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1428 J.M. Miret et al. / Journal of Symbolic Computation 44 (2009) 1425–1447

We define E nod as the subbundle of S 3 U ∗ | F whose fiber over (π, x b , u p ) ∈ F is the linear subspace

of forms ϕ ∈ S 3 U ∗ defined over π that have multiplicity at least 2 at x b and for which u p is a pair

of tangents (possibly coincident) at x b . In fact, given a point (π, x b , u p ) ∈ F and taking projective

coordinates x 0 , x 1 , x 2 , x 3 so that π = {x 3 = 0}, x b = [1, 0, 0, 0] and u p = {x 1 (b 1 x 1 + b 2 x 2 ) = 0},

b 1 , b 2 ∈ k, we can express the elements ϕ of the fiber of E nod over (π, x b , u p ) as follows:

ϕ = a 0 x 0 x 1 (b 1 x 1 + b 2 x 2 ) + a 1 x 3 1 + a 2x 2 1 x 2 + a 3 x 1 x 2 2 + a 4x 3 2 , (1)

where a i for i = 0, . . . , 4 are in k. Thus, E nod is a rank 5 subbundle of S 3 U ∗ | F .

In the next proposition we give a resolution of the vector bundle E nod over F. To do this, we consider

the natural inclusion map i : Q ∗ → U ∗ , the product map κ : Q ∗ ⊗ O F (−1) → S 3 Q ∗ | F , and the maps

h : U ∗ ⊗ O F (−1) → S 3 U ∗ | F and j : S 3 Q ∗ | F → S 3 U ∗ | F

whose images are clearly contained in E nod .

Proposition 1.2. The sequence

0 −→ Q ∗ ⊗ O F (−1)

α

−→ (U ∗ ⊗ O F (−1)) ⊕ S 3 Q ∗ β

| F −→ E nod −→ 0,

where α = ( i⊗1

−κ

)

and β = h + j, is an exact sequence of vector bundles over F.

Proof. It is similar to the one given in Proposition 1.1 of Hernández et al. (2007).

Let K nod be the projective bundle P(E nod ) over F. Then K nod is a non-singular variety of dimension

dim(K nod ) = dim(F) + rk(E nod ) − 1 = 11

whose points are pairs (f , (π, x b , u p )) ∈ P(S 3 U ∗ ) × Γ F such that the cubic f is contained in the plane

π, has a point of multiplicity at least 2 at x b and has u p as a pair of tangents (possibly coincident) at

x b . The generic points are those such that f is a non-degenerate cubic with a node at x b and nodal

tangents u p .

Notice that the variety K nod can be obtained as a blow up of the variety X nod , introduced in

Hernández et al. (2007), along the subvariety consisting of pairs (f , (π, x b )) ∈ X nod whose nodal cubic

f degenerates into three concurrent lines in π meeting at x b , so that its exceptional divisor is the

variety K trip given in Section 1.1.

Indeed, X nod is the projective bundle P(E ′ nod ), where E′ nod is the subbundle of S3 U| P(U) whose fiber

over (π, x b ) ∈ P(U) is the linear subspace of forms ϕ ∈ S 3 U defined over π that have multiplicity

al least 2 at x b . Then, the natural projection K nod → X nod is isomorphic to the blow-up of X nod along

P(S 3 Q ∗ ). It can be deduced from Proposition 4.1 in Hernández and Miret (2003), where the description

of the blow-up of a projective bundle P(E) along a projective subbundle P(F) is given in terms of the

quotient vector bundle E/F.

We will continue denoting by b and p the pullbacks to Pic(K nod ) of the classes b and p in Pic(F)

under the natural projection K nod → F. Since this projection is flat, b and p are the classes of the

hypersurfaces of K nod whose points (f , (π, x b , u p )) satisfy that x b is on a given plane and that u p

intersects a given line, respectively. Furthermore, by Lemma 1.1, the relation

ζ = ν − 3µ

holds in Pic(K nod ), where ζ is the hyperplane class of K nod and ν the class of the hypersurface of K nod

whose points (f , (π, x b , u p )) satisfy that f intersects a given line.

Proposition 1.3. The intersection ring A ∗ (K nod ) is isomorphic to the quotient of the polynomial ring

Z[µ, b, p, ν] by the ideal

〈µ 4 , b 3 − µb 2 + µ 2 b − µ 3 ,

p 3 − 3(µ + b)p 2 + 2(3µ 2 + 2µb + 3b 2 )p − 8µ 3 − 8µb 2 ,

ν 5 − (7µ + 5b + p)ν 4 + (27µ 2 + 22µb + 6µp + 15b 2 + 6bp)ν 3

−(57µ 3 + 49µ 2 b + 21µ 2 p + 47µb 2 + 22µbp + 15b 3 + 21b 2 p)ν 2

+(48µ 3 b + 36µ 3 p + 54µ 2 b 2 + 48µ 2 bp + 24µb 3 + 48µb 2 p + 36b 3 p)ν

−18µ 3 b 2 − 36µ 3 bp + 18µ 2 b 3 − 54µ 2 b 2 p − 36µb 3 p〉.

In particular, the Picard group Pic(K nod ) is a rank 4 free group generated by µ, b, p and ν.

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