# Relative node polynomials for plane curves.

1 Introduction and Main ResultsThe Severi degree [N.sup.d,[delta]] is the degree of the Severi variety of (possibly reducible) nodal plane curves of degree d with [delta] nodes. Equivalently, [N.sup.d,[delta]] is the number of such curves passing through (d+3)d/2 - [delta] generic points in the complex projective plane [CP.sup.2]. Severi varieties have received considerable attention since they were introduced by F. Enriques [Enr12] and F. Severi [Sev21] around 1915. Much later, in 1986, J. Harris [Har86] achieved a celebrated breakthrough by showing their irreducibility.

In 1994, P. Di Francesco and C. Itzykson [DFI95] conjectured that the numbers [N.sup.d,[delta]] are given by a polynomial in d, for a fixed number of nodes [delta], provided d is large enough. S. Fomin and G. Mikhalkin [FM10, Theorem 5.1] established this polynomiality in 2009. More precisely, they showed that there exists, for every [delta] [greater than or equal to] 1, a node polynomial [N.sub.[delta]](d) which satisfies [N.sup.d,[delta]] = [N.sub.[delta]] (d) for all d [greater than or equal to] 2[delta].

The polynomiality of [N.sup.d,[delta]] and the polynomials [N.sub.[delta]](d) were known in the 19th century for [delta] = 1, 2, 3. For [delta] = 4,5,6, this was only achieved by I. Vainsencher [Vai95] in 1995. In 2001, S. Kleiman and R. Piene [KP04] settled the cases [delta] = 7, 8. In [Blo11], the author computed [N.sub.[delta]](d) for [delta] [less than or equal to] 14 and improved the threshold of S. Fomin and G. Mikhalkin by showing that [N.sup.d,[delta]] = [N.sub.[delta]](d) provided d [greater than or equal to] [delta].

Severi degrees can be generalized to incorporate tangency conditions to a fixed line L [subset] [CP.sup.2]. More specifically, the relative Severi degree [N.sup.[delta].sub.[alpha][beta]] is the number of (possibly reducible) nodal plane curves with [delta] nodes which have tangency of order i to L at [[alpha].sub.i] fixed points (chosen in advance) and tangency of order i to L at [[beta].sub.i] unconstrained points, for all i [greater than or equal to] 1, and which pass through an appropriate number of generic points. Equivalently, is the degree of the generalized Severi variety studied in [CH98, Vak00]. By Bezout's Theorem, the degree of a curve with tangencies of order ([alpha], [beta]) equals d = [[summation].sub.i[greater than or equal to]1] i([[alpha].sub.i] + [[beta].sub.i])

The number of point conditions (for a potentially finite count) is (d+1)d/2 - [delta] + [[beta].sub.1] + [[beta].sub.2] + .... We recover non-relative Severi degrees by specializing to [alpha] = (0,0, ...) and [beta] = (d, 0,0, ...). The numbers [N.sup.[delta].sub.[alpha][beta]] are determined by the rather complicated Caporaso-Harris recursion [CH98].

In this paper, we show that much of the story of (non-relative) node polynomials carries over to relative Severi degrees. Our main result is that, up to a simple combinatorial factor and for fixed [delta] [greater than or equal to] 0, the relative Severi degrees [N.sup.[delta].sub.[alpha][beta]] are given by a multivariate polynomial in [[alpha].sub.1], [[alpha].sub.2], ..., [[beta].sub.1], [[beta].sub.2], ..., provided that [[beta].sub.1] + [[beta].sub.2] + ... is sufficiently large. For a sequence [alpha] = ([[alpha].sub.1], [[alpha].sub.2], ...) of non-negative integers with finitely many [[alpha].sub.i] non-zero we write

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Throughout the paper we use the grading deg([[alpha].sub.i]) = deg([[beta].sub.i]) = 1 (so that d and [absolute value of [beta]] are homogeneous of degree 1 ). The following is our main result.

Theorem 1.1 For every [delta] [greater than or equal to] 0, there is a combinatorially defined polynomial [N.sub.[delta]]([[alpha].sub.1], [[alpha].sub.2], ...; [[beta].sub.1], [[beta].sub.1], ...) of (total) degree 3[delta] such that, for all [[alpha].sub.1], [[alpha].sub.2], ..., [[beta].sub.1], [[beta].sub.2], ... with [absolute value of [beta]] [greater than or equal to] [delta], the relative Severi degree [N.sup.[delta].sub.[alpha],[beta]] is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.1)

We call [N.sub.[delta]]([alpha]; [beta]) the relative node polynomial and use the same notation as in the non-relative case if no confusion can occur. We do not need to specify the number of variables in light of the following stability condition.

