Characteristic numbers of rational cuspidal space curves

We solve the problem of computing characteristic numbers of rational space curves with a cusp, where there may or may not be a condition on the node. The solution is given in the form of effective recursions. We give explicit formulas when the dimension of the ambient projective space is at most 5. Many numerical examples are provided. A C++ program implementing most of the recursions is available upon request


introduction
Charateristic number of curves in projective spaces is a classical problem in algebraic geometry: how many curves in P r of given degree and genus that pass through a general set of linear subspaces, and are tangent to a general set of hyperplanes?The advent of stable maps has provided a powerful tool to attack such problem, especially in low genus.In this paper, we use stable maps to solve the problem of counting rational cuspidal space curves, where one may impose tangency conditions and an incident condition on the cusp.This is a quick application of the results in [N].
The number of rational cuspidal plane curves satisfying incidence conditions were compute in [P1].The method was to use intersections of divisors on M 0,n (2, d), as the locus of cuspidal plane curves is a divisor on M 0,n (2, d).This is no longer true for cuspidal space curves.The incidence-only characteristic numbers of rational cuspidal space curves were computed in [Z].There are a number of classical results regarding full characteristic numbers of some family of rational cuspdial curves.The numbers of cuspidal plane cubics satisfying incidence and tangency conditions, where the cusp may or may not be subject to an incident condition, were computed in [A].The analogous numbers for cuspidal plane cubics in P 3 were computed in [HM] Our approach is simple.We use the well-known divisorial relation on M 0,4 (r, d), that is the pull-back of the trivial relation on M 0,4 .We intersect this relation with a carefully choosen substack whose general members are maps that have a node in the image (rational nodal curves).Since cuspidal curves can be viewed as limits of rational nodal curves, the locus of cuspidal curves form a boundary component of the substack of rational nodal curves.Thus the above relation allows us to relate enumerative invariants of cuspidal curves to those of rational nodal curves, which can be computed from [N].
The main advantage of this method is that it is easy to implement and works well for all type of conditions.For example, there is no need to separate the family of rational cuspidal space curves into subfamilies where the cusp lie on general linear subspaces of different codimensions, as all characteristic numbers of such families can be computed using one common recursion.Both the counting of rational cuspidal space curves in this note, and the counting of rational nodal space curves in [N] will have application in couting elliptic space curves, in an upcoming paper by the author.
The content of the paper is as follows.In Section 2, we introduce the basic notations that will be used throughout this note.In Section 3, we propose and prove the recursion relating the enumerative geometry of cuspidal curves to that of rational nodal curves and that of reducible curves whose components are smooth rational intersecting at two points.Section 4 gives numerical examples and compare them to known results in literature.

Definitions and Notations
2.1.The moduli space of stable maps of genus 0 in P r .As usual, M 0,n (r, d) will denote the Kontsevich compactification of the moduli space of genus zero curves with n marked points of degree d in P r .We will also be using the notation M 0,S (r, d) where the markings are indexed by a set S. The following are Weil divisors on M 0,S (r, d): 2.2.The constraints and the ordering of constraints.We will be concerned with the number of curves passing through a constraint.Each constraint is denoted by a (r+1)−tuple ∆ as follows : (i) ∆( 0) is the number of hyperplanes that the curves need to be tangent to.
(ii) For 0 < i ≤ r, ∆(i) is the number of subspaces of codimension i that the curves need to pass through.
(iii) If the curves in consideration have a node (or cusp) and we place a condition on the node (cusp), that is the node (cusp) has to belong to a general codimension k linear subspace, then ∆ has r + 2 elements and the last element, ∆(r + 1), is k.
Note that because in general a curve of degree d will always intersect a hyperplane at d points, introducing an incident condition with a hyperplane essentially means multiplying the cycle class cut out by other conditions by d.For example, if we ask how many genus zero curves of degree 4 in P 3 that pass through the constraint ∆ = (1, 2, 3, 4, 0), that means we ask how many genus zero curves of degree 4 pass through three lines, four points, are tangent to one hyperplane, and then multiply that answer by 4 2 .We will also refer to ∆ as a set of linear spaces, hence we can say, pick a space a in ∆.
We consider the following ordering on the set of constraints, in order to prove that our algorithm will terminate later on.Let r(∆) = − i≤r i>1 ∆[i] • i 2 , and this will be our rank function.We compare two constraints ∆, ∆ using the following criteria, whose priority are in the following order : • If ∆(0) = ∆ (0) and ∆ has fewer non-hyperplane elements than ∆ does, then ∆ < ∆ .
Informally speaking, characteristic numbers where the constraints are more spread out at two ends are computed first in the recursion.We write ∆ = ∆ 1 ∆ 2 if ∆ = ∆ 1 + ∆ 2 as a parition of the set of linear spaces in ∆.
2.3.The stacks R, N , RR, RR 2 , S(r, d) .We list the following definitions of stacks of stable maps that will occur in our recursions.
1) Let R(r, d) be the usual moduli space of genus zero stable maps M 0,0 (r, d).
2) Let N (r, d) be the closure in M 0,{A,B} (r, d) of the locus of maps of smooth rational curves γ such that γ(A) = γ(B).Informally, N (r, d) parametrizes degree d rational nodal curves in P r .
where the fibre product is taken over evaluation maps ev C to P r .

