# View - Departament de Matemàtica Aplicada II - UPC

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View - Departament de Matemàtica Aplicada II - UPC

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

– τ : u s is a line through x b determined by the dependence relations with the remaining lines that

form such a degeneration.

– ψ: u s coincides 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 denoting 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 included 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 included 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 order to obtain them we can use the Wit

script 4.5.

A sample of them is given below. In this case, we include 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 considering 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 ))

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