Theorem 1.2 For [delta] [greater than or equal to] 0 and vectors [alpha] = ([[alpha].sub.1], ..., [[alpha].sub.m]), [beta] = ([[beta].sub.1], ..., [[beta].sub.m']) with [absolute value of [beta]] [greater than or equal to] [delta], it holds that

N[delta]([alpha], 0; [beta]) = N[delta]([alpha]; [beta]) and [N.sub.[delta]]([alpha]; [beta], 0) = [N.sub.[delta]]([alpha]; [beta])

as polynomials. Therefore, there exists a formal power series [N.sup.[infinity].sub.[delta]]([alpha]; [beta]) in infinitely many variables [[alpha].sub.1], [[alpha].sub.2], ..., [[beta].sub.1], [[beta].sub.2], ... which specializes to all relative node polynomials under [[alpha].sub.m+1] = [[alpha].sub.m+2] = ... = 0 and [[beta].sub.m'+1] = [[beta].sub.m'+2] = ... = 0, for various m, m' [greater than or equal to] 0.

In fact, even more is true.

Proposition 1.3 For [delta] [greater than or equal to] 0 the relative node polynomial [N.sub.[delta]]([alpha], [beta]) is a polynomial in d, [absolute value of [beta]], [[alpha].sub.1], ..., [[alpha].sub.[delta]], and [[beta].sub.1], ..., [[beta].sub.[delta]] where d = [[summation].sub.i [greater than or equal to] 1] i([[alpha].sub.i] + [[beta].sub.i]).

Using the combinatorial description we provide a method for computing the relative node polynomials for arbitrary [delta] (see Sections 3 and 4). We utilize it to compute [N.sub.[delta]]([alpha]; [beta]) for [delta] [less than or equal to] 6. The polynomials [N.sub.0] and [N.sub.1] already appeared (implicitly) in [FM10, Section 4.2].

Theorem 1.4 The relative node polynomials [N.sub.[delta]]([alpha]; [beta]), for 5 < 3 (resp., 5 < 6) are as listed in [Blo10, Appendix A] (resp., as provided in the ancillary files accompanying [Blo10]).

The polynomial [N.sub.[delta]]([alpha]; [beta]) is of degree 3[delta] by Theorem 1.1. We compute its terms of degree [greater than or equal to] 3[delta] - 2.

Theorem 1.5 The terms of [N.sub.[delta]]([alpha]; [beta]) of (total)

degree [greater than or equal to] 3[delta] - 2 are given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where d = [[summation].sub.i[greater than or equal to]1] i([[alpha].sub.i] + [[beta].sub.i]).

Theorem 1.5 can be extended to terms of [N.sub.[delta]]([alpha], [beta]) of degree [greater than or equal to] 3[delta] - 7 (see Remark 5.2). We observe that all coefficients of [N.sub.[delta]] ([alpha]; [beta]) in degree [greater than or equal to] 3[delta] - 2 are of the form [3.sup.[delta]]/[delta]! times a polynomial in [delta]. Without computating the coefficients, we can extended this further. It is conceivable to expect this property to hold for arbitrary degrees.

Proposition 1.6 Every coefficient of N[delta]([alpha]; [beta]) in degree [greater than or equal to] 3[delta] - 7 is given, up to a factor of [3.sup.[delta]]/[delta]!, by a polynomial in [delta] with rational coefficients.

Our approach to planar enumerative geometry is combinatorial and inspired by tropical geometry, a piecewise-linear analogue of algebraic geometry (see, for example, [Gat06]). By the celebrated Correspondence Theorem of G. Mikhalkin [Mik05, Theorem 1] one can replace the algebraic curve count in [CP.sup.2] by an enumeration of certain tropical curves. E. Brugalle and G. Mikhalkin [BM07, BM09] introduced a class of decorated graphs, called (marked) floor diagrams (see Section 2), which, if counted correctly, are equinumerous to such tropical curves. We use a version of these results which incorporates tangency conditions due to S. Fomin and G. Mikhalkin [FM10] (see Theorem 2.4). S. Fomin and G. Mikhalkin also introduced a template decomposition of floor diagrams which we extend to be suitable for the relative case. This decomposition is crucial in the proofs of all results in this paper, as is the reformulation of algebraic curve counts in terms of floor diagrams.

For some related work see [AB10], where F. Ardila and the author generalized the polynomiality of Severi degrees to a class of toric surfaces which contains [CP.sup.1] x [CP.sup.1] and Hirzebruch surfaces but which are non-smooth in general. A main feature is that we show polynomiality not only in the multi-degree of the curves but also in the parameters of the surface. In [BGM11], A. Gathmann, H. Markwig and the author defined Psi-floor diagrams which enumerate plane curves which satisfy point and tangency conditions, and conditions given by Psi-classes. We proved a Caporaso-Harris type recursion for Psi-floor diagrams, and show that relative descendant Gromov-Witten invariants equal their tropical counterparts.

