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# specialCubicFourfold -- make a special cubic fourfold

## Synopsis

• Usage:
specialCubicFourfold(S,X)
specialCubicFourfold(S,X,NumNodes=>n)
• Inputs:
• S, , an irreducible surface $S\subset\mathbb{P}^5$, which has as singularities only a finite number $n\geq 0$ of non-normal nodes (this number $n$ should be passed with the option NumNodes, otherwise it is obtained using a probabilistic method)
• X, , a smooth cubic fourfold $X\subset \mathbb{P}^5$ containing the surface $S$
• Optional inputs:
• InputCheck => ..., default value 1
• NumNodes => ..., default value null
• Verbose => ..., default value false, request verbose feedback
• Outputs:
• , the special cubic fourfold corresponding to the pair $(S,X)$

## Description

In the example below, we define a cubic fourfold containing a rational scroll of degree 7 with 3 nodes.

 i1 : K = ZZ/33331; x = gens ring PP_K^5; i3 : S = projectiveVariety ideal(x_0*x_2*x_3-2*x_1*x_2*x_3-x_1*x_3^2-x_2*x_3^2-x_0*x_1*x_4+2*x_1^2*x_4-x_1*x_2*x_4+x_2^2*x_4+2*x_0*x_3*x_4-x_1*x_3*x_4-x_1*x_4^2+x_1*x_3*x_5, x_1^2*x_3-4*x_1*x_2*x_3-x_0*x_3^2-3*x_1*x_3^2-2*x_2*x_3^2+2*x_0^2*x_4-9*x_0*x_1*x_4+11*x_1^2*x_4-x_0*x_2*x_4-2*x_1*x_2*x_4+2*x_2^2*x_4+12*x_0*x_3*x_4-7*x_1*x_3*x_4-4*x_3^2*x_4+x_0*x_4^2-6*x_1*x_4^2+4*x_2*x_4^2-2*x_3*x_4^2-2*x_4^3-x_0*x_1*x_5+x_1^2*x_5+2*x_1*x_2*x_5+3*x_0*x_3*x_5+2*x_1*x_3*x_5-x_3^2*x_5-x_0*x_4*x_5-4*x_1*x_4*x_5+3*x_2*x_4*x_5+2*x_3*x_4*x_5-x_1*x_5^2, x_0*x_1*x_3-7*x_1*x_2*x_3-3*x_0*x_3^2-4*x_1*x_3^2-3*x_2*x_3^2+x_3^3+3*x_0^2*x_4-14*x_0*x_1*x_4+17*x_1^2*x_4-x_0*x_2*x_4-3*x_1*x_2*x_4+3*x_2^2*x_4+19*x_0*x_3*x_4-9*x_1*x_3*x_4-x_2*x_3*x_4-6*x_3^2*x_4+x_0*x_4^2-9*x_1*x_4^2+6*x_2*x_4^2-3*x_3*x_4^2-3*x_4^3-2*x_0*x_1*x_5+2*x_1^2*x_5+4*x_1*x_2*x_5+5*x_0*x_3*x_5+4*x_1*x_3*x_5-2*x_3^2*x_5-2*x_0*x_4*x_5-7*x_1*x_4*x_5+5*x_2*x_4*x_5+3*x_3*x_4*x_5-2*x_1*x_5^2, x_0^2*x_3-12*x_1*x_2*x_3-6*x_0*x_3^2-6*x_1*x_3^2-5*x_2*x_3^2+2*x_3^3+5*x_0^2*x_4-24*x_0*x_1*x_4+29*x_1^2*x_4-x_0*x_2*x_4-5*x_1*x_2*x_4+5*x_2^2*x_4+32*x_0*x_3*x_4-14*x_1*x_3*x_4-2*x_2*x_3*x_4-10*x_3^2*x_4+x_0*x_4^2-15*x_1*x_4^2+10*x_2*x_4^2-5*x_3*x_4^2-5*x_4^3-3*x_0*x_1*x_5+3*x_1^2*x_5+6*x_1*x_2*x_5+8*x_0*x_3*x_5+7*x_1*x_3*x_5-3*x_3^2*x_5-3*x_0*x_4*x_5-11*x_1*x_4*x_5+8*x_2*x_4*x_5+5*x_3*x_4*x_5-3*x_1*x_5^2, x_1*x_2^2+6*x_1*x_2*x_3+2*x_0*x_3^2+3*x_1*x_3^2+2*x_2*x_3^2-x_3^3-3*x_0^2*x_4+12*x_0*x_1*x_4-14*x_1^2*x_4-2