This recursively tries to find an isomorphism between the base loci of the parameterizations.
In the following example, $X$ and $Y$ are two random rational normal curves of degree 6 in $\mathbb{P}^6\subset\mathbb{P}^8$, and $V$ (resp., $W$) is a random complete intersection of type (2,1) containing $X$ (resp., $Y$).
i1 : K = ZZ/10000019; |
i2 : (M,N) = (apply(9,i -> random(1,ring PP_K^8)), apply(9,i -> random(1,ring PP_K^8))); |
i3 : X = projectiveVariety(minors(2,matrix{take(M,6),take(M,{1,6})}) + ideal take(M,-2));
o3 : ProjectiveVariety, curve in PP^8
|
i4 : Y = projectiveVariety(minors(2,matrix{take(N,6),take(N,{1,6})}) + ideal take(N,-2));
o4 : ProjectiveVariety, curve in PP^8
|
i5 : ? X o5 = curve in PP^8 cut out by 17 hypersurfaces of degrees 1^2 2^15 |
i6 : time f = X ===> Y;
-- used 4.61149 seconds
o6 : MultirationalMap (automorphism of PP^8)
|
i7 : f X o7 = Y o7 : ProjectiveVariety, curve in PP^8 |
i8 : f^* Y o8 = X o8 : ProjectiveVariety, curve in PP^8 |
i9 : V = random({{2},{1}},X);
o9 : ProjectiveVariety, 6-dimensional subvariety of PP^8
|
i10 : W = random({{2},{1}},Y);
o10 : ProjectiveVariety, 6-dimensional subvariety of PP^8
|
i11 : time g = V ===> W;
-- used 4.86564 seconds
o11 : MultirationalMap (automorphism of PP^8)
|
i12 : g||W
o12 = multi-rational map consisting of one single rational map
source variety: 6-dimensional subvariety of PP^8 cut out by 2 hypersurfaces of degrees 1^1 2^1
target variety: 6-dimensional subvariety of PP^8 cut out by 2 hypersurfaces of degrees 1^1 2^1
o12 : MultirationalMap (rational map from V to W)
|
In the next example, $Z\subset\mathbb{P}^9$ is a random (smooth) del Pezzo sixfold, hence projectively equivalent to $\mathbb{G}(1,4)$.
i13 : A = matrix pack(5,for i to 24 list random(1,ring PP_K^9)); A = A - transpose A
5 5
o13 : Matrix (K[x0 ..x0 ]) <--- (K[x0 ..x0 ])
0 9 0 9
o14 = | 0
| 4699078x0_0-1746876x0_
| 27066x0_0+3423234x0_1+
| -3001071x0_0+3626292x0
| -1128116x0_0+2410838x0
-----------------------------------------------------------------------
1-380897x0_2+1422530x0_3+4811841x0_4-3896862x0_5-3722785x0_6-3815668x0_
230399x0_2+2782180x0_3+309050x0_4-1114049x0_5+2286765x0_6-2212565x0_7-7
_1+3927398x0_2-4508287x0_3-1613351x0_4-1776043x0_5+1219497x0_6-2150772x
_1+4011204x0_2-1473177x0_3+2441342x0_4-4718496x0_5+4295767x0_6+1349994x
-----------------------------------------------------------------------
7-3313914x0_8-4758195x0_9
4241x0_8-860866x0_9
0_7+2179139x0_8-692400x0_9
0_7-3469596x0_8-4256627x0_9
-----------------------------------------------------------------------