This paper is organized as follows. In Section 2 we review the definition of floor diagrams and their markings. In Section 3 we introduce a new decomposition of floor diagrams which is compatible with tangency conditions. In Section 4 we discuss Theorems 1.1,1.2, 1.4 and Proposition 1.3. In Section 5 we discuss Theorem 1.5 and Proposition 1.6. For complete proofs of all statements see [Blo10].

Acknowledgements

I thank the two anonymous referees for their valuable comments and suggestions.

2 Floor diagrams and relative markings

Floor diagrams are a class of decorated graphs which, if counted with appropriate multiplicity, enumerate plane curves with prescribed properties. They were introduced by E. Brugalle and G. Mikhalkin [BM07, BM09] in the non-relative case and generalized to the relative setting by S. Fomin and G. Mikhalkin [FM10]. We begin with a review of the relative setup following notation of [FM10].

Definition 2.1 A floor diagram D on a vertex set {1, ..., d} is a directed graph (possibly with multiple edges) with edge weights w(e) [member of] [Z.sub.>0] satisfying:

1. The edge directions preserve the vertex order, i.e., for each edge i [right arrow] j of D we have i < j.

2. (Divergence Condition) For each vertex j of D:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This means that at every vertex of D the total weight of the outgoing edges is larger by at most 1 than the total weight of the incoming edges.

The degree of a floor diagram D is the number of its vertices. It is connected if its underlying graph is. Note that in [FM10] floor diagrams are required to be connected. If D is connected its genus is the genus of the underlying graph (or the first Betti number of the underlying topological space). The cogenus of a connected floor diagram D of degree d and genus g is given by [delta](D) = (d-1)2(d-2)/2 - g. If D is not connected let [d.sub.1], [d.sub.2], ... and [[delta].sub.1], [[delta].sub.2], ... be the degrees and cogenera, respectively, of its connected components. Then the cogenus of D is [delta](D) = [[summation].sub.j] [[delta].sub.j] + [[summation].sub.j<j]' [d.sub.j][d.sub.j]. Via the correspondence between algebraic curves and floor diagrams [BM09, Theorem 2.5] these notions correspond literally to the respective analogues for algebraic curves. Connectedness corresponds to irreducibility. Lastly, a marked floor diagram D has multiplicity (i)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We draw floor diagrams using the convention that vertices in increasing order are arranged left to right. Edge weights of 1 are omitted.

Example 2.2 An example of a floor diagram of degree d = 4, genus g =1, cogenus 5 = 2, divergences 1,1,0, -2, and multiplicity [mu] = 4 is drawn below.

[ILLUSTRATION OMITTED]

To enumerate algebraic curves with floor diagrams we need the notion of markings of such diagrams. Our notation, which is more convenient for our purposes, differs slightly from [FM10] where S. Fomin and G. Mikhalkin define relative markings relative to the partitions [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. In the sequel, all sequences are sequences of non-negative integers with finite support.

Definition 2.3 For two sequences [alpha], [beta] we define an ([alpha], [beta])-marking of a floor diagram D of degree d = [[summation].sub.i[greater than or equal to]1]([[alpha].sub.i] + [[beta].sub.i]) by the following four step process which we illustrate in the case of Example 2.2 for [alpha] = (1, 0,0, ...) and [beta] = (1,1,0, 0, ...).

Step 1: Fix a pair of collections of sequences ({[[alpha].sup.i]}, {[[beta].sup.i]}), where i runs over the vertices of D, with:

1. The sums over each collection satisfy [[summation].sup.d.sub.i=1] [[alpha].sup.i] = [alpha] and [[summation].sup.d.sub.i=1] [[beta].sup.i] = [beta].

2. For all vertices i of d we have [[summation].sub.j[greater than or equal to]1]([[alpha].sup.i.sub.j] + [[beta].sup.i.sub.j]) = 1 - div(i).

The second condition says that the "degree of the pair ([[alpha].sup.i], [[beta].sup.i])" is compatible with the divergence at vertex i. Each such pair ({[[alpha].sup.i]}, {[[alpha].sup.i]}) is called compatible with d and ([alpha], [beta]). We omit writing down trailing zeros.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Step 2: For each vertex i of d and every j [greater than or equal to] 1 create [[beta].sup.i.sub.j] new vertices, called [beta]-vertices and illustrated as [??], and connected them to i with new edges of weight j directed away from i. For each vertex i of d and every j [greater than or equal to] 1 create [[alpha].sup.i.sub.j] new vertices, called [alpha]-vertices and illustrated as [dot encircle], and connected them to i with new edges of weight j directed away from i.

