-- An example of smooth rational surface S\subset P^8\subset P^9 of degree 9 and sectional genus 2 which is embedded in the Grassmannian G(1,4)\subset P^9. G = Grassmannian(1,4,CoefficientRing=>ZZ/10000019,Variable=>p) use ring G S = ideal(p_(0,1)+3482965*p_(0,2)-1747436*p_(1,2)+2559507*p_(0,3)+4310417*p_(1,3)+2857572*p_(2,3)-2638665*p_(0,4)+632328*p_(1,4)+74921*p_(2,4)+431195*p_(3,4),p_(2,3)*p_(1,4)-p_(1,3)*p_(2,4)+p_(1,2)*p_(3,4),p_(0,4)^2+1475121*p_(0,2)*p_(1,4)+2076647*p_(1,2)*p_(1,4)-4701152*p_(0,3)*p_(1,4)-478269*p_(1,3)*p_(1,4)-171317*p_(0,4)*p_(1,4)+4169313*p_(1,4)^2+2134310*p_(0,2)*p_(2,4)+1824437*p_(1,2)*p_(2,4)+2641996*p_(0,3)*p_(2,4)-1651510*p_(1,3)*p_(2,4)-3107857*p_(2,3)*p_(2,4)+3745569*p_(0,4)*p_(2,4)-2310438*p_(1,4)*p_(2,4)-965158*p_(2,4)^2-2139203*p_(0,2)*p_(3,4)+2800060*p_(1,2)*p_(3,4)+497820*p_(0,3)*p_(3,4)-1515499*p_(1,3)*p_(3,4)+464705*p_(2,3)*p_(3,4)-669990*p_(0,4)*p_(3,4)-758981*p_(1,4)*p_(3,4)+371416*p_(2,4)*p_(3,4)+4990785*p_(3,4)^2,p_(2,3)*p_(0,4)-p_(0,3)*p_(2,4)+p_(0,2)*p_(3,4),p_(1,3)*p_(0,4)-p_(0,3)*p_(1,4)-3482965*p_(0,2)*p_(3,4)+1747436*p_(1,2)*p_(3,4)-2559507*p_(0,3)*p_(3,4)-4310417*p_(1,3)*p_(3,4)-2857572*p_(2,3)*p_(3,4)+2638665*p_(0,4)*p_(3,4)-632328*p_(1,4)*p_(3,4)-74921*p_(2,4)*p_(3,4)-431195*p_(3,4)^2,p_(0,3)*p_(0,4)-1183046*p_(0,2)*p_(1,4)-1818289*p_(1,2)*p_(1,4)+2810442*p_(0,3)*p_(1,4)+912600*p_(1,3)*p_(1,4)-4663271*p_(0,4)*p_(1,4)-2136909*p_(1,4)^2+4180958*p_(0,2)*p_(2,4)+4364962*p_(1,2)*p_(2,4)-2583461*p_(0,3)*p_(2,4)+2900519*p_(1,3)*p_(2,4)-1897830*p_(2,3)*p_(2,4)-3196558*p_(0,4)*p_(2,4)-3836810*p_(1,4)*p_(2,4)-4884262*p_(2,4)^2+2968993*p_(0,2)*p_(3,4)-2845174*p_(1,2)*p_(3,4)+2040432*p_(0,3)*p_(3,4)-2660057*p_(1,3)*p_(3,4)-1689775*p_(2,3)*p_(3,4)+3283294*p_(0,4)*p_(3,4)-4812037*p_(1,4)*p_(3,4)-3226725*p_(2,4)*p_(3,4)-1218283*p_(3,4)^2,p_(1,2)*p_(0,4)-p_(0,2)*p_(1,4)-3482965*p_(0,2)*p_(2,4)+1747436*p_(1,2)*p_(2,4)-2559507*p_(0,3)*p_(2,4)-4310417*p_(1,3)*p_(2,4)-2857572*p_(2,3)*p_(2,4)+2638665*p_(0,4)*p_(2,4)-632328*p_(1,4)*p_