# View - Departament de Matemàtica Aplicada II - UPC

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View - Departament de Matemàtica Aplicada II - UPC

1446 J.M. Miret et al. / Journal of Symbolic Computation 44 (2009) 1425–1447

with (i,j,h) in (0..3,0..min(3,5-i),0..10-i-j ) };

4.8. Intersection numbers on ε

# Determination of number of cubics of degeneration \varepsilon

# involving the conditions m,b,n,r

PU=variety(5,{gcs=[b,m], monomial_values={m^3*b^2->1,m^2*b^3->1}});

Ud=sheaf(3,[m,m^2,m^3],PU);

Qd = Ud / o_(b,PU);

Qdb = Qd * o_(-m,PU);

# Fb is a projective bundle of planes over PU

# It has dimension 6 and its generating classes are a, m and l

Fb=variety(6, {gcs=[b,m,l]});

# The table of monomial values of Fb is computed as follows:

Fb(monomial_values)=

{m^i*b^j*l^(6-i-j) ->

integral(m^i*b^j*segre(5-i-j,Qdb),PU)

with (i,j) in (0..3,0..min(3,5-i))};

Ud1=sheaf(3,[m,m^2,m^3],Fb);

Udd=dual(Ud1);

Qlb= Udd / o_(l-m,Fb);

Qlbb=symm(4,Qlb);

# DegEps is a projective bundle of planes over Fb

# It has dimension 10 and its generating classes are b, m, l and r

DegEps=variety(10, {gcs=[b,m,l,r]});

# The table of monomial values of DegEps is computed as follows:

DegEps(monomial_values)=

{ m^i*b^j*n^h*r^(10-i-j-h) ->

integral(m^i*b^j*(3*l)^h*segre(6-i-j-h,Qlbb),Fb)

with (i,j,h) in (0..3,0..min(3,5-i),0..min(4,6-i-j)) };

4.9. Intersection numbers involving µ, b, ν and ρ

# Numbers of nodal cubics involving conditions m,b,n,r

Nr1={m^i*b^j*n^(11-i-j-h)*r ->

-6*integral(m^(i+1)*b^j*n^(11-i-j-h),Knods)

+ 4*integral(m^i*b^j*n^(12-i-j-h),Knods)

- 2*integral(m^i*b^(j+1)*n^(11-i-j-h),Knods)

- 2*integral(m^i*b^j*n^(11-i-j-h),NdegTh)

- 6*integral(m^i*b^j*n^(11-i-j-h),NdegEps)

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