[ILLUSTRATION OMITTED]

Step 3: Subdivide each edge of the original floor diagram D into two directed edges by introducing a new vertex for each edge. The new edges inherit weights and orientations. Call the resulting graph [??].

[ILLUSTRATION OMITTED]

Step 4: Linearly order the vertices of [??] extending the order of the vertices of the original floor diagram d such that, as in d, each edge is directed from a smaller vertex to a larger vertex. Furthermore, we require that the [alpha]-vertices are largest among all vertices, and for every pair of [alpha]-vertices i' > i, the weight of the i'-adjacent edge is larger than or equal to the weight of the i-adjacent edge.

[ILLUSTRATION OMITTED]

We call the extended graph [??], together with the linear order on its vertices, an ([alpha], [beta])-marked floor diagram, or an ([alpha], [beta])-marking of the floor diagram D.

We need to count ([alpha], [beta])-marked floor diagrams up to equivalence. Two ([alpha], [beta])-markings [[??].sub.1], [[??].sub.2] of a floor diagram D are equivalent if there exists a weight preserving automorphism of weighted graphs mapping [[??].sub.1] to [[??].sub.2] which fixes the vertices of D. The number of markings [v.sub.[alpha],[beta]](D) is the number of ([alpha], [beta])-marked floor diagrams [??] up to equivalence. Furthermore, we write [[mu].sub.[beta]](D) for the product [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] ... [mu](D). The next theorem follows from [FM10, Theorem 3.18] by a straight-forward extension of the inclusion-exclusion procedure of [FM10, Section 1] which was used to conclude [FM10, Corollary 1.9] (the non-relative count of reducible curves via floor diagrams) from [FM10, Theorem 1.6] (the non-relative count of irreducible curves via floor diagrams).

Theorem 2.4 For any [delta] [greater than or equal to] 1, the relative Severi degree [N.sup.[delta].sub.[alpha],[beta]] is given by

[N.sup.[delta].sub.[alpha],[beta]] = [summation over (D)][[mu].sub.[beta]](D)[v.sub.[alpha],[beta]](D),

where the sum is over (possibly disconnected) floor diagrams d of degree d = [[summation].sub.i[greater than or equal to]1] i([[alpha].sub.i] + [[beta].sub.i]) and cogenus [delta].

3 Relative Decomposition of Floor Diagrams

In this section we introduce a new decomposition of floor diagrams compatible with tangency conditions. It is crucial in the proofs of all results stated in Section 1. The new decomposition is an extension of ideas of S. Fomin and G. Mikhalkin [FM10]. We start out by reviewing their key gadget.

Definition 3.1 A template [GAMMA] is a directed graph (possibly with multiple edges) on vertices {0, ...,l}, where l [greater than or equal to] 1, and edge weights w(e) [member of] [Z.sub.>0], satisfying:

1. If i [right arrow] j is an edge then i < j.

2. Every edge [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] has weight w(e) [greater than or equal to] 2. (No "short edges.")

3. For each vertex j, 1 [less than or equal to] j [less than or equal to] l - 1, there is an edge "covering" it, i.e., there exists an edge i [right arrow] k with i < j < k.

Every template [GAMMA] comes with some numerical data associated with it. Its length l([GAMMA]) is the number of vertices minus 1. The product of squares of the edge weights is its multiplicity [mu]([GAMMA]). Its cogenus [delta]([GAMMA]) is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

For 1 [less than or equal to] j [less than or equal to] l([GAMMA]) let [[??].sub.j] = [[??].sub.j]([GAMMA]) denote the sum of the weights of edges [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] with i < j [less than or equal to] k and define

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

This makes [k.sub.min] ([GAMMA]) the smallest positive integer k such that [GAMMA] can appear in a floor diagram on {1, 2, ...} with left-most vertex k. See [FM10, Figure 10] for a list of all templates [GAMMA] with [delta]([GAMMA]) [less than or equal to] 2.

Our new decomposition of a floor diagram D depends on two (infinite) matrices A and B of non-negative integers. We require both to have only finitely many non-zero entries all of which lie above the respective dth row, where d is the degree of D.

