////////////////////////////////////////////////////////////////////////////////////////// // Refined Modifications for Type VI Graph: Non-generic x^3 with Vertical modifications // ////////////////////////////////////////////////////////////////////////////////////////// // 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 = (2+t^9); poly b4 = (1+t^8); poly b5 = (2+t^5); poly A2 = b2^2; poly A3 = b3^2; poly A4 = b4^2; poly A5 = -1*b5^2; // A5; // (-t^10-4*t^5-4) // A4; // (t^16+2*t^8+1) // A3; // (t^18+4*t^9+4) 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+4t9+2t8-4t5+1)*x4+(-t50-3t48-3t46-t44+t42-4t41-8t39-3t38-2t36+8t35-t34+4t33-t32-2t30-t26-4t25+4t23+4t21+4t19+5t18-8t17+16t14+8t13+5t10+12t9+20t5+16)*x3+(t66+2t64+t62-t60-t58+4t57+t56+4t55+t54-8t53-16t51-5t50-4t49-10t48+12t47-21t46-33t44-16t43-21t42-20t41-10t40-28t39-5t38-32t37-2t36-44t35-4t34-20t33-16t32-8t31-48t30-17t28-8t27-12t26-16t25-4t23-32t22-16t21-4t19-12t18-32t17-16t16-16t14-32t13-4t10-16t9-32t8-16t5-16)*x2+y2+(t76+2t74+t72+4t71+8t69+2t68+8t67+8t66+8t65+10t64+12t63+20t62+16t61+33t60+16t59+30t58+32t57+25t56+44t55+44t54+40t53+64t52+40t51+44t50+56t49+40t48+68t47+60t46+64t45+48t44+64t43+20t42+48t41+40t40+32t39+68t38+32t37+32t36+32t35+16t33+16t32+32t30+16t28)*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+4t9+2t8+5)*x4+(-t50-3t48-3t46-t44+t42-4t41-2t40-8t39-3t38-2t36-t34+4t33-3t32-4t31-10t30-10t27-17t26-4t25-16t22-12t21+2t19-15t18-16t17+8t14-24t13+t10+4t9-16t8+4t5)*x3+(-2t5-4)*x2y+(t70+t66+4t65+2t64+3t62+4t61+3t60-t58+14t57+17t56+4t55+2t54+20t52-5t50+4t49+30t48+36t47-21t46+4t45-4t44+48t43-5t42-18t41+22t40+32t39+27t38-32t37+12t36+40t35+64t34-20t33-8t32+40t31+32t30+32t29-16t28+32t27+80t26+8t23+32t22+64t21+32t18+32t13)*x2+(2t35+4t30+2t27+4t26+6t22+8t21+4t18+4t17+2t14+12t13+4t9+8t8+4t5+8)*xy+y2+(t76+2t74+t72+4t71+8t69+2t68+8t67+8t66+8t65+10t64+12t63+20t62+16t61+33t60+16t59+30t58+32t57+25t56+44t55+44t54+40t53+64t52+40t51+44t50+56t49+40t48+68t47+60t46+64t45+48t44+64t43+20t42+48t41+40t40+32t39+68t38+32t37+32t36+32t35+16t33+16t32+32t30+16t28)*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+4t9-8t8-4t5-4)*x4+(-t50+t48+5t46+3t44+t42-4t41+8t40-8t39+13t38+6t36+8t35+3t34+4t33-3t32+6t30+4t28+3t26+12t25-24t24+4t23-12t21+4t19+t18+24t17-36t16+16t14-24t13+t10+28t9-24t8+4t5+10)*x3+(-2t66-t64+4t62+2t60-4t58-8t57+7t56-20t55+22t54-8t53+12t52+8t51+t50-16t49+10t48-36t47+18t46