-4699078x0_0+1746876x0_1+380897x0_2-1422530x0_3-4811841x0_4+3896862x0_
0
-4775990x0_0-1733951x0_1+2685339x0_2-101690x0_3-1299785x0_4-3383627x0_
-301724x0_0+3056350x0_1-4261084x0_2+1869785x0_3-1725095x0_4+4002080x0_
2255774x0_0+152589x0_1-2805551x0_2-411254x0_3+2029789x0_4-2013016x0_5-
-----------------------------------------------------------------------
5+3722785x0_6+3815668x0_7+3313914x0_8+4758195x0_9
5+1688069x0_6+4817905x0_7-2628713x0_8-4634439x0_9
5+3630364x0_6+522185x0_7+3993769x0_8+117133x0_9
421034x0_6+4901792x0_7-4988209x0_8+494257x0_9
-----------------------------------------------------------------------
-27066x0_0-3423234x0_1-230399x0_2-2782180x0_3-309050x0_4+1114049x0_5-
4775990x0_0+1733951x0_1-2685339x0_2+101690x0_3+1299785x0_4+3383627x0_
0
4824428x0_0-260873x0_1-3590724x0_2-438108x0_3+4564938x0_4-837765x0_5-
1232609x0_0+3795926x0_1+3890630x0_2-3831039x0_3-1495247x0_4-1456108x0
-----------------------------------------------------------------------
2286765x0_6+2212565x0_7+74241x0_8+860866x0_9
5-1688069x0_6-4817905x0_7+2628713x0_8+4634439x0_9
4403511x0_6+1089387x0_7+1483485x0_8-4660338x0_9
_5+976817x0_6-2292406x0_7+4444574x0_8-4380563x0_9
-----------------------------------------------------------------------
3001071x0_0-3626292x0_1-3927398x0_2+4508287x0_3+1613351x0_4+1776043x0_
301724x0_0-3056350x0_1+4261084x0_2-1869785x0_3+1725095x0_4-4002080x0_5
-4824428x0_0+260873x0_1+3590724x0_2+438108x0_3-4564938x0_4+837765x0_5+
0
1260139x0_0-2367455x0_1+2074452x0_2-1540641x0_3-2096244x0_4-604376x0_5
-----------------------------------------------------------------------
5-1219497x0_6+2150772x0_7-2179139x0_8+692400x0_9
-3630364x0_6-522185x0_7-3993769x0_8-117133x0_9
4403511x0_6-1089387x0_7-1483485x0_8+4660338x0_9
-115065x0_6-2900230x0_7-1708776x0_8+3939426x0_9
-----------------------------------------------------------------------
1128116x0_0-2410838x0_1-4011204x0_2+1473177x0_3-2441342x0_4+4718496x0_5
-2255774x0_0-152589x0_1+2805551x0_2+411254x0_3-2029789x0_4+2013016x0_5+
-1232609x0_0-3795926x0_1-3890630x0_2+3831039x0_3+1495247x0_4+1456108x0_
-1260139x0_0+2367455x0_1-2074452x0_2+1540641x0_3+2096244x0_4+604376x0_5
0
-----------------------------------------------------------------------
-4295767x0_6-1349994x0_7+3469596x0_8+4256627x0_9 |
421034x0_6-4901792x0_7+4988209x0_8-494257x0_9 |
5-976817x0_6+2292406x0_7-4444574x0_8+4380563x0_9 |
+115065x0_6+2900230x0_7+1708776x0_8-3939426x0_9 |
|
5 5
o14 : Matrix (K[x0 ..x0 ]) <--- (K[x0 ..x0 ])
0 9 0 9
|
i15 : Z = projectiveVariety pfaffians(4,A); o15 : ProjectiveVariety, 6-dimensional subvariety of PP^9 |