The triple (D, A, B) decomposes as follows. Let l(A) and l(B) be the largest row indices such that A and B have a non-zero entry in this row, respectively. After we remove all "short edges" from D, i.e., all edges of weight 1 between consecutive vertices, the resulting graph is an ordered collection of templates ([[GAMMA].sub.1], ..., [[GAMMA].sub.r]), listed left to right. Let [k.sub.s] be the smallest vertex in D of each template [[GAMMA].sub.s]. Record all pairs ([[GAMMA].sub.s], [k.sub.s]) which satisfy [k.sub.s] + l([[GAMMA].sub.s]) [less than or equal to] d - max(l(A), l(B)). Record the remaining templates together with all vertices i, for i [greater than or equal to] max(l(A), l(B)) in one graph A on vertices 0, ..., l by shifting the vertex labels by d - l. See Example 3.4 for an example of this decomposition. Furthermore, by construction, if m is the number of recorded pairs ([[GAMMA].sub.s], [k.sub.s]), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (3.1)

Fix a floor diagram D. A partitioning of [alpha] and [beta] into a compatible pair of collections ({[[alpha].sup.i], [[beta].sup.i]}) (see Step 1 in Definition 2.3), where i runs over the vertices of D, determines a pair of matrices A, B, if [[alpha].sub.1], [[alpha].sub.2], ..., [[beta].sub.1], [[beta].sub.2], ... are large enough as follows. Define the ith row [a.sub.i] resp. [b.sub.i] of the matrices A resp. B to be the sequence [[alpha].sup.d-i] resp. [[beta].sup.d-i], for i [greater than or equal to] 1, where d is the degree of D. (If d - i [less than or equal to] 0, set

[a.sub.i] = [b.sub.i] = 0.) In other words, the jth entry [a.sub.ij] in row i of A is the number of [alpha]-edges of weight i adjacent to the (j + 1)st vertex of [LAMBDA], counted from the right, and similarly for B (see Example 3.2). The sequences [[alpha].sup.d] and [[beta].sup.d] (which are attached to the right-most vertex of D) satisfy

[[alpha].sup.d] = [alpha] - [summation over (i[greater than or equal to]1)] [a.sub.i] and [[beta].sup.d] = [beta] - [summation over (i[greater than or equal to]1)] [b.sub.i] (3.2)

if both expression are (component-wise) non-negative.

Example 3.2 For [alpha] = (1,1), [beta] = (4,1) and the floor diagram D pictured below, the partitioning determines the matrices

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

determines the matrices

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In light of (3.2) we consider, for given tangency conditions [alpha] and [beta], only triples (D, A, B) with

[summation over (i[greater than or equal to]1)] [a.sub.i] [less than or equal to] [alpha] and [summation over (i[greater than or equal to]1)] [b.sub.i] [less than or equal to] [beta] (always component-wise), (3.3)

For fixed d, the decomposition

(D,A,B) [right arrow] ({([[GAMMA].sub.s], [k.sub.s])}, [LAMBDA], A, B). (3.4)

is reversible if the data on the right-hand side satisfies (3.1) and the tuple ([LAMBDA], A, B) is an "extended template."

Definition 3.3 A tuple ([LAMBDA], A, B) is an extended template of length l = l(A) = l([LAMBDA], A, B) if [LAMBDA] is a directed graph (possibly with multiple edges) on vertices {0, ..., l}, where l [greater than or equal to] 0, with edge weights w(e) [member of] [Z.sub.>0], satisfying:

1. If i [right arrow] j is an edge then i < j.

2. Every edge [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] has weight w(e) [greater than or equal to] 2. (No "short edges.")

Moreover, we require A and B to be (infinite) matrices with non-negative integral entries and finite support, and we write l(A) and l(B) for the respective largest row indices of A and B of a non-zero entry. Additionally, we demand that l([LAMBDA]) [greater than or equal to] max(l(A), l(B)) and that, for each 1 [less than or equal to] j < l - max(l(A), l(B)), there is an edge i [right arrow] k of [LAMBDA] with i < j < k.

Example 3.4 An example of a decomposition of a floor diagram D subject to the matrices A and B of Example 3.2 is pictured below. Once we fix the degree of the floor diagram the decomposition is reversible (here d = 8).

[ILLUSTRATION OMITTED]

The cogenus of an extended template ([LAMBDA], A, B) is the sum of the cogenera [delta]([LAMBDA]), [delta](A) and [delta](B), where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and similarly for B. It is not hard to see that the correspondence (3.4) is cogenus preserving in the sense that

[delta](D) = ([m.summation over (i=1)][delta]([[GAMMA].sub.i])) + [delta]([LAMBDA]) + [delta](A) + [delta](B).

