//////////////////////////////////////////////////////////////// // Modifications for Type VI Graph with Vertical modification // //////////////////////////////////////////////////////////////// // The following contains an example for the modified Type VI graph in R^5 (with one genus 1 component with a node). The curve can be visualized by means of projections, after a sequence of 2 vertical modifications // Since our modifications involve square roots, we take our branch points to be squares, resp. negative of squares, for computational convenience. // The parameters w2,..., w5 satisfy w2 < w3 = w4 = w5 with no cancellations. LIB "all.lib"; LIB "poly.lib"; LIB "all.lib"; LIB "tropical.lib"; LIB "elim.lib"; ring rr = (0,t), (a2,a3,a4,a5, x,y,z),dp; poly f=y^2-x*(x-a2)*(x-a3)*(x-a4)*(x-a5); // f; // -a2*a3*a4*a5*x+a2*a3*a4*x^2+a2*a3*a5*x^2+a2*a4*a5*x^2+a3*a4*a5*x^2-a2*a3*x^3-a2*a4*x^3-a3*a4*x^3-a2*a5*x^3-a3*a5*x^3-a4*a5*x^3+a2*x^4+a3*x^4+a4*x^4+a5*x^4-x^5+y^2 setring(rr); poly b2 = (t^14+t^16); poly b3 = (5+t^9); poly b4 = (3+t^8); poly b5 = (1+t^5); poly A2 = b2^2; poly A3 = b3^2; poly A4 = b4^2; poly A5 = -1*b5^2; map P2 = rr, A2, A3, A4, A5, x, y, z; poly ff = P2(f); ring r = (0,t),(x,y),dp; map newP2 = rr, 0,0,0,0,x,y,0; poly f2 = newP2(ff); // f2; // -x5+(t32+2t30+t28+t18+t16-t10+10t9+6t8-2t5+33)*x4+(-t50-3t48-3t46-t44+t42-10t41-4t40-20t39-11t38-8t37-6t36+4t35-t34+2t33-33t32-66t30-32t28-5t26-10t25+2t23+2t21+10t19-2t18-60t17-24t16+20t14+12t13+34t10-80t9-144t8+68t5-191)*x3+(t66+2t64+t62-t60+3t58+10t57+9t56+18t55+5t54+4t53-16t51+2t50+38t49+28t48+110t47+30t46+48t45-17t44-24t43-54t42+68t41+76t40+158t39+254t38+12t37+138t36-146t35-t34-68t33+191t32-12t31+362t30+182t28-60t27-31t26-10t25-18t23-120t22-50t21-90t19-159t18-60t17-25t16-180t14-300t13-225t10-90t9-150t8-450t5-225)*x2+y2+(t76+2t74+t72+2t71+4t69+6t68+12t67+13t66+20t65+8t64+22t63+21t62+24t61+49t60+72t59+69t58+130t57+71t56+98t55+151t54+96t53+240t52+208t51+279t50+290t49+343t48+210t47+389t46+360t45+385t44+600t43+405t42+390t41+600t40+180t39+525t38+540t37+150t36+900t35+450t33+225t32+450t30+225t28)*x drawTropicalCurve(f2,"max"); /////////////////// // XZ-projection // /////////////////// poly b2 = newP2(b2); poly b3 = newP2(b3); poly b4 = newP2(b4); poly b5 = newP2(b5); poly g2 = substitute(f2, y, y+b3*b4*b5*x-b5*x^2); // g2; // -x5+(t32+2t30+t28+t18+t16+10t9+6t8+34)*x4+(-t50-3t48-3t46-t44+t42-10t41-4t40-20t39-11t38-8t37-6t36+4t35-t34+2t33-33t32-66t30-32t28-2t27-5t26-10t25+2t23-4t22+2t21+4t19-12t18-62t17-24t16+8t14-8t13+4t10-86t9-154t8+8t5-221)*x3+(-2t5-2)*x2y+(t66+2t64+t62-t60+3t58+10t57+9t56+18t55+5t54+4t53-16t51+2t50+38t49+28t48+110t47+30t46+48t45-16t44-24t43-54t42+68t41+76t40+160t39+254t38+12t37+144t36-136t35-68t33+191t32+382t30+191t28)*x2+(2t22+2t17+6t14+10t13+6t9+10t8+30t5+30)*xy+y2+(t76+2t74+t72+2t71+4t69+6t68+12t67+13t66+20t65+8t64+22t63+21t62+24t61+49t60+72t59+69t58+130t57+71t56+98t55+151t54+96t53+240t52+208t51+279t50+290t49+343t48+210t47+389t46+360t45+385t44+600t43+405t42+390t41+600t40+180t39+525t38+540t37+150t36+900t35+450t33+225t32+450t30+225t28)*x drawTropicalCurve(g2,"max"); ///////////////////// // Z_4Y-projection // ///////////////////// poly b2 = newP2(b2); poly b3 = newP2(b3); poly b4 = newP2(b4); poly b5 = newP2(b5); poly A4 = newP2(A4); poly g2 = substitute(f2, x, x+A4); // g2; // -x5+(t32+2t30+t28+t18-4t16-t10+10t9-24t8-2t5-12)*x4+(-t50+t48+5t46+3t44+t42-10t41+20t40-20t39+37t38-8t37+18t36+4t35+3t34+2t33-3t32+6t30+4t28+15t26+30t25-72t24+2t23-6t21+10t19+10t18+180t17-252t16+20t14-36t13-2t10+280t9-216t8-4t5+187)*x3+(-2t66-t64+4t62+2t60-12t58-20t57+15t56-42t55+62t54-20t53+36t52-4t51-4t50-136t49+96t48-250t47+201t46-96t45+76t44+48t43+6t42-136t41-50t40-376t39+65t38-210t37+48t36-8t35+128t34+346t33-682t32+24t31-388t30-72t29-196t28+120t27+224t26+1640t25-1296t24+36t23+240t22-224t21+180t19+237t18+3360t17-922t16+360t14-48t13+207t10+2610t9+2244t8+414t5+3366)*x2+y2+(-t82-t80+t78-13t74-10t73-5t72-22t71+25t70-12t69+12t68-8t67-54t66-138t65+18t64-262t63+159t62-120t61+63t60-24t59-106t58-636t57+49t56-1176t55+299t54-530t53+24t52+18t51-134t50-1082t49-598t48-2346t47-504t46-1260t45-412t44-48t43+118t42+376t41-2652t40-1692t39-2472t38-1408t37-1176t36-36t35+1029t34+5232t33-3845t32+216t31-1800t30-480t29-1359t28+1080t27+1701t26+12690t25-348t24+162t23+2160t22+270t21+810t19+2052t18+15660t17+10098t16+1620t14+2484t13+1296t10+8100t9+20196t8+2592t5+12960)*x drawTropicalCurve(g2,"max"); ///////////////////// // Z_4Z-projection // ///////////////////// poly gz4z = substitute(substitute(f2, y, y+b3*b4*b5*x-b5*x^2),x, x + A4); // gz4z; // -x5+(t32+2t30+t28+t18-4t16+10t9-24t8-11)*x4+(-t50+t48+5t46+3t44+t42-10t41+20t40-20t39+37t38-8t37+18t36+4t35+3t34+2t33-3t32+6t30+4t28-2t27+19t26+30t25-72t24+2t23-4t22+2t21+4t19+24t18+178t17-248t16+8t14-8t13+4t10+274t9-202t8+8t5+193)*x3+(-2t5-2)*x2y+(-2t66-t64+4t62+2t60-12t58-20t57+15t56-42t55+62t54-20t53+36t52-4t51-4t50-136t49+96t48-250t47+201t46-96t45+77t44+42t43+12t42-136t41-50t40-374t39+53t38-198t37+54t36-52t35+171t34+340t33-676t32+36t31-476t30+12t29