Rational cuspidal curves in a moving family of ℙ2


 In this paper we obtain a formula for the number of rational degree d curves in ℙ3 having a cusp, whose image lies in a ℙ2 and that passes through r lines and s points (where r + 2s = 3
 d + 1). This problem can be viewed as a family version of the classical question of counting rational cuspidal curves in ℙ2, which has been studied earlier by Z. Ran ([13]), R. Pandharipande ([12]) and A. Zinger ([16]). We obtain this number by computing the Euler class of a relevant bundle and then finding out the corresponding degenerate contribution to the Euler class. The method we use is closely based on the method followed by A. Zinger ([16]) and I. Biswas, S. D’Mello, R. Mukherjee and V. Pingali ([1]). We also verify that our answer for the characteristic numbers of rational cuspidal planar cubics and quartics is consistent with the answer obtained by N. Das and the first author ([2]), where they compute the characteristic number of δ-nodal planar curves in ℙ3 with one cusp (for δ ≤ 2).


Introduction
A classical question in enumerative algebraic geometry is:

Question. What is N d , the number of rational (genus zero) degree d curves in P that pass through d − generic points?
Although the computation of N d is a classical question, a complete solution to the above problem was unknown until the early s when Ruan-Tian ( [14]) and ) obtained a formula for N d . Generalization of this question to enumerate rational curves with higher singularities (such as cusps, tacnodes and higher order cusps) have been studied by Z. Ran ([13]), R. Pandharipande ([12]) and A. Zinger ([16], [17] and [15]). These results have also been generalized to other surfaces (such as P × P ) by J. Kock ([8]) and more generally for del-Pezzo surfaces by I. Biswas, S. D'Mello, R. Mukherjee and V. Pingali ( [1]). The problem of enumerating elliptic cuspidal curves has been solved by Z. Ran ([13]), and more recently a solution to this question in any genus has been obtained by Y. Ganor and E. Shustin ( [4]) using methods from Tropical Geometry.
A natural generalization of problems in enumerative geometry (where one studies curves inside some xed ambient surface such as P ) is to consider a family version of the same problem. This generalization is considered by S. Kleiman and R. Piene ( [7]) and more recently by T. Laarakker ([10]) where they study the enumerative geometry of nodal curves in a moving family of surfaces.
Motivated by this generalization, A. Paul and the authors of this paper studied a family version of computing N d in ( [11]); there the authors nd a formula for the characteristic number of rational planar curves in P (i.e. curves in P that lie inside a P ). In this paper we build up on the results of ( [11]) to nd the characteristic number of rational planar curves in P having a cusp.
Before stating the main result of our paper, let us explain a few computational things. Let N P ,Planar d (r, s, θ) be the number as de ned in equation 3.2; this number was computed in [11]. Using this number, we can compute the number Φ d (i, j, r, s, θ) via Lemma 5.1, 5.2 and 5.3 (for i ≤ ). Using the values of Φ d (i, j, r, s, θ) (for i ≤ ), we can compute the e (Euler class) via equation (4.7). Finally, we can compute the number B (boundary contribution) via equations (4.6) and (5.7). With this computation, we can state the main result of the paper. Theorem 1.1. Let C P ,Planar d (r, s) be the number of genus zero, degree d curves in P having a cusp, whose image lies in a P , intersecting r generic lines and s generic points (where r+ s = d+ ) and let e and B be as computed above. Then Remark. Note that when s ≥ , C P ,Planar d (r, s) is automatically zero, since four generic points do not lie in a plane. We also note that when s = , C P ,Planar d (r, s) is equal to the number of rational cuspidal curves in P through d − points; this is because three generic points determine a unique plane. Hence, s = reduces to the result of Z. Ran ([13]), R. Pandharipande ([12]) and A. Zinger ([16]).
We have written a mathematica program to implement our formula; the program is available on our web page https://www.sites.google.com/site/ritwik371/home. In section 7, we subject our formula to several low degree checks; in particular, we verify that our numbers are logically consistent with those obtained by N. Das and the rst author ( [2]).
Let us now give a brief overview of the method we use in this paper; we closely adapt the method applied by A. Zinger ([16]) and I. Biswas, S. D'Mello, R. Mukherjee and V. Pingali ( [1]). We express our enumerative number as the number of zeros of a section of an appropriate vector bundle (restricted to an open dense set of an appropriate moduli space). As is usually the case, the Euler class of this vector bundle is our desired enumerative number, plus an extra boundary contribution. In ( [16]) and ( [1]) the method of "dynamic intersections" (cf. Chapter 11 in [3]) is used to compute the degenerate contribution to the relevant Euler class. We argue in section 6 how the multiplicity computation in ( [16]) and ( [1]) implies the multiplicity of the degenerate locus that occurs in our case. Finally, computation of the Euler class involves the intersection of tautological classes on the moduli space of planar degree d curves; this in turn involves the characteristic number of rational planar curves in P , for which we use the result of our paper [11]. Hence, we can compute both the Euler class and the degenerate contribution, which gives us our desired number C P ,Planar d (r, s).