With an extended template ([LAMBDA], A, B) we associate the following numerical data. For 1 [less than or equal to] j [less than or equal to] l([LAMBDA]) let [[??].sub.j]([LAMBDA]) denote the sum of the weights of edges i [right arrow] k of [LAMBDA] with i < j [less than or equal to] k. Define [d.sub.min]([LAMBDA], A, B) to be the smallest positive integer d such that ([LAMBDA], A, B) can appear (at the right end) in a floor diagram on {1, 2, ..., d}. We will see later that [d.sub.min] is given by an explicit formula. For a matrix A = ([a.sub.ij]) of non-negative integers with finite support define the "weighted lower sum sequence" wls(A) by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (3.5)

We now define the number of "markings" of templates and extended templates and relate them to the number of ([alpha], [beta])-markings of the corresponding floor diagrams. To each template r we associate a polynomial as follows. For k [greater than or equal to] [k.sub.min]([GAMMA]) let [[GAMMA].sub.(k)] denote the graph obtained from [GAMMA] by first adding k + i - 1 - [[??].sub.i] short edges connecting i - 1 to i, for 1 [less than or equal to] i [less than or equal to] l([GAMMA]), and then subdividing each edge of the resulting graph by introducing one new vertex for each edge. By [FM10, Lemma 5.6] the number of linear extensions (up to equivalence, see the paragraph after Definition 2.3) of the vertex poset of the graph [[GAMMA].sub.(k)] extending the vertex order of r is given by a polynomial [P.sub.[GAMMA]](k) in k, if k [greater than or equal to] [k.sub.min]([GAMMA])(see [FM10, Figure 10]).

For each pair of sequences ([alpha], [beta]) and each extended template ([LAMBDA], A, B) satisfying (3.3) and d [greater than or equal to] [d.sub.min], where d = [[summation].sub.i[greater than or equal to]1] i([[alpha].sub.i] + [[beta].sub.i]), we define its "number of markings" as follows. Write l = l([LAMBDA]) and let P([LAMBDA], A, B) be the poset obtained from [LAMBDA] by

1. first creating an additional vertex l + 1 (> l),

2. then adding [b.sub.ij] edges of weight j between l - i and l + 1, for all 1 [greater than or equal to] i [less than or equal to] l and j [greater than or equal to] 1,

3. then adding [[beta].sub.j] - [[summation].sub.[greater than or equal to]1] [b.sub.ij] edges of weight j between l and l + 1, for j [greater than or equal to] 1,

4. then adding

d - l([LAMBDA]) + i - 1 - [[??].sub.i]([LAMBDA]) - wls[(A).sub.i+1-i] - wls[(B).sub.i+1-i] (3.6)

("short") edges of weight 1 connecting i - 1 and i, for 1 [less than or equal to] i [less than or equal to] l, and finally

5. subdividing all edges of the resulting graph by introducing a midpoint vertex for each edge.

Denote by [Q.sub.([LAMBDA],A,B)]([alpha]; [beta]) the number of linear orderings on P([LAMBDA], A, B) (up to equivalence) which extend the linear order on [LAMBDA]. As d [greater than or equal to] [d.sub.min]([LAMBDA], A, B) if and only if (3.6) is non- negative, for 1 [less than or equal to] i [less than or equal to] l, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

For sequences s, [t.sub.1], [t.sub.2], ... with s [greater than or equal to] [[summation].sub.i][t.sub.i] (component- wise) we denote by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] the multinomial coefficient of sequences.

We obtain all ([alpha], [beta])-markings of the floor diagram D that come from a compatible pair of sequences ({[[alpha].sup.i]}, {[[beta].sup.i]}) by independently ordering the [alpha]-vertices and the non-a-vertices. The number such markings is given (via the correspondence (3.4)) by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (3.7)

where [[alpha].sup.T.sub.1], [[alpha].sup.T.sub.2], ... are the column vectors of A. We conclude this section by recasting relative Severi degrees in terms of templates and extended templates.

Proposition 3.5 For any [delta] [greater than or equal to] 1, the relative Severi degree [N.sup.[delta].sub.[alpha],[beta]] is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (3.8)

where the first sum is over all collections ([[GAMMA].sub.1], ..., [[GAMMA].sub.m]) of templates and all extended templates ([LAMBDA], A, B) satisfying (3.3), d [greater than or equal to] [d.sub.min]([LAMBDA], A, B) and

[m.summation over (i=1)][delta]([[GAMMA].sub.i]) + [delta]([LAMBDA]) + [delta](A) + [delta](B) = [delta],

and the second sum is over all positive integers [k.sub.1], ..., [k.sub.m] which satisfy (3.1).

4 Relative Severi Degrees and Polynomiality

We now turn to the discussion of the proofs of our main results by first mentioning a number of technical lemmata whose proofs can be found in [Blo10]. For a graph G, we denote by #E(G) the number of edges of G. We write [parallel]A[[parallel].sub.1] = [[summation].sub.i,j[greater than or equal to]1] [a.sub.ij] for the 1-norm of a (possibly infinite) matrix A = ([a.sub.ij]).

