),B ) denotes the length of the local ring along the subvariety π(W ) of B.
As in the introduction, we will use the notation Now assume that B is equidimensional, and let s be given as in (6). Then we have .
is the class of a natural cycle with support B(δ).
Proof Assume that B is equidimensional. Without loss of generality, we may assume that B is connected so that satisfies AMP and we have the following diagram with Cartesian squares: f π π f s in which C and C are the universal divisors on S |L /B | and S |L/B| in the linear systems |L /B | and |L/B| respectively.
Note that ( f π ) * [|L /B |] = [|L/B|]. By Lemma 3.4 and Proposition 3.1 we have with α a sum of classes with codimension > δ. We have (set-theoretically) be an open over which ρ is an isomorphism. Then is an isomorphism. Since W is irreducible and maps onto W , we have and the morphism W → W is generically of degree 1. We have now proved . It follows that Hence, by Lemma 4.1 we have If C satisfies DIM, it follows that s(B) and |L/B|(δ) intersect properly in |L/B|, i.e. in dimension dim B − δ, and we have set theoretically To see this, note that since |L/B|(δ) has codimension δ in |L/B|, an irreducible component of s −1 |L/B|(δ) has codimension ≤ δ. On the other hand, it consists of curves with δ-invariant ≥ δ. So by DIM, it has pure codimension δ in B. Moreover, the set consists of curves with δ-invariant ≥ δ that are not δ-nodal. By DIM, it has codimension > δ. Hence B(δ) lies dense in s −1 |L/B|(δ) , proving equation (7).
is a the class of a natural effective cycle with support equal to B(δ). Now let B be any complete scheme and let V → B be a morphism from an n-dimensional variety V . By the above we have and hence we have an equality of bivariant classes with α ∈ A * (B) a sum of classes of degree > δ.

Universality: relative EGL
Let B be a scheme, and let C be a relative effective divisor on a smooth family of surfaces q : S → B. The arithmetic genus of a curve in the family p : C → B is denoted by g, which we view as a locally constant function on B. We consider the transformation of power series (2) and rewrite it as follows: It follows that we can write in which the a i are polynomials of degree δ − i in g, depending only on δ and i. In fact, a i = a iδ can be found by inverting the upper triangular matrix with 1's on the diagonal Proposition 5. 1 The class γ (C), can be expressed universally as a polynomial of degree δ in the classes in which q * denotes the Gysin push-forward.
We prove the existence of a universal polynomial in Lemma 5.2. In Lemma 5.4 we compute its degree.

Lemma 5.2 There exists a polynomial as in the proposition of degree ≤ δ in the classes
(a, b, c).
Proof We can view g as an element of A 0 (B). In fact we have By the Eq. (8), it suffices therefore to prove that the degree-δ parts of the classes can be expressed universally as polynomials of degree i in the classes (a, b, c). By Lemma 2.7, we have the equality and hence the lemma follows by the following generalisation of Theorem 2.5. Proof The argument given in [8] directly generalises to the relative case. For the case S = S × B, see also [22], Section 4. In fact, Proposition 3.
It follows that q n * P is a sum of classes q m * P m for m = 1, . . . , n and polynomials P m in the Chern classes of the bundles pr * i T S/B and pr * i F i , pulled back along the several projections S m B → S. Now use the fact that for classes α 1 , . . . , α m ∈ A * (S) we have in A * (B).

Lemma 5.4 The degree of the polynomial of Lemma 5.2 is δ.
Proof By Lemma 5.2, the class γ (C) can be expressed universally as polynomial γ in classes (a, b, c) of degree ≤ δ. Now let C be the universal curve in a complete linear system |L| on a surface S, and let P δ ⊂ |L| be a general linear system. Let ω ∈ A 1 (P δ ) be the class of a hyperplane. As explained in the proof of [21,Thm. 4.1], the algorithm of [8] applied to the right hand side of (9) for i = δ produces a term c 2 (S) δ /δ! coming from the term As noted in Remark 3.3, the integrals in [21] differ by a factor c(T P δ ). However, this does not affect the term c 2δ (T S [δ] )ω δ . It follows that γ is a polynomial of degree δ in classes the classes (a, b, c).