(2,4)-74921*p_(2,4)^2-431195*p_(2,4)*p_(3,4),p_(0,2)*p_(0,4)-2805216*p_(0,2)*p_(1,4)+2879992*p_(1,2)*p_(1,4)+2277418*p_(0,3)*p_(1,4)+3329877*p_(1,3)*p_(1,4)-2580434*p_(0,4)*p_(1,4)+3018867*p_(1,4)^2-1973175*p_(0,2)*p_(2,4)+4772064*p_(1,2)*p_(2,4)+2582254*p_(0,3)*p_(2,4)-4180706*p_(1,3)*p_(2,4)+4643788*p_(2,3)*p_(2,4)-1442183*p_(0,4)*p_(2,4)-1112502*p_(1,4)*p_(2,4)+309497*p_(2,4)^2-2279509*p_(0,2)*p_(3,4)+3223677*p_(1,2)*p_(3,4)+2097709*p_(0,3)*p_(3,4)-936409*p_(1,3)*p_(3,4)-4316155*p_(2,3)*p_(3,4)-1277168*p_(0,4)*p_(3,4)-2100160*p_(1,4)*p_(3,4)-3697451*p_(2,4)*p_(3,4)-1459052*p_(3,4)^2,p_(1,2)*p_(2,3)-1250988*p_(0,3)*p_(2,3)-1831540*p_(1,3)*p_(2,3)-2897678*p_(2,3)^2-1097101*p_(0,2)*p_(2,4)-4114665*p_(1,2)*p_(2,4)+3297274*p_(0,3)*p_(2,4)+3225199*p_(1,3)*p_(2,4)-3943976*p_(2,3)*p_(2,4)+146963*p_(0,4)*p_(2,4)+4897264*p_(1,4)*p_(2,4)+1563584*p_(2,4)^2-1541825*p_(0,2)*p_(3,4)+3548396*p_(1,2)*p_(3,4)+2707167*p_(0,3)*p_(3,4)+988300*p_(1,3)*p_(3,4)+2403092*p_(2,3)*p_(3,4)-2542139*p_(0,4)*p_(3,4)-2725529*p_(1,4)*p_(3,4)-2624686*p_(2,4)*p_(3,4)-3078776*p_(3,4)^2,p_(0,2)*p_(2,3)+2905955*p_(0,3)*p_(2,3)+2625769*p_(1,3)*p_(2,3)-642115*p_(2,3)^2+4200099*p_(0,2)*p_(2,4)+1619008*p_(1,2)*p_(2,4)+2226635*p_(0,3)*p_(2,4)+774517*p_(1,3)*p_(2,4)-2642263*p_(2,3)*p_(2,4)-514992*p_(0,4)*p_(2,4)+1360438*p_(1,4)*p_(2,4)-3381448*p_(2,4)^2+3889066*p_(0,2)*p_(3,4)+4278570*p_(1,2)*p_(3,4)-4618770*p_(0,3)*p_(3,4)+2025969*p_(1,3)*p_(3,4)+2758187*p_(2,3)*p_(3,4)-1505647*p_(0,4)*p_(3,4)+3398409*p_(1,4)*p_(3,4)-4764348*p_(2,4)*p_(3,4)+3318085*p_(3,4)^2,p_(1,3)^2+3878593*p_(0,3)*p_(2,3)-4007402*p_(1,3)*p_(2,3)-4852892*p_(2,3)^2-1786311*p_(0,2)*p_(1,4)+4033218*p_(1,2)*p_(1,4)-1163171*p_(0,3)*p_(1,4)-4069753*p_(1,3)*p_(1,4)+1473028*p_(0,4)*p_(1,4)+2828290*p_(1,4)^2+3242868*p_(0,2)*p_(2,4)-706684*p_(1,2)*p_(2,4)+3586346*p_(0,3)*p_(2,4)-675928*p_(1,3)*p_(2,4)-1126115*p_(2,3)*p_(2,4)+1871136*p_(0,4)*p_(2,4)+4476644*p_(1,4)*p_(2,4)-2725228*p_(2,4)^2-3183943*p_(0,2)*p_(3,4)-1389611*p_(1,2)*p_(3,4)-2395447*p