Future directions
In this section, we will explore a few natural questions that would occur to the reader and that we hope to explore in the future. The eld of Enumerative Geometry and Gromov-Witten Theory studies the following type of question: we have a xed surface X (say P ) and we consider an appropriate class of curves in X of a given homology class β. The basic goal of this eld is to study this space of curves (moduli space) and compute intersection numbers.
The aim of the papers [7], [10] and [2] is to study a family version of the classical question of enumerating curves in a xed linear system. The goal of our earlier paper [11] and the present paper is to develop a family version of the study of intersection theory of Moduli space of stable maps. We feel the most natural example is the study of planar curves in P ; this is a family version of studying curves in P . However, the reader might wonder what other examples one might study with the methods developed in this paper? Let us consider an obvious generalization: nd the characteristic number of planar rational curves in P n for any n (and a similar question for cuspidal curves). We believe this question is doable, but it is likely to get computationally out of hand very quickly as n increases; even for n = this is likely to be a formidable question (from a computational point of view). The reason is as follows. To consider planar curves in P n , we rst of all have to consider the space of planes in P n ; this is the Grassmannian G( , n + ). When n = , this space is much more tractable since it is simply the projective space! For higher n, one can derive formulas (analogous to what we did in our paper [11]) for the characteristic number of rational planar curves. However, computationally it would get out of hand since one would have to compute intersections of Schubert cells in the Grassmannian (something which is doable in principle, but quite laborious in practice). Nevertheless, we do intend to investigate this question in future.
More generally, we believe one can study the following situation: let S −→ M be a bre bundle over M, with bres F. Consider the space of curves into S, whose image lies inside some F. Study the intersection theory of this space. In our case, S is as de ned in (4.2). But more generally, S could be the projectivization of some vector bundle. Or S could be some other ber bundle (whose bres are di erent from the projective space). The methods of our paper [11] and this paper ought to be applicable as long as the intersection theory of the base space M, the bre F and the total space of the bre bundle S is tractable.
A second question that the reader might wonder is if one might be able to extend the result of this paper to other types of singularities that have been worked out for P (for example tacnode, triple point or E -singularities¹ as worked out in [15] and [17] for P ). We believe the answer to this question is yes; we are in fact trying to understand how Zinger obtained the formulas for rational curves with E singularities, tacnodes and triple points and trying to extend it for planar curves in P .
We end this section with one last question. The contents of [11] and this paper are about genus zero curves. It is natural to wonder if one might be able to obtain higher genus analogues of this result; for example can one compute the characteristic number of planar elliptic curves in P ? This is a far more non-trivial question. We expect that the methods employed in [5] can be generalized to enumerate elliptic planar curves in P ; we intend to pursue this in the future.