Multiplicativity
We will check that the class γ has the expected multiplicative behaviour, cf. [19] and [11]. Let B be a base scheme. For k = 1, 2, let S k → B be proper and smooth of relative dimension 2, and let C k be a relative effective divisor on S k , and write p k : C k → B for the morphism to B. Let C be the union We have the following relations: For k = 1, 2, let g k be the arithmetic genus of a curve in the family C k → B, so we have with g the genus of a curve in the family C → B. It follows easily from (10) that we have the identity n cl g−δ (C) = i+ j=δ n cl For any i ≥ 0, let γ i be the degree-i part of n cl g−i . We record the following lemma.

Lemma 5.5
Let B be complete and let C 1 and C 2 be given as above. Assume that for k = 1, 2, the line bundle O(C k ) on S k satisfies AMP. Then we have the relation Proof By Proposition 4.4, it follows directly from (11), by taking degree-δ parts on both sides of the equation.

Remark 5.6
As was shown in [11], this gives, in the case of the Göttsche conjecture, the generating series γ = γ δ q δ in terms of the Bell polynomials cf. [19]. In this case, there are four power series, corresponding to L 2 , K L, K 2 and c 2 (S), that determine the generating series. In the general case, one needs to determine the coefficients of the classes (a, b, c) with a + b + 2c = δ + 2 for each δ, rather than just the ones with b + 2c ≤ 2. Expectedly, it suffices to evaluate γ on a sufficiently large class of examples to obtain the qualitative result.
6 Application: plane curves in P 3 We will apply the results to the problem of counting δ-nodal plane curves of degree d in P 3 . As we will see below, the space of such curves has dimension Let N δ,d denote the number of δ-nodal plane curves of degree d that intersect general lines 1 , . . . , n ⊂ P 3 . The main result of this section is that for each δ, and d ≥ δ, the numbers N δ,d are given by polynomial of degree ≤ 2δ + 9 in d. Let Gr := Gr(2, P 3 ) be the Grassmannian of planes in P 3 and let U be the tautological vector bundle on Gr. Let O Gr (1) be the bundle corresponding to the hyperplane class via the identification Gr =P 3 . These two bundles are related via the tautological short exact sequence Moreover, it follows that B parametrizes planes in P 3 , together with a degree d curve on that plane. As a planar curve in P 3 of degree > 1 lies in a unique plane, the variety in fact parametrizes planar curves in P 3 .
Let p : C → B be the universal curve. Then C is a relative effective divisor on the family S B = S × Gr B and we have

Remark 6.3
It should be noted that for δ > 0, the scheme B 1 ,..., n is singular. In fact, for every i, a local computation shows that the singular locus of the variety B i is the divisor of curves C ∈ B i such that i lies in the plane spanned by C.
Proof We will prove the first two statements by an argument in the spirit of Lemma 4.7 in [18]. We have n = r + 2 − δ = dim B − δ. It follows that, cf. loc. cit., the expected dimension of Let Gr(1, P 3 ) be the Grassmannian of lines in P 3 , and let L → Gr(1, P 3 ) be the universal line. Let P be the limit of the following diagram: in which we use the notation Then P parametrises the following data: • lines 1 , . . . n ⊂ P 3 ; • points p 1 , . . . , p n ∈ P 3 ; • a plane V ⊂ P 3 ; • a curve C ∈ B(δ); subject to the following conditions: The horizontal maps in the diagram are flat with relative dimensions 2n and n respectively. Since B(δ) has dimension n, it follows that dim(P) = 4n = dim(Gr(1, P 3 ) n ) .
A curve C ∈ B(δ) has a dense smooth open subset C • (as it is nodal). Therefore also the universal curve C → B restricted to B(δ) has this property (the latter being reduced). As P → C n B | B(δ) is smooth (with fibre ∼ = (P 2 ) n ), we see that P is generically smooth. In particular, the singular locus P sing of P has dimension < 4n. Consider the morphism φ : P → Gr(1, P 3 ) n .
Let U 1 = Gr(1, P 3 ) n \φ(P sing ) be the complement of the image of the singular locus in Gr(1, P 3 ) n . As Gr(1, P 3 ) n has dimension 4n, the open U 1 ⊂ Gr(1, P 3 ) n is nonempty. Since φ −1 (U 1 ) is smooth, there is a non-empty open U 2 ⊂ U 1 such that the morphism φ is finite and reduced over U 2 . Moreover, the closed subsets is non-empty. Now let = ( 1 , . . . , n ) ∈ U be an n-tuple of lines and let P = φ −1 ( ) be the fibre over . Consider the morphism