-187t28+18t27+309t26+1596t25-1254t24+54t23+36t22-66t21+108t19+234t18+3258t17-843t16+216t14-72t13+108t10+2538t9+2232t8+216t5+3267)*x2+(2t22-4t21+2t17-4t16+6t14-14t13+6t9-14t8-6t5-6)*xy+y2+(-t82-t80+t78-13t74-10t73-5t72-22t71+25t70-12t69+12t68-8t67-54t66-138t65+18t64-262t63+159t62-120t61+65t60-30t59-102t58-636t57+49t56-1172t55+287t54-522t53+48t52-52t51-90t50-1088t49-594t48-2298t47-644t46-1176t45-304t44-348t43+282t42+306t41-2610t40-1476t39-3072t38-1128t37-960t36-576t35+1197t34+4932t33-3705t32+648t31-2880t30-360t29-1197t28+810t27+1377t26+12150t25-288t24+486t23+1620t22-810t21+972t19+1296t18+15390t17+9558t16+1944t14+648t13+972t10+8262t9+19278t8+1944t5+12636)*x+(2t38-2t37+2t33-2t32+18t30-14t29+18t25-14t24+54t22-18t21+54t17-18t16+54t14+54t13+54t9+54t8+108t5+108)*y+(t76-2t75+t74+2t71-4t70+2t69+18t68-32t67+15t66-2t65+t64+36t63-64t62+28t61+135t60-198t59+85t58-32t57+14t56+270t55-396t54+134t53+540t52-540t51+207t50-198t49+67t48+1080t47-1080t46+144t45+1215t44-270t43+135t42-540t41+72t40+2430t39-540t38-810t37+1458t36+1944t35-27t34-270t33-405t32+2916t31+3888t30-2484t29+729t28+4374t27+1215t26+1944t25-1242t24+1458t23+8748t22-486t21+2916t19+3645t18+4374t17-243t16+5832t14+5832t13+2916t10+2916t9+2916t8+5832t5+2916) drawTropicalCurve(gz4z,"max"); ///////////////////// // Z_3Y-projection // ///////////////////// poly A3 = newP2(A3); poly h2 = substitute(f2, x, x+A3); // h2; // -x5+(t32+2t30+t28-4t18+t16-t10-40t9+6t8-2t5-92)*x4+(3t50+5t48+t46-t44+t42+30t41-4t40+60t39-11t38+32t37-12t36+4t35+3t34+2t33+67t32+134t30+64t28-120t27+19t26+30t25-6t23+2t21-30t19-870t18+180t17+76t16-60t14+12t13-66t10-2760t9+456t8-132t5-3141)*x3+(3t68+4t66-t64-2t62+2t60+60t59-9t58+100t57-24t56+24t55-17t54-14t53+3t52+20t51+428t50-82t49+805t48-220t47+363t46-252t45+49t44+36t43+81t42+1322t41-224t40+2684t39-571t38+1362t37-1770t36+544t35+452t34+82t33+1346t32+24t31+2972t30+1034t28-9400t27+2762t26+1520t25-864t23+240t22+100t21-1320t19-33582t18+9120t17+1925t16-2640t14+600t13-1425t10-62820t9+11550t8-2850t5-47050)*x2+y2+(t86+t84-t82-t80+t78+30t77-5t76+40t75-13t74-8t73-8t72-18t71+t70+18t69+361t68-92t67+571t66-240t65+98t64-182t63-84t62+6t61+141t60+2266t59-669t58+4052t57-1721t56+1512t55-1737t54+204t53+136t52+592t51+7836t50-2278t49+15422t48-5710t47+7170t46-10000t45+3396t44+1920t43+1220t42+13628t41-2860t40+29620t39-7400t38+13140t37-43741t36+17220t35+9525t34+800t33+5760t32+1800t31+21800t30+2375t28-151820t27+57775t26+19250t25-16050t23+6000t22+1250t21-14250t19-357400t18+115500t17+16250t16-28500t14+7500t13-10000t10-470500t9+97500t8-20000t5-260000)*x