Lemma 4.1 For every extended template ([LAMBDA], A, B) there is a polynomial [q.sub.([LAMBDA],A,B)] in [[alpha].sub.1], [[alpha].sub.2], ..., [[beta].sub.1], [[beta].sub.2], ... of degree #E([LAMBDA]) + [parallel]B[[parallel].sub.1] + [delta](B) such that for all [alpha] and [beta] satisfying (3.3) the number [Q.sub.([LAMBDA],A,B)]([alpha]; [beta]) of linear orderings (up to equivalence) of the poset P ([LAMBDA], A, B) is given by

[Q.sub.([LAMBDA],A,B)]([alpha]; [beta]) = ([absolute value of [beta]] - [delta](B))!/[beta]! x [q.sub.([LAMBDA],A,B)] ([alpha]; [beta])

provided [[summation].sub.i[greater than or equal to]1i([[alpha].sub.i] + [[beta].sub.i]) [greater than or equal to] [d.sub.min] ([LAMBDA],A,B).

Recall that, for an extended template ([LAMBDA], A, B), we defined [d.sub.min] = [d.sub.min]([LAMBDA], A, B) to be the smallest d [greater than or equal to] 1 such that d - l([LAMBDA]) + i - 1 [greater than or equal to] [[??].sub.i]( [LAMBDA]) + wls[(A).sub.l([LAMBDA])+1-i] + wls[(B).sub.l([LAMBDA])+1-i] for all 1 [less than or equal to] i [less than or equal to] l([LAMBDA]). Let [i.sub.0] be the smallest i for which equality is attained (it is easy to see that equality is attained for some i). Define the quantity s(A, A, B) to be the number of edges of [LAMBDA] from [i.sub.0] - 1 to [i.sub.0] (of any weight). See [Blo10, Figure 2] for examples.

Lemma 4.2 For any extended template ([LAMBDA], A, B) and any [alpha], [beta] [greater than or equal to] 0 (component- wise) with

[d.sub.min]([LAMBDA], A, B) - s([LAMBDA], A, B) [less than or equal to][summation over (i[greater than or equal to]1)] i([[alpha].sub.i] + [[beta].sub.i]) [less than or equal to] [d.sub.min]([LAMBDA], A, B) - 1

we have [q.sub.([LAMBDA],A,B)]([alpha]; [beta]) = 0, where [q.sub.([LAMBDA],A,B)] is the polynomial of Lemma 4.1.

The next lemma specifies which extended templates are compatible with a given degree.

Lemma 4.3 For every extended template (A, A, B) we have

[d.sub.min] ([LAMBDA], A, B) - s([LAMBDA], A, B) [less than or equal to] [delta]([LAMBDA]) + [delta](A) + [delta](B) + 1.

Proof of Theorems 1.1 and 1.2: The basic idea of the proof is to show that all summands of (3.8) are polynomial in [[alpha].sub.1], [[alpha].sub.2], ..., [[beta].sub.1], [[beta].sub.2], ... (up to the combinatorial factor), thus contribute polynomially to the relative node polynomial N[delta]([alpha]; [beta]). The first factor of each summand is an iterated "discrete integral" of polynomials and thus polynomial in d. For the second factor we use Lemma 4.1. Then we use Lemmata 4.2 and 4.3 to reduce the polynomiality threshold. For a detailed proof see [Blo10].

Remark 4.4 Expression (3.8) gives, in principle, an algorithm to compute the relative node polynomial [N.sub.[delta]]([alpha]; [beta]), for any [delta] [greater than or equal to] 1. In [Blo11, Section 3] we explain how to generate all templates of a given cogenus, and how to compute the first factor in (3.8). The generation of all extended templates of a given cogenus from the templates is straightforward, as is the computation of the second factor in (3.8).

Proof of Theorem 1.4: Proposition 3.5 gives a combinatorial description of relative Severi degrees. The proof of Lemma 4.1 (see [Blo10]) provides a method to calculate the polynomial [Q.sub.([LAMBDA],A,B)]([alpha]; [beta]). All terms of expression (3.8) are explicit or can be evaluated using the techniques of [Blo11, Section 3]. This reduces the calculation to a (non-trivial) computer calculation.

5 Coefficients of Relative Node Polynomials

We now turn towards the computation of the coefficients of the relative node polynomial [N.sub.[delta]]([alpha]; [beta]) of large degree for any 5. We propose a method to compute all terms of [N.sub.[delta]]([alpha]; [beta]) of degree [greater than or equal to] 3[delta] - t, for any given t [greater than or equal to] 0. This method was used (with t = 2) to compute the terms in Theorem 1.5.

The main idea of the algorithm is that, even for general [delta], only a small number of summands of (3.8) contribute to the terms of [N.sub.[delta]]([alpha]; [beta]) of large degree. A summand of (3.8) is indexed by a collection of templates [??] = {[[GAMMA].sub.s]} and an extended template ([LAMBDA], A, B). To determine whether this summand might contribute to N[delta]([alpha]; [beta]) we define the (degree) defects

* of the collection of templates [??] by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and

* of the extended template ([LAMBDA], A, B) by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The following lemma restricts the indexing set of (3.8) to the relevant terms, if only the leading terms of [N.sub.[delta]]([alpha]; [beta]) are of interest. For a proof see [Blo10].