For a point [C] ∈ B(δ), the fibre of P over [C] is the scheme
Let V ⊂ P 3 be the plane spanned by C. By definition of U , we have i ⊂ V for the lines 1 , . . . , n in . It follows that the intersection is a reduced point, if it is non-empty. Hence P maps isomorphically to its image in B.
Finally, we will show that in fact To see this, first note that ψ(P ) lies in the open subset W ⊂ B consisting of curves C with i ⊂ V for i = 1, . . . , n and V the plane spanned by C. As noted before, for a curve C ∈ W , the scheme C ∩ i is empty or consists of a single reduced point. It follows that over W , the scheme As the intersection B(δ) ∩ B 1 ,..., n is finite and reduced, it is transverse, by (14). It follows by The following lemma is essentially [32], Exercise 3.4. See also [10], Example 3.2.22. For completeness, we will include the proof.

Lemma 6.4 For a line ⊂ P 3 , the closed subvariety B ⊂ B is a divisor, cut out by a section of the line bundle
Proof It suffices to construct the section outside the codimension two subvariety Z ⊂ Gr of planes [V ] ∈ Gr containing the line . Let U := Gr − Z be the complement. Consider the fibre product S of the following diagram: Then the morphism S → Gr has fibre V ∩ ⊂ P 3 over a point [V ] ∈ Gr. In particular, it restricts to an isomorphism over U . Now consider the fibre product C of the diagram C P 3 .
The morphism C → B has scheme theoretic fibre C ∩ over a point [C] ∈ B. It follows that C is mapped onto B and the morphism restricts to an isomorphism over B × Gr U .
As noted before, C is the zero locus of a canonical section of the line bundle The following corollary of Proposition 6.1 is the first part of Theorem B, our second main result.

Corollary 6.5 For every
Proof By Proposition 6.1, we need to compute the integral Let H = c 1 (O Gr (1)) and ξ = c 1 (O B (1)). Then, by the Lemma 6.4, we have for general lines 1 , . . . , n ⊂ P 3 the equation On the other hand, we know by Theorem A that γ (C) is a polynomial of degree δ in classes  1)). Then we have