drawTropicalCurve(h2,"max"); ///////////////////// // Z_3Z-projection // ///////////////////// poly gz3z = substitute(substitute(f2, y, y+b3*b4*b5*x-b5*x^2),x, x + A3); // gz3z; // -x5+(t32+2t30+t28-4t18+t16-40t9+6t8-91)*x4+(3t50+5t48+t46-t44+t42+30t41-4t40+60t39-11t38+32t37-12t36+4t35+3t34+2t33+67t32+134t30+68t28-122t27+19t26+30t25+2t23-4t22+2t21+4t19-876t18+178t17+76t16+8t14-8t13+4t10-2726t9+446t8+8t5-3071)*x3+(-2t5-2)*x2y+(3t68+4t66-t64-2t62+2t60+60t59-9t58+100t57-24t56+24t55-17t54-14t53+3t52+20t51+428t50-82t49+805t48-220t47+369t46-258t45+50t44+36t43+81t42+1334t41-236t40+2686t39-571t38+1464t37-1848t36+548t35+453t34+82t33+1550t32-144t31+2992t30+1673t28-9688t27+2703t26+1530t25+414t23-540t22+150t21+420t19-33543t18+8730t17+1950t16+840t14-600t13+300t10-61080t9+10950t8+600t5-45325)*x2+(-4t23+2t22-4t18+2t17-34t14+10t13-34t9+10t8-70t5-70)*xy+y2+(t86+t84-t82-t80+t78+30t77-5t76+40t75-13t74-8t73-8t72-18t71+t70+18t69+361t68-92t67+571t66-240t65+102t64-188t63-82t62+6t61+141t60+2274t59-681t58+4056t57-1721t56+1614t55-1871t54+238t53+138t52+592t51+8040t50-2554t49+15502t48-5710t47+8238t46-11158t45+3558t44+1960t43+1220t42+15764t41-5380t40+30220t39-7400t38+19000t37-48373t36+16960t35+9825t34+800t33+17480t32-9600t31+23800t30+20075t28-158710t27+53325t26+20250t25+19350t23-19500t22+3750t21+13500t19-350950t18+102750t17+17500t16+27000t14-15000t13+7500t10-442750t9+86250t8+15000t5-242500)*x+(-2t41+2t40-2t36+2t35-34t32+30t31-34t27+30t26-210t23+150t22-210t18+150t17-550t14+250t13-550t9+250t8-500t5-500)*y+(t82-2t81+t80+2t77-4t76+2t75+34t73-63t72+28t71+t70+68t68-128t67+60t66+499t64-836t63+311t62+30t61+998t59-1740t58+750t57+4120t55-6001t54+1630t53+375t52+8240t50-13000t49+5000t48+20875t46-24630t45+2875t44+2500t43+41750t41-57500t40+18750t39+66250t37-54125t36-10000t35+9375t34+132500t32-150000t31+37500t30+128125t28-40000t27-59375t26+18750t25+256250t23-212500t22+31250t21+137500t19+65625t18-106250t17+15625t16+275000t14-125000t13+62500t10+137500t9-62500t8+125000t5+62500) drawTropicalCurve(gz3z,"max"); /////////////////// // ZY-projection // /////////////////// ring s = (0,t),(x,y,z),dp; map P = r, x,y; poly ff= P(f2); poly b3 = P(b3); poly b4 = P(b4); poly b5 = P(b5); ideal I = (ff,z-y+b3*b4*b5*x-b5*x^2); poly fnox= eliminate(I,x)[1]; // Replace z by y and y by x so that the pictures are not flipped. ring r2 = (0,t),(x,y),dp; setring(r2); map PP = s,0,x,y; poly newfnox = PP(fnox); // newfnox; // 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drawTropicalCurve(newfnox,"max");