*x_2^2*x_4-15*x_0*x_3*x_4+6*x_1*x_3*x_4+x_2*x_3*x_4+5*x_3^2*x_4+x_0*x_4^2+8*x_1*x_4^2-5*x_2*x_4^2+2*x_3*x_4^2+2*x_4^3+x_0*x_1*x_5-2*x_1^2*x_5-4*x_1*x_2*x_5-4*x_0*x_3*x_5-3*x_1*x_3*x_5+2*x_3^2*x_5+2*x_0*x_4*x_5+7*x_1*x_4*x_5-4*x_2*x_4*x_5-2*x_3*x_4*x_5+2*x_1*x_5^2, x_0*x_2^2+10*x_1*x_2*x_3+3*x_0*x_3^2+5*x_1*x_3^2+4*x_2*x_3^2-x_3^3-5*x_0^2*x_4+19*x_0*x_1*x_4-22*x_1^2*x_4-x_0*x_2*x_4+3*x_1*x_2*x_4-4*x_2^2*x_4-24*x_0*x_3*x_4+9*x_1*x_3*x_4+x_2*x_3*x_4+8*x_3^2*x_4+2*x_0*x_4^2+11*x_1*x_4^2-7*x_2*x_4^2+4*x_3*x_4^2+3*x_4^3+2*x_0*x_1*x_5-4*x_1^2*x_5-7*x_1*x_2*x_5-7*x_0*x_3*x_5-5*x_1*x_3*x_5-x_2*x_3*x_5+3*x_3^2*x_5+4*x_0*x_4*x_5+12*x_1*x_4*x_5-7*x_2*x_4*x_5-3*x_3*x_4*x_5+4*x_1*x_5^2, x_1^2*x_2+17*x_1*x_2*x_3+6*x_0*x_3^2+9*x_1*x_3^2+7*x_2*x_3^2-2*x_3^3-9*x_0^2*x_4+36*x_0*x_1*x_4-44*x_1^2*x_4+3*x_0*x_2*x_4+5*x_1*x_2*x_4-7*x_2^2*x_4-47*x_0*x_3*x_4+21*x_1*x_3*x_4+2*x_2*x_3*x_4+16*x_3^2*x_4+24*x_1*x_4^2-16*x_2*x_4^2+7*x_3*x_4^2+7*x_4^3+3*x_0*x_1*x_5-6*x_1^2*x_5-9*x_1*x_2*x_5-12*x_0*x_3*x_5-8*x_1*x_3*x_5+5*x_3^2*x_5+5*x_0*x_4*x_5+19*x_1*x_4*x_5-12*x_2*x_4*x_5-7*x_3*x_4*x_5+5*x_1*x_5^2, x_0*x_1*x_2+29*x_1*x_2*x_3+11*x_0*x_3^2+15*x_1*x_3^2+12*x_2*x_3^2-4*x_3^3-16*x_0^2*x_4+62*x_0*x_1*x_4-74*x_1^2*x_4+5*x_0*x_2*x_4+9*x_1*x_2*x_4-12*x_2^2*x_4-80*x_0*x_3*x_4+35*x_1*x_3*x_4+4*x_2*x_3*x_4+27*x_3^2*x_4+40*x_1*x_4^2-27*x_2*x_4^2+12*x_3*x_4^2+12*x_4^3+5*x_0*x_1*x_5-10*x_1^2*x_5-16*x_1*x_2*x_5-21*x_0*x_3*x_5-14*x_1*x_3*x_5+9*x_3^2*x_5+9*x_0*x_4*x_5+33*x_1*x_4*x_5-21*x_2*x_4*x_5-12*x_3*x_4*x_5+9*x_1*x_5^2, x_0^2*x_2+49*x_1*x_2*x_3+19*x_0*x_3^2+25*x_1*x_3^2+20*x_2*x_3^2-7*x_3^3-28*x_0^2*x_4+106*x_0*x_1*x_4-124*x_1^2*x_4+8*x_0*x_2*x_4+16*x_1*x_2*x_4-20*x_2^2*x_4-134*x_0*x_3*x_4+58*x_1*x_3*x_4+7*x_2*x_3*x_4+45*x_3^2*x_4+66*x_1*x_4^2-45*x_2*x_4^2+20*x_3*x_4^2+20*x_4^3+9*x_0*x_1*x_5-18*x_1^2*x_5-28*x_1*x_2*x_5-37*x_0*x_3*x_5-23*x_1*x_3*x_5+16*x_3^2*x_5+16*x_0*x_4*x_5+57*x_1*x_4*x_5-36*x_2*x_4*x_5-20*x_3*x_4*x_5+16*x_1*x_5^2, x_1^3+47*x_1*x_2*x_3+18*x_0*x_3^2+23*x_1*x_3^2+19*x_2*x_3^2-7*x_3^3-24*x_0^2*x_4+97*x_0*x_1*x_4-117*x_1^2*x_4+8*x_0*x_2*x_4+16*x_1*x_2*x_4-19*x_2^2*x_4-127*x_0*x_3*x_4+54*x_1*x_3*x_4+7*x_2*x_3*x_4+42*x_3^2*x_4-x_0*x_4^2+62*x_1*x_4^2-42*x_2*x_4^2+19*x_3*x_4^2+19*x_4^3+9*x_0*x_1*x_5-16*x_1^2*x_5-25*x_1*x_2*x_5-33*x_0*x_3*x_5-23*x_1*x_3*x_5+14*x_3^2*x_5+14*x_0*x_4*x_5+51*x_1*x_4*x_5-33*x_2*x_4*x_5-19*x_3*x_4*x_5+14*x_1*x_5^2, x_0*x_1^2+79*x_1*x_2*x_3+29*x_0*x_3^2+40*x_1