_(0,3)*p_(3,4)+4715253*p_(1,3)*p_(3,4)+2240336*p_(2,3)*p_(3,4)+4801481*p_(0,4)*p_(3,4)-3279091*p_(1,4)*p_(3,4)-4039790*p_(2,4)*p_(3,4)-2708674*p_(3,4)^2,p_(0,3)*p_(1,3)-3566236*p_(0,3)*p_(2,3)+2334127*p_(1,3)*p_(2,3)+3305057*p_(2,3)^2+2763693*p_(0,2)*p_(1,4)+1233799*p_(1,2)*p_(1,4)-1182907*p_(0,3)*p_(1,4)+4739918*p_(1,3)*p_(1,4)-3080755*p_(0,4)*p_(1,4)-3893092*p_(1,4)^2-2479024*p_(0,2)*p_(2,4)-1288544*p_(1,2)*p_(2,4)+3376464*p_(0,3)*p_(2,4)+2745701*p_(1,3)*p_(2,4)+3564096*p_(2,3)*p_(2,4)+4033833*p_(0,4)*p_(2,4)-3580982*p_(1,4)*p_(2,4)+205236*p_(2,4)^2+3704367*p_(0,2)*p_(3,4)+3196359*p_(1,2)*p_(3,4)-499428*p_(0,3)*p_(3,4)-868235*p_(1,3)*p_(3,4)+3969863*p_(2,3)*p_(3,4)+594957*p_(0,4)*p_(3,4)-4758026*p_(1,4)*p_(3,4)-3112886*p_(2,4)*p_(3,4)+2986069*p_(3,4)^2,p_(1,2)*p_(1,3)+4761359*p_(0,3)*p_(2,3)+3308243*p_(1,3)*p_(2,3)-135391*p_(2,3)^2-4720092*p_(0,2)*p_(1,4)-3243388*p_(1,2)*p_(1,4)+3481179*p_(0,3)*p_(1,4)+1250790*p_(1,3)*p_(1,4)+4510695*p_(0,4)*p_(1,4)-799880*p_(1,4)^2-2211591*p_(0,2)*p_(2,4)+2427081*p_(1,2)*p_(2,4)+1593655*p_(0,3)*p_(2,4)+1715496*p_(1,3)*p_(2,4)+2107139*p_(2,3)*p_(2,4)+1007136*p_(0,4)*p_(2,4)+1538306*p_(1,4)*p_(2,4)+2677807*p_(2,4)^2+1174685*p_(0,2)*p_(3,4)+184371*p_(1,2)*p_(3,4)-2203362*p_(0,3)*p_(3,4)-4612985*p_(1,3)*p_(3,4)-3674636*p_(2,3)*p_(3,4)-1376309*p_(0,4)*p_(3,4)-3529599*p_(1,4)*p_(3,4)+296988*p_(2,4)*p_(3,4)+1455698*p_(3,4)^2,p_(0,2)*p_(1,3)+1732206*p_(0,3)*p_(2,3)-3831578*p_(1,3)*p_(2,3)+496895*p_(2,3)^2+3103811*p_(0,2)*p_(1,4)-4614982*p_(1,2)*p_(1,4)+2757127*p_(0,3)*p_(1,4)-505391*p_(1,3)*p_(1,4)+3961590*p_(0,4)*p_(1,4)-2066461*p_(1,4)^2+2413499*p_(0,2)*p_(2,4)-820541*p_(1,2)*p_(2,4)-2058607*p_(0,3)*p_(2,4)-3615410*p_(1,3)*p_(2,4)-673922*p_(2,3)*p_(2,4)+3527933*p_(0,4)*p_(2,4)-1310800*p_(1,4)*p_(2,4)+157092*p_(2,4)^2+1913647*p_(0,2)*p_(3,4)-1478564*p_(1,2)*p_(3,4)-1238402*p_(0,3)*p_(3,4)-4012165*p_(1,3)*p_(3,4)-637907*p_(2,3)*p_(3,4)+339067*p_(0,4)*p_(3,4)+4836137*p_(1,4)*p_(3,4)+3920872*p_(2,4)*p_(3,4)-947398*p_(3,4)^2,p_(0,3)^2-1718