Problem 12

Give an example of a smooth tropical cubic surface in \(\mathbb{R}^3\) that contains 27 distinct tropical lines. How many other lines does each line intersect? Can we draw a picture of that cubic surface together with the 27 lines?

Solution

In Polymake:

application "tropical";
$H = new Hypersurface<Max>( POLYNOMIAL=>toTropicalPolynomial("max(-49/1000+3*x0 ,
-523/250+2*x0+x1 , -501/250+2*x0+x2 , -2+2*x0+x3 , -801/100+x0+2*x1 , -499/100+x0+x1+x2 ,
-5997/1000+x0+x1+x3 , -7919/1000+x0+2*x2 , -5981/1000+x0+x2+x3 , -799/100+x0+2*x3 ,
-18023/1000+3*x1 , -6021/500+2*x1+x2 , -1759/125+2*x1+x3 , -12091/1000+x1+2*x2 ,
-2761/250+x1+x2+x3 , -6983/500+x1+2*x3 , -17933/1000+3*x2 , -6979/500+2*x2+x3 ,
-7023/500+x2+2*x3 , -2249/125+3*x3)"));
$L = lines_in_cubic($H->POLYNOMIAL);
print $L->N_ISOLATED;
print $L->N_FAMILIES;
visualize_in_surface($H,$L->all_isolated);