It will follow that γ (C) is a polynomial in
in which we use the short exact sequences and (13). The structure of the Chow ring of S B is given by and the push-forward is computed by repeatedly substituting the equation and taking the coefficient of η 2 . Since H 4 = 0, the substitution procedure terminates after three steps. For fixed a, b an c, we obtain a polynomial in H , ξ and d. It follows that γ (C) can be written as a polynomial in H , ξ and d.
Note that the coefficient of H i in (a, b, c) has degree at most 2 +i as a polynomial in d . As γ (C) is a polynomial of degree δ in classes (a, b, c), the coefficient of H i in this γ (C) has degree at most 2δ + i, as a polynomial in d.
We consider the class in A 0 (B). We will show that its degree is a polynomial in d. The Chow ring of B = P(q * L) is given by We have and hence its Chern class is a polynomial in d and H . In fact, we have Note that coefficient of H i is a polynomial in d of degree 3i. We can compute the degree Recall that we write π : B → Gr for the projection. Then the class is a product of classes of the following types: • The coefficients of γ (C) as a polynomial in ξ ; • Classes of the form n i (d H) i ; • Polynomials in the Chern classes of q * L.
As remarked before, the coefficient of H i in γ (C) has degree 2δ + i in d. In other words, every factor d 2δ+i appearing in the terms of γ (C) is accompanied by a factor H i . Similarly, the coefficients of H i of classes of the second and third type, are polynomials of degree 3i in d, so every factor d j appearing in the terms of these classes is accompanied by a factor H j/3 . It follows that u H 3 has degree at most 2δ + 9 in d.

Torus localization
As in the previous section, let Gr = Gr(2, P 3 ) be the Grassmannian of planes in P 3 , with universal plane q : S = P(U) → Gr, in which U is the tautological vector bundle on Gr. On S we have defined the line bundle L = O S (d), which we use to construct the projective bundle B = P(q * L) over Gr parametrizing planar curves of degree d in P 3 , with universal curve C → B. The variety B has dimension r + 2, with the rank of q * L. Finally, we define and write B 1 ,..., n for the locus of curves intersecting general lines 1 , . . . , n . Rather than using the algorithm of [8], we can use the Bott residue formula to evaluate the integral of Proposition 6.1. By the Lemmas 6.4 and 2.7, we need to compute for i = 1, . . . , δ. We have equations and It follows that the class on the right hand side of (17) is a polynomial in classes pulled back from S [i] Gr , and the first Chern class of the line bundle O B (1). We will continue to use the notation ξ = c 1 (O B (1)) and H = c 1 (O Gr (1)). We can rewrite the factors involving the bundle O B (1) in the integral as follows: In the last expression, α ∈ A * (S [i] B ) is a sum of classes of degree > 3 + 2i + δ. It follows that α ξ n−3 = 0 , as we have Hence we can compute (17) by integrating the class As in the proof of Corollary 6.5, the push-forward along the projection Gr can be computed by substituting the equation and taking the coefficient of ξ r −1 . Hence we can rewrite (17) as an integral in which P is a polynomial 3 in d and the Chern classes of the bundles in the brackets. We can evaluate this integral using the Bott residue formula. For notation and definitions, see [7]. Recall that for a torus T acting on a smooth variety X , the fixed locus X T is smooth [15], so a connected component F ⊂ X T has normal bundle N F X of rank equal to the codimension d F of F in X .)

Theorem 7.1 (Bott residue formula ([7])) Let E 1 , . . . , E r be T -equivariant vector bundles on a complete, smooth n-dimensional variety X with a torus action by T . Let p(E) be a polynomial in the Chern classes of the bundles E i . Then
, in which we sum over the connected components F of the fixed locus X T of the torus action.
Consider the natural action of the torus T = (G m ) 4 on P 3 given by It induces a dual action of T on the Grassmanian Gr =P 3 which lifts to an equivariant structure on O Gr (1). The action onP 3 × P 3 restricts to an action on S (which is simply the incidence variety). This action, in turn, lifts to an equivariant structure on the line bundle O S (1). Moreover, we obtain actions on the Hilbert schemes S

Lemma 7.2 The fixed locus of the action of T on S [i]
Gr is finite and reduced.