4) Similarly we can define
(the projections are evaluation maps e C ) of the locus of maps γ such that γ(A) = γ(B).We call maps in this family rational two-nodal reducible curves.6) For d > 0 let S(r, d) be the closure in M 0,{A} (r, d) of the locus of maps of smooth domains γ such that the differential vanishes at A. Informally, §(r, d) parametrizes degree d rational cuspidal curves in P r .2.4.Stacks of stable maps with constraints.Let F be a stack of stable maps of curves into P r .For a constraint ∆, we define (F, ∆) be the closure in F of the locus of maps that satisfy the constraint ∆.If the stack of maps F has two marked points A and B, we define (F, L u A L v B ) to be the closure in F of the locus of maps γ such that γ(A) lies on u general hyperplanes, and that γ(B) lies on v general hyperplanes.
If F has one marked point A then we define (F, L u A W v A ) to be the closure of maps γ such that γ(A) lies on u general hyperplanes, and that the image of γ is smooth at γ(A) and the tangent line to the image of γ at γ(A) passes through v general codimension 2 subspaces.
If a stack of F consists of a finite number of points then we denote #F to be the stacktheoretic length of F.
If F is a closed substack of the stacks N R, RR then we denote (F, Γ 1 , Γ 2 , k) to be the closure in F of the locus of maps γ such that the restriction of γ on the i−th component satisfies constraint Γ i and that γ(C) lies on k general hyperplanes.We use the notation (F, ∆, k) if we don't want to distinguish the conditions on each component.
) to be the closure in F of the locus of maps γ such that the restriction of γ on the i−th component satisfies constraint Γ i and that γ(C) lies on l general hyperplanes, and that γ(A) = γ(B) lies on k general hyperplanes.Similary, we use the notation (F, ∆, k, l) if we don't want to distinguish the conditions on each component.
Note that for maps of reducible source curves, tangency condition include the case where the image of the node lies on the tangency hyperplane, as the intersection multiplicity is 2 in this case.

Counting Rational Cuspidal Curves via Rational Nodal Curves
We need a result to establish the locus of rational cuspidal curves as a boundary component of the locus of rational nodal curves (both are substacks of the genus 0 stable map space).
Lemma 3.1.Let N (r, d) be as in Section 2. Let K be the closure in N (r, d) of maps γ such that : (i) The domain has two components and both A, B are on the same component (ii) γ contracts that component.Let C be the node of the source curve.Then if γ is a general map in K then the restriction of γ on the other component has differential vanished at C.