524*p_(0,3)*p_(2,3)-796436*p_(1,3)*p_(2,3)-4331063*p_(2,3)^2-647681*p_(0,2)*p_(1,4)-2834097*p_(1,2)*p_(1,4)-2299587*p_(0,3)*p_(1,4)+1792979*p_(1,3)*p_(1,4)-189750*p_(0,4)*p_(1,4)+1911105*p_(1,4)^2-3890850*p_(0,2)*p_(2,4)+939418*p_(1,2)*p_(2,4)-4940314*p_(0,3)*p_(2,4)+279052*p_(1,3)*p_(2,4)+3125567*p_(2,3)*p_(2,4)+1451186*p_(0,4)*p_(2,4)-2627890*p_(1,4)*p_(2,4)+3468518*p_(2,4)^2+2429383*p_(0,2)*p_(3,4)+143021*p_(1,2)*p_(3,4)+1635542*p_(0,3)*p_(3,4)-4854981*p_(1,3)*p_(3,4)+4106871*p_(2,3)*p_(3,4)+2603722*p_(0,4)*p_(3,4)+1498238*p_(1,4)*p_(3,4)-4426727*p_(2,4)*p_(3,4)+2785824*p_(3,4)^2,p_(1,2)*p_(0,3)-3187904*p_(0,3)*p_(2,3)-1142737*p_(1,3)*p_(2,3)-4949420*p_(2,3)^2+3103811*p_(0,2)*p_(1,4)-4614982*p_(1,2)*p_(1,4)+2757127*p_(0,3)*p_(1,4)-505391*p_(1,3)*p_(1,4)+3961590*p_(0,4)*p_(1,4)-2066461*p_(1,4)^2+2572898*p_(0,2)*p_(2,4)-3248057*p_(1,2)*p_(2,4)+3275700*p_(0,3)*p_(2,4)-1901998*p_(1,3)*p_(2,4)-262088*p_(2,3)*p_(2,4)-124265*p_(0,4)*p_(2,4)-1530583*p_(1,4)*p_(2,4)-278422*p_(2,4)^2-309039*p_(0,2)*p_(3,4)-2731692*p_(1,2)*p_(3,4)+2988138*p_(0,3)*p_(3,4)+3981317*p_(1,3)*p_(3,4)-2030838*p_(2,3)*p_(3,4)+1264907*p_(0,4)*p_(3,4)+393986*p_(1,4)*p_(3,4)+208512*p_(2,4)*p_(3,4)+4813233*p_(3,4)^2,p_(0,2)*p_(0,3)-2576487*p_(0,3)*p_(2,3)-3894809*p_(1,3)*p_(2,3)+541987*p_(2,3)^2-2507531*p_(0,2)*p_(1,4)+1947211*p_(1,2)*p_(1,4)-1546402*p_(0,3)*p_(1,4)+3544069*p_(1,3)*p_(1,4)+4528365*p_(0,4)*p_(1,4)-3798237*p_(1,4)^2-1627549*p_(0,2)*p_(2,4)+551707*p_(1,2)*p_(2,4)+3167295*p_(0,3)*p_(2,4)+514752*p_(1,3)*p_(2,4)+4280533*p_(2,3)*p_(2,4)-3055419*p_(0,4)*p_(2,4)-4582797*p_(1,4)*p_(2,4)+2871699*p_(2,4)^2-216831*p_(0,2)*p_(3,4)-471881*p_(1,2)*p_(3,4)+2738510*p_(0,3)*p_(3,4)-4818192*p_(1,3)*p_(3,4)+2093679*p_(2,3)*p_(3,4)-1995368*p_(0,4)*p_(3,4)-2464915*p_(1,4)*p_(3,4)-3655171*p_(2,4)*p_(3,4)-2971937*p_(3,4)^2,p_(1,2)^2-2696653*p_(0,3)*p_(2,3)-2766654*p_(1,3)*p_(2,3)+1337901*p_(2,3)^2+3603593*p_(0,2)*p_(1,4)-3252178*p_(1,2)*p_(1,4)+2412*p_(0,3)*p_(1,4)-299405*p_(1,3)*p_(1,4)-2699701*p_(0,4)*p_(1,4