Notation
Let us de ne a planar curve in P to be a curve, whose image lies inside a P . We will now develop some notation to describe the space of planar curves of a given degree d.
Let us denote the dual of P by P ; this is the space of P inside P . An element of P can be thought of as a non zero linear functional η : C −→ C upto scaling (i.e., it is the projectivization of the dual of C ). Given such an η, we de ne the projectivization of its zero set as P η . In other words, Note that this P η is a subset of P . Next, we de ne the moduli space of planar degree d curves into P as a bre bundle over P . More precisely, we de ne Here we are using the standard notation to denote M ,k (X, β) to be the moduli space of genus zero stable maps, representing the class β ∈ H (X, Z) and M ,k (X, β) to be its stable map compacti cation. Since the dimension of a ber bundle is the dimension of the base, plus the dimension of the ber, we conclude that the dimension of M Planar ,k (P , d) is d + + k. Next, we note that there is a natural forgetful map where one forgets the plane P η and simply thinks about the stable map to P . When d ≥ , the map π F is injective when restricted to the open dense subspace of non multiply covered curves (from a smooth domain). This is because every planar degree d degree curve lies in a unique plane, when d ≥ . When d = , this map is not injective since a line is not contained in a unique plane. Infact we note that the space of lines is dimensional, while the dimension of M Planar , denote the universal tangent line bundle over the marked point (the ber over each point is the tangent space over the given marked point). This line bundle will pullback to a line bundle over M Planar , (P , d) via the map π F ; we will denote it by the same symbol L (we will in general avoid writing the pullback symbol π * F if there is no cause of confusion).
Let us now de ne a few cycles in M (P , d) by the same symbol H L and Hp. We will also denote H and a to be the standard generators of H * (P ; Z) and H * ( P ; Z) respectively. As π is a projection map from M Planar , (P , d) to P ; we denote the pullback π * a by the same symbol a. Finally, there is an evaluation map from ev : M Planar , We will denote the pullback of H via this map to be ev * H. This is the one case where we explicitly write the pullback map (since it will avoid confusion; in the remaining cases omitting to write down the pullback map causes no confusion). We will now de ne a few numbers by intersecting cycles on M Planar ,k (P , d). We will use the convention that We note that using the results of our paper ( [11]), the numbers N P ,Planar d (r, s, θ) are all computable. In section 5, a formula is given to compute the relevant Φ d (i, j, r, s, θ) necessary to obtain the main result of this paper. We note that using the convention introduced in equation (3.1)

Euler class computation
We will now describe the basic method by which we compute the characteristic number of planar rational cuspidal curves in P . We will express this number as the number of zeros of a section of an appropriate bundle restricted to an open dense subspace of the moduli space of planar curves in P . Before we do that, let us make a few abbreviations that we will often use. We denote Let γ P −→ P and γ P −→ P be the tautological line bundles over P and P respectively. We now note that any linear functional η : C −→ C induces a section η : P −→ γ * P of the dual bundle, given by Note that we are making a slight abuse of notation here by denoting the linear functional and the induced section by the same letter η.
Let q ∈ P , such that η(q) = (i.e. q ∈ P η ). Let us now consider the vertical derivative, namely The vertical derivative can be de ned by writing down the section η in a trivialization and taking the derivative along the ber; this operation will be well de ned if η(q) = . If η is a non zero functional, then the induced section is transverse to zero. Hence the vertical derivative will be surjective. Let us de ne Wq := Ker∇η|q .
We note that TP η |q is a subset of the kernel of ∇η|q. Since ∇η|q is surjective, we conclude that TP η is precisely equal to Wq. Hence, the following sequence is exact, where the rst map is the inclusion map. Let us now de ne An element of S denotes a plane P η in P together with a marked point q that lies in the plane. We will now de ne W −→ S to be the rank two vector bundle, where the bre over each point ([η], q) is Wq (i.e. it is TqP η , the tangent space of P η at the point q). Hence, from equation (4.1), we conclude that over S we have the following exact sequence of vector bundles, where π P : S −→ P denotes the projection to P . Hence, from equation Let us denote this section as ψ. In section 6 we show that when restricted to M, ψ is transverse to zero. However, the section will also vanish on the boundary ∂M. The following Lemma precisely identi es the part of the boundary stratum on which the section vanishes.