Proof As the variety S [i]
Gr is smooth, so is the fixed locus, as remarked above. Hence it suffices to show that the underlying set is finite. The fixed points of Gr are the four planes for k = 0, . . . , 3, given by the vanishing of a coordinate. Note that the morphism Gr → Gr is equivariant for the T -action. It follows that T acts on the fibres V [i] k . Now we follow [9], Section 4. Let Z ⊂ V 0 be a subscheme of length i, fixed under the action of T . Then Z is supported on the T -invariant locus {P 1 , P 2 , P 3 } ⊂ V 0 , with As Z is invariant, so is the ideal It follows that I(Z 1 ) is generated by monomials in u and v. The coordinate ring C[u, v]/I(Z 1 ) is spanned by the i 1 monomials u k v l not contained in I(Z 1 ). For every k ≥ 0, define It is easy to see that the d k define a partition of length l 1 . Similarly, we get partitions P Z 2 and P Z 3 . Conversely, any tripartition (P 1 , P 2 , P 3 ) of length i, consisting of three partitions P 1 , P 2 , P 3 with |P 1 | + |P 2 | + |P 3 | = i, corresponds to a T -invariant length-i subscheme of V 0 . It is clear that there are finitely many such tripartitions. By repeating this argument for the other planes V k , the result follows.
We will apply Theorem 7.1 to the integral (18 are T -representations. Lemma 3 of [7] gives the equivariant Chern classes explicitly as polynomials in the characters of the torus. The details are similar to the computation in [9].

Proof Theorem B, second part
We have performed the calculation using Maple. We have computed the polynomials N δ up to δ = 12. The results are printed in Appendix A.

Remark 7.3
Up to δ = 7, our answers are in agreement with polynomials communicated by Ritwik Mukherjee, which he calculated using the methods from [3,4] and [33] and verified by means of the algorithm of [19].

Remark 7.4
This method of calculating node polynomials seems to be quite efficient. For example, consider the integral It is easy to see that it computes the number of δ-nodal curves of degree d intersecting n − 3 general points in a fixed plane P 2 . By an minor adaptation of our code, we were able to compute the node polynomials up to δ = 15, finding agreement with the polynomials up to δ = 14, published by Block in [6]. However, Göttsche has computed the polynomials up to δ ≤ 28 [11]. The polynomial for δ = 15 is given in Appendix B.

Low degree checks
Let δ, d ≥ 1. We want to determine the contribution of reducible curves to the number N δ,d of planar δ-nodal curves of degree d in P 3 intersecting general lines 1 , . . . , n ⊂ P 3 . For certain δ and d, all curves contributing to N δ,d are reducible, thereby giving consistency checks of our formulae. If in addition the irreducible components of these reducible curves are smooth or 1-nodal, these can be calculated by classical methods.
Let C be a curve in our counting problem, and assume we can write C as the union of irreducible curves C = C 1 ∪ · · · ∪ C r . The curves C i are necessarily nodal (if singular), and intersect transversely. For i = 1, . . . , r , let δ i be the number of nodes of C i , and d i its degree. As C lies in a plane, two curves C i and C j with 1 ≤ i < j ≤ r intersect in d i d j points. We have Moreover, there is a partition (20) such that C i intersects the lines in i ⊂ { 1 , . . . , n }.
Conversely, choose a partition as in (20) and integers d i and δ i for i = 1, . . . , r such that the equations (19) hold. We will determine the number of curves contributing to N δ,d that decompose as described above, with these fixed data.
For i ∈ {1, . . . , r }, let B i := P(Sym d i (U * )) π i − → Gr = Gr(2, P 3 ) be the projective bundle parametrizing planar curves of degree d i . Consider the locus We have the following lemma. such that for an n-tuple ( 1 , . . . , n ) ∈ U , the statement of the lemma holds.
Proof The argument is similar to the proof of Proposition 6.1, so we will not give all the details. For i = 1, . . . , r , let n i = # i and form the limit P i as in the diagram (15). We have natural morphisms φ i : P i → Gr(1, P 3 ) n i . Consider the morphism Gr(1, P 3 ) n i = Gr(1, P 3 ) n .
As in Proposition 6.1, φ is finite and smooth over a non-empty open Here we use the fact due to Severi, that for the line bundles O(d) on P 2 , the locus of irreducible δ-nodal curves in |O(d)|, if non-empty, has codimension δ [29]. There is a non-empty open U 1 ⊂ U 0 , such that for a point = ( 1 , . . . , r ) ∈ U 1 , the fibre over is Finally, there is an non-empty open U 2 ⊂ U 1 , such that for ∈ U 2 , and any point (C 1 , . . . , C r ) ∈ φ −1 ( ), the curves C 1 , . . . , C r intersect transversely, i.e.
is a nodal curve, in which the union is taken in the plane V ⊂ P 3 corresponding to the image of (C 1 , . . . , C r ) in Gr. By a count of dimensions, the sets π i (W i ) intersect properly in Gr. It follows that the contribution to N δ,d by curves of this type is given by We formulate the conclusion of the discussion above in the following proposition.