*x_3^2+32*x_2*x_3^2-11*x_3^3-41*x_0^2*x_4+164*x_0*x_1*x_4-196*x_1^2*x_4+14*x_0*x_2*x_4+26*x_1*x_2*x_4-32*x_2^2*x_4-214*x_0*x_3*x_4+92*x_1*x_3*x_4+11*x_2*x_3*x_4+71*x_3^2*x_4-2*x_0*x_4^2+105*x_1*x_4^2-71*x_2*x_4^2+32*x_3*x_4^2+32*x_4^3+14*x_0*x_1*x_5-26*x_1^2*x_5-41*x_1*x_2*x_5-55*x_0*x_3*x_5-38*x_1*x_3*x_5+23*x_3^2*x_5+23*x_0*x_4*x_5+85*x_1*x_4*x_5-55*x_2*x_4*x_5-32*x_3*x_4*x_5+23*x_1*x_5^2, x_0^2*x_1+133*x_1*x_2*x_3+48*x_0*x_3^2+68*x_1*x_3^2+54*x_2*x_3^2-18*x_3^3-70*x_0^2*x_4+278*x_0*x_1*x_4-330*x_1^2*x_4+24*x_0*x_2*x_4+44*x_1*x_2*x_4-54*x_2^2*x_4-361*x_0*x_3*x_4+156*x_1*x_3*x_4+18*x_2*x_3*x_4+120*x_3^2*x_4-4*x_0*x_4^2+177*x_1*x_4^2-120*x_2*x_4^2+54*x_3*x_4^2+54*x_4^3+23*x_0*x_1*x_5-44*x_1^2*x_5-69*x_1*x_2*x_5-93*x_0*x_3*x_5-63*x_1*x_3*x_5+39*x_3^2*x_5+39*x_0*x_4*x_5+144*x_1*x_4*x_5-93*x_2*x_4*x_5-54*x_3*x_4*x_5+39*x_1*x_5^2, x_0^3+224*x_1*x_2*x_3+80*x_0*x_3^2+115*x_1*x_3^2+91*x_2*x_3^2-30*x_3^3-119*x_0^2*x_4+470*x_0*x_1*x_4-555*x_1^2*x_4+41*x_0*x_2*x_4+75*x_1*x_2*x_4-91*x_2^2*x_4-608*x_0*x_3*x_4+263*x_1*x_3*x_4+30*x_2*x_3*x_4+202*x_3^2*x_4-8*x_0*x_4^2+297*x_1*x_4^2-202*x_2*x_4^2+91*x_3*x_4^2+91*x_4^3+39*x_0*x_1*x_5-76*x_1^2*x_5-118*x_1*x_2*x_5-158*x_0*x_3*x_5-105*x_1*x_3*x_5+67*x_3^2*x_5+68*x_0*x_4*x_5+245*x_1*x_4*x_5-158*x_2*x_4*x_5-91*x_3*x_4*x_5+67*x_1*x_5^2); o3 : ProjectiveVariety, surface in PP^5 i4 : X = projectiveVariety ideal(x_1^2*x_3+x_0*x_2*x_3-6*x_1*x_2*x_3-x_0*x_3^2-4*x_1*x_3^2-3*x_2*x_3^2+2*x_0^2*x_4-10*x_0*x_1*x_4+13*x_1^2*x_4-x_0*x_2*x_4-3*x_1*x_2*x_4+3*x_2^2*x_4+14*x_0*x_3*x_4-8*x_1*x_3*x_4-4*x_3^2*x_4+x_0*x_4^2-7*x_1*x_4^2+4*x_2*x_4^2-2*x_3*x_4^2-2*x_4^3-x_0*x_1*x_5+x_1^2*x_5+2*x_1*x_2*x_5+3*x_0*x_3*x_5+3*x_1*x_3*x_5-x_3^2*x_5-x_0*x_4*x_5-4*x_1*x_4*x_5+3*x_2*x_4*x_5+2*x_3*x_4*x_5-x_1*x_5^2); o4 : ProjectiveVariety, hypersurface in PP^5 i5 : time F = specialCubicFourfold(S,X,NumNodes=>3); -- used 0.0160302s (cpu); 0.015543s (thread); 0s (gc) o5 : ProjectiveVariety, cubic fourfold containing a surface of degree 7 and sectional genus 0 i6 : time describe F warning: clearing value of symbol x to allow access to subscripted variables based on it : debug with expression debug 9868 or with command line option --debug 9868 -- used 1.43142s (cpu); 0.327634s (thread); 0s (gc) o6 = Special cubic fourfold of discriminant 26 containing a 3-nodal surface of degree 7 and sectional genus 0 cut out by 13 hypersurfaces of degree 3 i7 : assert(F == X)