-12t44+32t43-9t42+4t41-16t40-40t39+22t38-44t37+16t36-8t35+14t34+40t33-73t32+16t31+6t30-48t29-11t28+16t27+18t26+104t25-80t24+8t23+64t22-40t21+8t19+15t18+112t17-28t16+32t14+16t13+5t10+44t9+40t8+20t5+28)*x2+y2+(-t82-t80+t78-5t74-4t73-t72-12t71+9t70-8t69+4t68-9t66-20t65-44t63+8t62-16t61-17t60+16t59-23t58-36t57-9t56-88t55-2t54-44t53-40t52+12t51-28t50-24t49-47t48-84t47-11t46-88t45-27t44-16t43+6t42+40t41-88t40-16t39+9t38-76t37-22t36+35t34+132t33-69t32+16t31+66t30-16t29-14t28+16t27+31t26+156t25+8t24+4t23+64t22+36t21+4t19+15t18+88t17+68t16+16t14+40t13+3t10+20t9+56t8+12t5+15)*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+4t9-8t8)*x4+(-t50+t48+5t46+3t44+t42-4t41+8t40-8t39+13t38+6t36+8t35+3t34+4t33-3t32+6t30+4t28-2t27+7t26+12t25-24t24+4t23-8t22+4t21+2t19+5t18+16t17-20t16+8t14-8t13+t10+20t9-8t8+4t5+10)*x3+(-2t5-4)*x2y+(-2t66-t64+4t62+2t60-4t58-8t57+7t56-20t55+22t54-8t53+12t52+8t51+t50-16t49+10t48-36t47+18t46-11t44+26t43-3t42+4t41-16t40-36t39-2t38-20t37+18t36-22t35+30t34+16t33-49t32+24t31-50t30-10t28+6t27+30t26+48t25-32t24+12t23+24t22-24t21+6t19+15t18+72t17-12t16+24t14+3t10+36t9+24t8+12t5+20)*x2+(2t22-4t21+4t17-8t16+2t14-4t13+4t9-8t8)*xy+y2+(-t82-t80+t78-5t74-4t73-t72-12t71+9t70-8t69+4t68-9t66-20t65-44t63+8t62-16t61-15t60+10t59-19t58-36t57-9t56-80t55-26t54-28t53-32t52-10t51-8t50-48t49-31t48-52t47-99t46-40t45-15t44-44t43+46t42-48t41-40t40+32t39-103t38-44t37-14t36-12t35+75t34+20t33-37t32+48t31+18t30-48t29-12t28+18t27+51t26+108t25-24t24+12t23+72t22-12t21+6t19+19t18+96t17+20t16+24t14+24t13+3t10+28t9+40t8+12t5+15)*x+(2t38-2t37+4t33-4t32+6t30-4t29+12t25-8t24+6t22+12t17+2t14+4t13+4t9+8t8+2t5+4)*y+(t76-2t75+t74+4t71-8t70+4t69+6t68-10t67+8t66-8t65+4t64+24t63-40t62+16t61+15t60-18t59+28t58-40t57+16t56+60t55-72t54+16t53+20t52-10t51+56t50-72t49+16t48+80t47-40t46-16t45+15t44+10t43+70t42-40t41-16t40+60t39+40t38-40t37+6t36+18t35+56t34+40t33-40t32+24t31+72t30-16t29+t28+10t27+28t26+72t25-16t24+4t23+40t22+16t21+2t19+8t18+40t17+16t16+8t14+16t13+t10+8t9+16t8+4t5+4) drawTropicalCurve(gz4z,"max"); ///////////////////// // Z_3Y-projection // ///////////////////// poly A3 = newP2(A3); poly h2 = substitute(f2, x, x+A3); // h2; // -x5+(t32+2t30+t28-4t18+t16-t10-16t9+2t8-4t5-19)*x4+(3t50+5t48+t46-t44+t42+12t41+24t39-3t38+16t37-8t36+8t35+3t34+4t33+15t32+30t30+12t28-48t27+7t26+12t25-12t23+4t21-12t19-151t18+24t17+16t16-48t14+8t13-11t10-228t9+32t8-44t5-128)*x3+(3t68+4t66-t64-2t62+2t60+24t59-t58+40t57-8t56+16t55-9t54+4t53+3t52+8t51+76t50-4t49+140t48-24t47+81t46-72t45+60t44+8t43+39t42+100t41-10t40+248t39-41t38+124t37-269t36+112t35+80t34+28t33-28t32+16t31+168t30-2t28-696t27+168t26+128t25-280t23+64t22+32t21-88t19-1152t18+256t17+80t16-352t14+64t13-40t10-1024t9+160t8-160t5-368)*x2+y2+(t86+t84-t82-t80+t78+12t77-t76+16t75-5t74-4t72-4t71+t70+4t69+61t68-8t67+94t66-32t65+36t64-44t63+23t62-4t61+23t60+164t59-30t58+308t57-97t56+160t55-205t54+152t53+88t51+220t50-48t49+600t48-164t47+289t46-596t45+344t44+128t43+124t42+4t41+24t40+640t39-116t38+168t37-1312t36+512t35+336t34+48t33-512t32+192t31+320t30-120t28-2176t27+688t26+320t25-864t23+256t22+64t21-160t19-2384t18+640t17+128t16-640t14+128t13-48t10-1472t9+256t8-192t5-384)*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+t16-16t9+2t8-15)*x4+(3t50+5t48+t46-t44+t42+12t41+24t39-3t38+16t37-8t36+8t35+3t34+4t33+15t32+30t30+16t28-50t27+7t26+12t25+4t23-8t22+4t21+2t19-139t18+16t17+16t16+8t14-8t13+t10-172t9+16t8+4t5-80)*x3+(-2t5-4)*x2y+(3t68+4t66-t64-2t62+2t60+24t59-t58+40t57-8t56+16t55-9t54+4t53+3t52+8t51+76t50-4t49+140t48-24t47+87t46-78t45+61t44+8t43+39t42+124t41-34t40+252t39-41t38+166t37-279t36+92t35+84t34+28t33+140t32-120t31+184t30+107t28-592t27+36t26+144t25+156t23-192t22+48t21+36t19-756t18+96t16+144t14-96t13+12t10-528t9+48t5-160)*x2+(-4t23+2t22-8t18+4t17-14t14+4t13-28t9+8t8-12t5-24)*xy+y2+(t86+t84-t82-t80+t78+12t77-t76+16t75-5t74-4t72-4t71+t70+4t69+61t68-8t67+94t66-32t65+40t64-50t63+25t62-4t61+23t60+180t59-54t58+316t57-97t56+202t55-245t54+144t53+8t52+88t51+388t50-272t49+664t48-164t47+471t46-636t45+168t44+192t43+124t42+732t41-808t40+832t39-116t38+584t37-968t36-256t35+528t34+48t33+1152t32-1344t31+576t30+408t28-864t27-816t26+576t25+1248t23-1152t22+192t21+192t19-400t18-768t17+256t16+768t14-384t13+48t10-64t9-256t8+192t5)*x+(-2t41+2t40-4t36+4t35-14t32+12t31-28t27+24t26-36t23+24t22-72t18+48t17-40t14+16t13-80t9+32t8-16t5-32)*y+(t82-2t81+t80+4t77-8t76+4t75+14t73-22t72+4t71+4t70+56t68-104t67+48t66+85t64-88t63-44t62+48t61+340t59-576t58+240t57+292t55-100t54-416t53+240t52+1168t50-1760t49+640t48+620t46+368t45-1520t44+640t43+2480t41-3200t40+960t39+832t37+1616t36-3008t35+960t34+3328t32-3456t31+768t30+688t28+2816t27-3392t26+768t25+2752t23-2048t22+256t21+320t19+2624t18-2048t17+256t16+1280t14-512t13+64t10+1280t9-512t8+256t5+256) 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");