Proposition 8.4 The number of δ-nodal plane curves in
μ( ) ν d 1 ,δ 1 ,n 1 · · · ν d r ,δ r ,n r in which the second sum is taken over unordered r -tuples (or multisets) of triples

Remark 8.5
The generality condition in the proposition means that the lines have to be general in the sense of Lemma 8.1, for every triple (d,δ,n) appearing in the second sum.
Using the proposition, we will compute the numbers N δ,d , for 0 ≤ δ ≤ 6 and δ = 8 and certain low d. We will compare results with with the numbers N δ (d), with N δ the node polynomial as computed in the previous section, and given in the appendix for δ ≤ 12. In these cases, we can choose d in such a way that the irreducible components of the curves are smooth or 1-nodal.

Remark 8.7
See Tables 1, 2 and 3. Since we can apply Theorem A only under the assumption d ≥ δ, we have not proved that the value of the polynomial N δ (d) equals the curve count N δ,d in the cases indicated with *. However, as we will prove below, in these cases the polynomials give the right numbers. In general, the Göttsche threshold d ≥ δ/2 + 1 for nodal curves in P 2 , determined in [20], seems to hold also in our case, i.e. that the node polynomials N δ have value N δ,d in these d.
It follows that the number of nodes of an irreducible plane curve of degree d is bounded by its arithmetic genus (d−1)(d−2) 2 . Now use the equations (19).
We have the following elementary lemma.
Hence we can compute the classes by the methods of the previous sections. In the case that δ = 0, 1, however, the classes can be computed by elementary means. Let δ = 0. The locus of curves in B = P(Sym d (U * )) intersecting a line, is cut out by a section of O Gr (d) ⊗ O B (1). Note that a general such curve is smooth. It follows that we have the equation the right hand side of which can easily be calculated. Now let δ = 1. For a curve C ⊂ P 2 , given by a degree d polynomial f , the singular locus is given by the equations of bundles on S × Gr B, in which S = P(U) → Gr is the universal P 2 -bundle over the Grassmannian. The homomorphisms simultaneously vanish on the singular locus of the fibres of the universal curve C → B. As curves with one node lie dense in this locus, it follows that the class of the closure of the locus of 1-nodal curves in B is given by We conclude that Again, by a straight-forward computation, we obtain the numbers in the third column.
has the following interpretation: it is the number of planar curves C in P 3 of degree d, with δ nodes, intersecting general lines 1 , . . . , n−i ⊂ P 3 , such that the plane of the curve contains general points P 1 , . . . , P i ∈ P 3 . For certain cases, this enumerative problem has already been studied by Schubert using his calculus introduced in [28]. E.g. he treats conics, planar and twisted cubics and planar quartic curves in P N that intersect points, lines and planes. The curves are allowed to have nodal singularities, or a cusp in the case of the planar cubic. The degrees of the classes in the second and third column of Lemma 8.8 can be found in §20 and §24 of loc. cit. respectively. By Proposition 8.4 it follows that the numbers in Tables 1, 2 and 3 can computed by 19th century geometry and some elementary combinatorics.