)+292089*p_(1,4)^2-1470264*p_(0,2)*p_(2,4)+3664595*p_(1,2)*p_(2,4)+4821817*p_(0,3)*p_(2,4)-3732321*p_(1,3)*p_(2,4)+522866*p_(2,3)*p_(2,4)+1447372*p_(0,4)*p_(2,4)-384498*p_(1,4)*p_(2,4)+2592467*p_(2,4)^2+1237395*p_(0,2)*p_(3,4)-3307469*p_(1,2)*p_(3,4)+3061676*p_(0,3)*p_(3,4)-439735*p_(1,3)*p_(3,4)+4417071*p_(2,3)*p_(3,4)+4017887*p_(0,4)*p_(3,4)+4014701*p_(1,4)*p_(3,4)-3127404*p_(2,4)*p_(3,4)+4801292*p_(3,4)^2,p_(0,2)*p_(1,2)-3772350*p_(0,3)*p_(2,3)+3507649*p_(1,3)*p_(2,3)+310312*p_(2,3)^2-4144050*p_(0,2)*p_(1,4)+1742799*p_(1,2)*p_(1,4)-1683808*p_(0,3)*p_(1,4)-680358*p_(1,3)*p_(1,4)+514522*p_(0,4)*p_(1,4)+3984458*p_(1,4)^2+3106555*p_(0,2)*p_(2,4)-2396497*p_(1,2)*p_(2,4)+2121775*p_(0,3)*p_(2,4)+4974196*p_(1,3)*p_(2,4)-2284139*p_(2,3)*p_(2,4)+4648991*p_(0,4)*p_(2,4)-3673825*p_(1,4)*p_(2,4)-873084*p_(2,4)^2-955197*p_(0,2)*p_(3,4)+933500*p_(1,2)*p_(3,4)-185043*p_(0,3)*p_(3,4)+4984600*p_(1,3)*p_(3,4)-1922316*p_(2,3)*p_(3,4)+4320507*p_(0,4)*p_(3,4)-2430410*p_(1,4)*p_(3,4)+3211102*p_(2,4)*p_(3,4)+4158377*p_(3,4)^2,p_(0,2)^2+629509*p_(0,3)*p_(2,3)+259050*p_(1,3)*p_(2,3)+3261792*p_(2,3)^2-834064*p_(0,2)*p_(1,4)-4113557*p_(1,2)*p_(1,4)+2716321*p_(0,3)*p_(1,4)+279515*p_(1,3)*p_(1,4)+4885172*p_(0,4)*p_(1,4)-4798770*p_(1,4)^2-4002189*p_(0,2)*p_(2,4)+2388866*p_(1,2)*p_(2,4)+724119*p_(0,3)*p_(2,4)+4509748*p_(1,3)*p_(2,4)+2884613*p_(2,3)*p_(2,4)+3143881*p_(0,4)*p_(2,4)-3483466*p_(1,4)*p_(2,4)+346665*p_(2,4)^2-962905*p_(0,2)*p_(3,4)-105798*p_(1,2)*p_(3,4)-2871127*p_(0,3)*p_(3,4)-191145*p_(1,3)*p_(3,4)+3557240*p_(2,3)*p_(3,4)-3703125*p_(0,4)*p_(3,4)+4597513*p_(1,4)*p_(3,4)-274959*p_(2,4)*p_(3,4)-2242218*p_(3,4)^2); assert(isSubset(G,S) and dim S == 3 and degree S == 9 and (genera S)_1 == 2); -- A parameterization of S (a,b,c) := (local a,local b,local c); ringP2 := (coefficientRing ring G)[a,b,c]; f = map(ringP2,ring