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1434 J.M. Miret et al. / Journal of Symbolic Computation 44 (2009) 1425–1447

– τ : u s is a line through x b **de**termined by the **de**pen**de**nce relations with the remaining lines that

form such a **de**generation.

– ψ: u s coinci**de**s with one of the lines that constitute the nodal cubic, different from the nodal

tangents.

– τ ′ : u s is a new line in the plane π.

Once again, using the projection formula and **de**noting by g the pullback to K z,s

nod

of the hyperplane

class g of G z,s , we get:

∫

∫

K z,s

nod

µ i b j p h z r g t ν k =

G z,s

µ i b j p h z r g t s 6−i−j−h−r (E z,s

nod ⊗ O Γ (−3)).

Then, computing the Segre classes E z,s

z,s

nod

from (7), we can calculate all the intersection numbers of Knod

in the conditions µ, b, p, z, g and ν. The script for this computation is inclu**de**d in 4.3.

Together with Miret et al. (2003), where the fundamental numbers of cuspidal cubics were

computed, and taking into account relation (6), all intersection numbers of K z,s

nod

in the conditions µ,

b, p, z, s and ν can be obtained in the following way:

∫

∫

µ i b j p h z r s t ν k = µ i b j p h z r s t−1 (g − p)ν k + 2

∫K h µ i b j q h z r+t−1 ν k ,

ncusp

K z,s

nod

K z,s

nod

where q is the condition on K ncusp that the cuspidal tangent intersects a given line. The Wit script for

the calculation of the numbers of K ncusp is inclu**de**d in 4.4.

Now, from the intersection numbers of the varieties K z,s

nod and K ncusp, we can compute the numbers

of nodal plane curves with the condition µ, b, p, z, s and ν. In or**de**r to obtain them we can use the Wit

script 4.5.

A sample of them is given below. In this case, we inclu**de** those numbers involving the conditions

µ, b, s and ν.

µ 3 sν 7 = 18 µ 2 sν 8 = 296 µsν 9 = 2560 sν 10 = 14760

µ 3 bsν 6 = 11 µ 2 bsν 7 = 164 µbsν 8 = 1284 bsν 9 = 6560

µ 3 b 2 sν 5 = 2 µ 2 b 2 sν 6 = 32 µb 2 sν 7 = 254 b 2 sν 8 = 1256

µ 2 b 3 sν 5 = 2 µb 3 sν 6 = 21 b 3 sν 7 = 108

µ 3 s 2 ν 6 = 25 µ 2 s 2 ν 7 = 374 µs 2 ν 8 = 2948 s 2 ν 9 = 15280

µ 3 bs 2 ν 5 = 20 µ 2 bs 2 ν 6 = 263 µbs 2 ν 7 = 1822 bs 2 ν 8 = 8012

µ 3 b 2 s 2 ν 4 = 4 µ 2 b 2 s 2 ν 5 = 56 µb 2 s 2 ν 6 = 391 b 2 s 2 ν 7 = 1642

µ 2 b 3 s 2 ν 4 = 4 µb 3 s 2 ν 5 = 36 b 3 s 2 ν 6 = 153

µ 2 s 3 ν 6 = 50 µs 3 ν 7 = 712 s 3 ν 8 = 5304

µ 2 bs 3 ν 5 = 40 µbs 3 ν 6 = 504 bs 3 ν 7 = 3316

µ 2 b 2 s 3 ν 4 = 8 µb 2 s 3 ν 5 = 108 b 2 s 3 ν 6 = 718

µb 3 s 3 ν 4 = 8 b 3 s 3 ν 5 = 68

µs 4 ν 6 = 60 s 4 ν 7 = 676

µbs 4 ν 5 = 40 bs 4 ν 6 = 482

µb 2 s 4 ν 4 = 8 b 2 s 4 ν 5 = 104

b 3 s 4 ν 4 = 8.

(8)

Among these numbers, only two of them, µ 3 sν 7 = 18 and µ 3 s e ν 6 = µ 3 s 2 ν 6 = 25, were given by

Schubert (see Schubert (1879), p. 160).

2.3. The condition v

We construct now another compactification of U nod by consi**de**ring the closure K v nod

of the graph

of the rational map that assigns the triplet of flexes x v to a given nodal cubic (f , (π, x b , u p , u z , u s )) of

K z,s

nod . Notice that the generic points of this new variety K v nod consist of pairs (f , (π, x b, u p , u z , u s , x v ))