Proof.
We can look at a general 1−dimensional family in N (r, d).Let C 0 be a general curve in N (r, d) that has non-empty intersection with K. First we blow down the contracted components of the fibres to get the family C where the marked points may cross.The family π : S → C has two sections s A , s B : C → S corresponding two preimages of the node, and there is a map µ : S → P r .The two sections cross at a point P .Let Q = π(P ) and F be the fibre over Q.We need to show that µ |F is not unramified at P .This is easy to see.Let µ(s A ) = µ(s B ) = G be the nodes of the image.For a general point t ∈ G, the inverse image is a scheme of length 2 (in fact, it is a sum of two reduced points).Thus µ −1 (µ(P )) is of length at least 2. That means the map µ |F is not unramified at P .We now give recursions to compute characteristic numbers of rational cuspidal curves, with condition on the cusp.Let p, q be two hyperplanes in P r .Consider the moduli space X = M 0,S (r, d) where S = {A, B, P, Q}.Let N S (r, d) be the closure in X of locus of maps γ with γ(A) = γ(B).We will exploit the fact that if K is a boundary divisor of X such that a general map in K contracts a component containing both A and B, then N S (r, d) ∩ K consists of rational cuspidal curves, with the cusp at the image of A (and B).
Warning : if ∆(0) = 0 then those summands above involving reducible curves contain (twice) the case where the node is mapped to a tangency hyperplane.Also, in computing those summands, one needs to consider all possible splitting of constraints over two components.
Proof.Let ∆ be the same as ∆ except that ∆ contains the two hyperplanes p, q.Let Y be the one-dimensional family in N S (r, d) of maps γ that satisfy ∆ and also γ(P ) ∈ p and γ(Q) ∈ q.We then intersect Y with the two rationally equivalent divisors: First we intersect Y with the left hand side of the equation.A general map γ in ({A, B} || {P, Q}) has two-component source curves.Let the component containing A, B be C 1 , the other C 2 .We consider cases: • γ |C 1 has degree 0. By lemma 3.1, we get d 2 S(r, d) • ∆.The product is understood as intersecting cycle classes representing constraint ∆ on S(r, d).The factor d 2 comes from two hyperplane conditions on P and Q.Since we are intersecting cycle classes, tangency to a hyperplane means either ordinary tangency condition where the cuspidal curve is tangent to the hyperplane at a smooth point, or it could mean the cusp lies on the hyperplane.If we replane l ordinary tangency conditions by l hyperplane conditions through the cusp, we get ∆(0) l #(S(r, d), ∆ l ).Thus the total contribution of this case is • γ |C 2 has degree 0. Two hyperplane conditions on P and Q become the top incident condition (that explain why we need ∆ (2) = ∆(2) + 1).In this case we got #(N (r, d), ∆ ) • γ has positive degree on each component.Let d i be the degree of γ on C i .The total contribution is The coefficient d 2 2 comes from two hyperplane conditions on P and Q of the second component.Now we intersection Y with the right hand side.We consider 2 cases: • γ |C 1 has degree 0 or γ |C 2 has degree 0. In each case, the contribution is d#(N (r, d), ∆ ).
The total contribution from both cases is 2d#(N (r, d), ∆ ).• γ has positive degree on each component.The contribution is Rearranging the terms we have the desired equation.
We have some flexibility in choosing the marked points A, B, P, Q to invoke the WDVV relation.For example, we can impose (non-hyperplane) incident conditions on P and Q and then invoke WDVV.In fact we have a simpler and faster recursion this way with the price of not being to compute all characteristic numbers (for example, when all conditions are tangency ).
Let ∆ be a constraint such that ∆(0) = 0. Let p, q be two linear spaces in ∆.Consider the moduli space X = M 0,S (r, d) where S = {A, B, P, Q}.Let N S (r, d) be the closure in X of locus of maps γ with γ(A) = γ(B).The following proposition produces a different formula for incidence-only rational cuspidal curve numbers.
Proposition 3.3.Let k = ∆(r + 1).Let ∆ be derived from ∆ by removing p, q and add p ∩ q.Let ∆ p be derived from ∆ by removing p and add codim(p) to ∆(r + 1).Let ∆ q be derived from d by removing q and add codim(q) to ∆(r + 1).Let ∆ be derived from ∆ by removing p, q.If Γ is a constraint and X is a set of linear spaces then Γ (X) is the constraint derived from Γ by adding linear spaces in X.The following equality holds if both sides are finite: Proof.The proof is identical to that of Theorem 3.2.
We implemented both recursions and confirm that they gave the same numbers in case of incidence-only constraints, which is good check of the method.
Formula for plane curves.For plane curves with incident conditions, the formula simplify enough to get something computable by hands.We use the following standard notations.Let C d be the number of rational cuspidal plane curves, passing through 3d − 2 general points.Let R d be the number of rational plane curves passing through 3d − 1 points.Let N d , N L d , N P d be the number of rational plane curves with a choice of a node, where the node moves freely, on a line, on a point, respectively that pass through 3d − 1, 3d − 2, 3d − 3 points respectively.Now we restrict our formulas in Theorem 3.2 and Proposition 3.3 to plane curves.Applying Proposition 3.3 and Theorem 3.2 gives the following recursions for rational cuspidal plane curves with incident conditions.Corollary 3.4.
The formula for full chacracteristic numbers even for plane curves is unfortunately still rather involved (it may contain, for example, intersection numbers on Bl D (P 2 ×P 2 ), the blow up of product of projective planes along the diagonal, see [N]) and is best left to a computer program.

Numerical Examples
Let C be the family of rational cuspidal curves.Let C f , C b , C s , C l , C p be the same families except that the cusp is required to lie on a 4−space, 3−space, a plane, a line, and a point respectively.The tables below list the characteristic numbers for such family.In particular, we recover all previously known results, such as in [A], [P1], [HM].In some of the tables, we impose some point incident conditions on the families to make the numbers small enough to fit into the tables (tables 10 and 12).All other conditions are tangency and top incident conditions (incident with a codimension 2 linear subspace).We were also able to recover the results in table 5 of [Z] r, d) of the locus of curves with two components such that U ∪ V = S is a partition of the marked points over the two components.• The divisor (d 1 , d 2 ) is the closure in M 0,S (r, d) of the locus of curves with two components, sucht that d 1 + d 2 = d is the degree partition over the two components.• The divisor (U, d 1 || V, d 2 ) is the closure in M 0,S (r, d) of the locus of curves with two components, where U ∪ V = S and d 1 + d 2 = d are the partitions of markings and degree over the two components respectively.

Fig 1 .
Fig 1. Pictorial description of a general curve in the stacks R, N , RR, N R, RR 2 , S(r, d)

Table 5 .
using either of the recursions (in Theorem 3.2 or Proposition 3.3).Cubics in P 3 .