G,{3084721*a^4-1274471*a^3*b+1850478*a^2*b^2-644227*a*b^3+853050*b^4+4342427*a^3*c+202876*a^2*b*c+727363*a*b^2*c-1844616*b^3*c-3636782*a^2*c^2-1937207*a*b*c^2-1044302*b^2*c^2+3519941*a*c^3+1157920*b*c^3-1164441*c^4, -589454*a^4-3676020*a^3*b-3551879*a^2*b^2-4573037*a*b^3-136179*b^4+3998272*a^3*c+2624786*a^2*b*c-2084709*a*b^2*c+4018489*b^3*c-4867850*a^2*c^2-121748*a*b*c^2+3207294*b^2*c^2+3923517*a*c^3-981230*b*c^3+372263*c^4, -3131257*a^4+1906364*a^3*b+4570345*a^2*b^2-460104*a*b^3-4547147*b^4-2319233*a^3*c+264815*a^2*b*c+3046300*a*b^2*c+2899492*b^3*c+1775378*a^2*c^2+3806391*a*b*c^2-4559194*b^2*c^2+826616*a*c^3-2746753*b*c^3+588946*c^4, -3981821*a^4-2601439*a^3*b+4979009*a^2*b^2+1295163*a*b^3-3481479*b^4+2389775*a^3*c-489592*a^2*b*c+1217647*a*b^2*c+4057368*b^3*c-4264931*a^2*c^2-1860750*a*b*c^2-3352352*b^2*c^2+1972080*a*c^3-3172476*b*c^3-2074995*c^4, -1265320*a^4+4045041*a^3*b+672432*a^2*b^2+4113477*a*b^3-4426196*b^4-3991144*a^3*c-4763215*a^2*b*c-1747226*a*b^2*c+3743122*b^3*c+2128685*a^2*c^2-2655426*a*b*c^2-656611*b^2*c^2-3695597*a*c^3+328250*b*c^3+4894411*c^4, -3281157*a^4+632560*a^3*b+1902214*a^2*b^2+949139*a*b^3+3108730*b^4+15764*a^3*c+3496613*a^2*b*c+2758850*a*b^2*c-4322126*b^3*c+3346151*a^2*c^2-810308*a*b*c^2+382870*b^2*c^2+2853137*a*c^3-3703899*b*c^3-2689100*c^4, 3171041*a^4+2378472*a^3*b+449548*a^2*b^2-981364*a*b^3+4342173*b^4+1531240*a^3*c+3901902*a^2*b*c-1846558*a*b^2*c-3791466*b^3*c+3282812*a^2*c^2-2531412*a*b*c^2-4165545*b^2*c^2-3885927*a*c^3+2352289*b*c^3+4735110*c^4, -2654679*a^4-4655005*a^3*b+2255039*a^2*b^2+1439304*a*b^3+515591*b^4-2688956*a^3*c+480133*a^2*b*c-418067*a*b^2*c-2111825*b^3*c-2184609*a^2*c^2-2422998*a*b*c^2+1936337*b^2*c^2-1851735*a*c^3+4305481*b*c^3-379504*c^4, -2217766*a^4+3757541*a^3*b+1338719*a^2*b^2-3711065*a*b^3-284780*b^4-1039675*a^3*c-1827043*a^2*b*c-160230*a*b^2*c-3058165*b^3*c-1256752*a^2*c^2+1908609*a*b*c^2+3229738*b^2*c^2-4846975*a*c^3+4572641*b*c^3+595114*c^4, -978745*a^4+1544872*a^3*b+317063*a^2*b^2+4206456*a*b^3+1861848*b^4-3433714*a^3*c-1471211*a^2*b*c+4709432*a*b^2*c+1782765*b^3*c-4987932*a^2*c^2-937077*a*b*c^2+1195176*b^2*c^2-1884026*a*c^3+1464503*b*c^3+2814397*c^4}); assert(S == kernel f);