Lemma 4.1. Let Y denote the following subset of ∂M: stable map from a wedge of three spheres of degree d , and d , where the marked point lies on the degree zero component (which is also called a ghost bubble).
Restricted to ∂M, the section ψ vanishes precisely on Y, i.e.
Proof: An element of Y is of the above type (see the picture). We rst note that the section ψ vanishes identically on Y since taking the derivative of a constant map is zero. Hence, the right hand side of equation This number is strictly less than d − + ; hence this con guration can not pass through r lines and s points (where r + s = d + ).
Finally, we also observe that although the section vanishes on multiply covered curves, such curves will not pass through r lines and s points; this can be seen by considering the dimensions of the space of the underlying reduced curve and observing that the dimension is strictly less than d + . Hence, the only con gurations on which the section vanishes are those that split as a d curve, a ghost bubble and a d curve, with the last marked point lying on a ghost bubble (which is Y).
Next, we need to compute the cardinality of the set Y. This cardinality is computed in ( [11]) (in the proof of Theorem 3.3); it is given by  ( , , r, s, ). (4.7) The numbers Φ d (i, j, r, s, θ) that arise in the right hand side of equation (

Intersection of Tautological Classes
We will now give a formula for the relevant Φ d (i, j, r, s, θ) that are necessary to compute the Euler class. We will often refer to Φ d (i, j, r, s, θ) as a level i number.  ( , j, r, s, θ ( , j, r, s, θ) for j > and Φ d ( , j, r, s, θ) for j > can be computed without any further e ort; we have not presented the formulas since they are not needed for the Euler class computation.
Before we start proving these Lemmas, let us rst recall an important result about c (L * ).

Lemma 5.4. On M
Planar , (P , d), the following equality of divisors holds: We note that the degree of this map is one since in the rst case we are considering a curve and a marked point that goes to a line (H ), while in the second case we are considering a curve whose image intersects a line. Hence Φ d ( , , r, s, θ) = (ev * H ), M Planar , Finally, let us consider the case when j = . Let us assume r + s + θ + = d + (otherwise the number is zero). Let us consider the forgetful map We note that the degree of this map is one since in the rst case we are considering a curve and a marked point that goes to a point (H ), while in the second case we are considering a curve whose image passes through a point. Hence

Low degree checks
In this section we subject our formula to certain low degree checks. All these numbers have been computed using our mathematica program. We will abbreviate C P ,Planar d (r, s) by C d (r, s). First of all our formula gives us the value of zero for C d (r, s) when d = . This is as geometrically as expected since there are no conics with a cusp.
Next, in ( [2]), N. Das and the rst author compute the following numbers: what is N d (A δ A , r, s), the number of planar degree d curves in P , passing through r lines and s points, that have δ (ordered) nodes and one cusp, for all δ ≤ . Note that here r + s = δ + . For d = , and δ = , this number should be the same as the characteristic number of genus zero planar cubics in P with a cusp, i.e. C d (r, s). We have veri ed that is indeed the case. We tabulate the numbers for the readers convenience: These numbers are the same as N d (A δ A , r, s) for d = and δ = .
Next, we note that when d = and δ = , the number δ! N d (A δ A , r, s) is same as the characteristic number of genus zero planar quartics in P with a cusp, i.e. C d (r, s). We have veri ed that fact. The numbers are C ( , ) = , C ( , ) = , C ( , ) = and C ( , ) = .
These numbers are the same as ! N d (A δ A , r, s) for d = and δ = . We have to divide out by a factor of δ! because in the de nition of N d (A δ A , r, s), the nodes are ordered.