i16 : ? Z o16 = 6-dimensional subvariety of PP^9 cut out by 5 hypersurfaces of degree 2 |
i17 : time h = Z ===> GG_K(1,4)
-- used 10.7862 seconds
o17 = h
o17 : MultirationalMap (isomorphism from PP^9 to PP^9)
|
i18 : h || GG_K(1,4)
o18 = multi-rational map consisting of one single rational map
source variety: 6-dimensional subvariety of PP^9 cut out by 5 hypersurfaces of degree 2
target variety: GG(1,4) ⊂ PP^9
o18 : MultirationalMap (rational map from Z to GG(1,4))
|
i19 : show oo
o19 = -- multi-rational map --
source: subvariety of Proj(K[x0 , x0 , x0 , x0 , x0 , x0 , x0 , x0 , x0 , x0 ]) defined by
0 1 2 3 4 5 6 7 8 9
{
2 2 2 2 2 2 2 2
x0 x0 + 1177025x0 - 1095541x0 x0 - 2449657x0 x0 - 3243503x0 x0 - 1061060x0 - 2783252x0 x0 - 3274449x0 x0 + 3386662x0 x0 + 4880814x0 x0 + 3733032x0 + 1640949x0 x0 - 329156x0 x0 - 2083947x0 x0 - 2312431x0 x0 - 1936834x0 x0 + 1149240x0 - 4060388x0 x0 - 3611959x0 x0 + 3508392x0 x0 - 537860x0 x0 - 4281865x0 x0 - 4451400x0 x0 + 2528307x0 - 4230765x0 x0 - 2408221x0 x0 - 1286535x0 x0 - 4336603x0 x0 - 4220128x0 x0 + 749283x0 x0 + 3981741x0 x0 + 3038844x0 + 4740906x0 x0 + 4843505x0 x0 - 3886471x0 x0 + 696812x0 x0 - 4988346x0 x0 + 2077593x0 x0 - 2859698x0 x0 - 1518929x0 x0 - 3232058x0 + 3121313x0 x0 - 4756567x0 x0 - 3127167x0 x0 - 1627093x0 x0 + 2611828x0 x0 + 4968424x0 x0 - 64090x0 x0 + 2180367x0 x0 + 2238599x0 x0 + 2439061x0 ,
1 2 2 0 3 1 3 2 3 3 0 4 1 4 2 4 3 4 4 0 5 1 5 2 5 3 5 4 5 5 0 6 1 6 2 6 3 6 4 6 5 6 6 0 7 1 7 2 7 3 7 4 7 5 7 6 7 7 0 8 1 8 2 8 3 8 4 8 5 8 6 8 7 8 8 0 9 1 9 2 9 3 9 4 9 5 9 6 9 7 9 8 9 9
2 2 2 2 2 2 2 2
x0 x0 + 4570442x0 - 95455x0 x0 + 4540866x0 x0 - 3666920x0 x0 - 2236818x0 - 2988265x0 x0 + 2811128x0 x0 + 1843189x0 x0 + 2468602x0 x0 + 90348x0 - 899258x0 x0 - 4198922x0 x0 + 2215739x0 x0 - 1414418x0 x0 - 4001772x0 x0 + 1783295x0 - 1769084x0 x0 + 4558141x0 x0 - 1508432x0 x0 - 81167x0 x0 - 625040x0 x0 + 2937038x0 x0 + 4090836x0 + 3729984x0 x0 + 4292910x0 x0 - 1067772x0 x0 - 2704646x0 x0 - 4942689x0 x0 - 3684702x0 x0 + 2267021x0 x0 + 1087738x0 - 2739302x0 x0 + 1562924x0 x0 - 2847121x0 x0 - 4121261x0 x0 + 2174897x0 x0 + 90757x0 x0 - 2407863x0 x0 - 2324974x0 x0 - 174089x0 + 4436267x0 x0 + 487180x0 x0 - 4151396x0 x0 - 1383729x0 x0 + 773635x0 x0 + 3984741x0 x0 + 3303811x0 x0 - 4267053x0 x0 - 4832450x0 x0 - 4362709x0 ,
0 2 2 0 3 1 3 2 3 3 0 4 1 4 2 4 3 4 4 0 5 1 5 2 5 3 5 4 5 5 0 6 1 6 2 6 3 6 4 6 5 6 6 0 7 1 7 2 7 3 7 4 7 5 7 6 7 7 0 8 1 8 2 8 3 8 4 8 5 8 6 8 7 8 8 0 9 1 9 2 9 3 9 4 9 5 9 6 9 7 9 8 9 9
2 2 2 2 2 2 2 2 2