Lemma 5.1 The summand of (3.8) indexed by [??] and ([DELTA], A, B) is of the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where P ([alpha]: [beta]) is a polynomial in [[alpha].sub.1], [[alpha].sub.2], ..., [[beta].sub.1], [[beta].sub.2], ... of degree [less than or equal to] 3[delta] - def([??]) - def ([DELTA], A, B).

Therefore, to compute the coefficients of degree [greater than or equal to] 3[delta] - t of N[delta]([alpha]; [beta]) for some t [greater than or equal to] 0, it suffices to consider only summands of (3.8) with def ([??]) [less than or equal to] t and def ([DELTA], A, B) [less than or equal to] t.

One can proceed as follows. First, we can compute, for some formal variable [??], the terms of degree [greater than or equal to] 2[??] - t of the first factor of (3.8) to [N.sub.[??]]([alpha]; [beta]), that is the terms of degree [greater than or equal to] 2[??] - t of

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (5.1)

where the first sum is over all collections of templates [??] = ([[GAMMA].sub.1], ..., [[GAMMA].sub.m]) with [delta]([??]) = [??]. The leading terms of [R.sub.[??]](d) can be computed with a slight modification of [Blo11, Algorithm 2] (by replacing, in the notation of [Blo11], [C.sup.end] by C and [M.sup.end] by M). The algorithm relies on the polynomiality of solutions of certain polynomial difference equations, which has been verified for t [less than or equal to] 7, see [Blo11, Section 5] for more details.

Finally, to compute the coefficients of degree [greater than or equal to] 3[delta] - t, it remains to compute all extended templates ([LAMBDA], A, B) with def ([LAMBDA], A, B) [less than or equal to] t and collect the terms of degree [greater than or equal to]3[delta] - t of the polynomial

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (5.2)

where, as before, [a.sup.T.sub.1], [a.sup.T.sub.2], ... denote the column vectors of the matrix A, [q.sub.([LAMBDA],A,B)]([alpha]; [beta]) is the polynomial of Lemma 4.1, and [??] = [delta] - [delta]([LAMBDA], A, B). Notice that, for an indeterminant x and integers c [greater than or equal to] 0 and [delta] [greater than or equal to] 1, we have the expansion

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where s(n, m) is the Stirling number of the first kind [Sta97, Section 1.3] for integers n, m [greater than or equal to] 0. Furthermore, with [delta]' = [delta] - c the coefficients s([delta]', [delta]' - t) of the sum equal [delta]'([delta]' - 1) ... ([delta]' - t) x [S.sub.t]([delta]'), where [S.sub.t] is the t-th Stirling polynomial [GKP94, (6.45)], for t [greater than or equal to] 0, and thus are polynomial in [delta]'. Therefore, we can compute the leading terms of the product in (5.2) by collecting the leading terms in the sum expansion above. Theorem 1.5 is proved by an implementation of this method.

Proof of Proposition 1.6: Using [Blo11, Algorithm 2] we can compute the terms of the polynomial [R.sub.[??]] (d) of degree [greater than or equal to] 2[??] - 7 (see [Blo11, Section 5]) and observe that all coefficients are polynomial in [??]. By the previous paragraph the coefficients of the expansion of the sum of (5.2) are polynomial in [delta]. This completes the proof.

Remark 5.2 It is straight-forward to compute the coefficients of [N.sub.[delta]]([alpha]; [beta]) of degree [greater than or equal to] 3[delta] - 7 (and thereby to extend Theorem 1.5). Algorithm 3 of [Blo11] computes the coefficients of the polynomials [R.sub.[??]](d) of degree [greater than or equal to] 2[??] - 7, and thus the desired terms can be collected from (5.2). We expect this method to compute the leading terms of [N.sub.[delta]]([alpha]; [beta]) of degree [greater than or equal to] 3[delta] - t for arbitrary t [greater than or equal to] 0 (see [Blo11, Section 5], especially Conjecture 5.5).

References

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(i) If floor diagrams are viewed as floor contractions of tropical plane curves this corresponds to the notion of multiplicity of tropical plane curves.

Florian Block ([dagger])

Department of Mathematics, University of Michigan, Ann Arbor, MI, USA

([dagger]) The author was partially supported by a Rackham One-Term Dissertation Fellowship & by the NSF grant DMS- 055588.

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Author: | Block, Florian |
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Publication: | DMTCS Proceedings |

Article Type: | Report |

Geographic Code: | 1USA |

Date: | Jan 1, 2011 |

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