x0 - 4983412x0 + 4394308x0 x0 + 2815203x0 x0 - 331579x0 x0 + 2764751x0 - 2805962x0 x0 + 668466x0 x0 - 2554152x0 x0 + 4279022x0 x0 - 2965749x0 - 1508144x0 x0 - 1365547x0 x0 + 4996438x0 x0 - 3145140x0 x0 + 1783510x0 x0 + 2519033x0 + 3099034x0 x0 + 4073779x0 x0 - 385562x0 x0 + 2573406x0 x0 + 3053509x0 x0 - 1162366x0 x0 + 4302327x0 + 117773x0 x0 + 1743374x0 x0 + 2462036x0 x0 - 4097671x0 x0 - 1750639x0 x0 - 418086x0 x0 - 369814x0 x0 - 1611145x0 - 1066092x0 x0 + 1462981x0 x0 - 897233x0 x0 - 4918316x0 x0 - 442190x0 x0 + 1847233x0 x0 + 3694786x0 x0 + 1873500x0 x0 + 747368x0 + 3942073x0 x0 + 1611533x0 x0 - 1063893x0 x0 + 3564330x0 x0 - 1700115x0 x0 - 410720x0 x0 + 2237535x0 x0 + 316734x0 x0 - 4299028x0 x0 - 1057166x0 ,
1 2 0 3 1 3 2 3 3 0 4 1 4 2 4 3 4 4 0 5 1 5 2 5 3 5 4 5 5 0 6 1 6 2 6 3 6 4 6 5 6 6 0 7 1 7 2 7 3 7 4 7 5 7 6 7 7 0 8 1 8 2 8 3 8 4 8 5 8 6 8 7 8 8 0 9 1 9 2 9 3 9 4 9 5 9 6 9 7 9 8 9 9
2 2 2 2 2 2 2 2
x0 x0 + 4945366x0 + 122400x0 x0 + 4613283x0 x0 + 3607605x0 x0 - 3797186x0 + 880351x0 x0 - 4741277x0 x0 - 1205877x0 x0 + 3486734x0 x0 + 2898287x0 + 3006671x0 x0 + 3243145x0 x0 - 1623964x0 x0 + 4521923x0 x0 + 4756659x0 x0 - 4484352x0 - 3236288x0 x0 - 695220x0 x0 + 970619x0 x0 - 3098637x0 x0 - 2898498x0 x0 - 3885949x0 x0 + 1379926x0 + 452843x0 x0 - 602246x0 x0 - 2706941x0 x0 - 290203x0 x0 - 1724577x0 x0 + 123395x0 x0 + 583922x0 x0 + 115646x0 + 2172520x0 x0 + 1125647x0 x0 - 2817246x0 x0 + 4180857x0 x0 - 3227735x0 x0 - 2538539x0 x0 - 1811940x0 x0 - 3865917x0 x0 - 3160054x0 - 738144x0 x0 + 2508981x0 x0 + 2624313x0 x0 + 2088816x0 x0 + 2135052x0 x0 + 140567x0 x0 - 3482168x0 x0 + 3607342x0 x0 - 205597x0 x0 - 3307x0 ,
0 1 2 0 3 1 3 2 3 3 0 4 1 4 2 4 3 4 4 0 5 1 5 2 5 3 5 4 5 5 0 6 1 6 2 6 3 6 4 6 5 6 6 0 7 1 7 2 7 3 7 4 7 5 7 6 7 7 0 8 1 8 2 8 3 8 4 8 5 8 6 8 7 8 8 0 9 1 9 2 9 3 9 4 9 5 9 6 9 7 9 8 9 9
2 2 2 2 2 2 2 2 2
x0 - 2084490x0 + 3965062x0 x0 + 3920717x0 x0 - 748684x0 x0 - 3601351x0 - 4434745x0 x0 + 2429080x0 x0 + 4533987x0 x0 - 3525867x0 x0 + 316102x0 - 989703x0 x0 + 2718669x0 x0 - 2734172x0 x0 - 926135x0 x0 + 1668848x0 x0 - 3637865x0 + 2187085x0 x0 - 1706603x0 x0 + 1210817x0 x0 + 993302x0 x0 - 3101990x0 x0 - 1214647x0 x0 + 4207879x0 + 1139881x0 x0 + 1251151x0 x0 + 1168855x0 x0 - 912010x0 x0 - 3811666x0 x0 - 3437312x0 x0 + 1947189x0 x0 + 4881584x0 - 4659830x0 x0 + 3854863x0 x0 + 701591x0 x0 + 3308028x0 x0 + 2182166x0 x0 + 3628590x0 x0 + 1667603x0 x0 - 1656132x0 x0 - 4157810x0 - 2253747x0 x0 - 1789013x0 x0 + 3891052x0 x0 - 4072307x0 x0 - 3350837x0 x0 + 4382093x0 x0 + 348517x0 x0 - 3484961x0 x0 + 1064900x0 x0 - 2493096x0
0 2 0 3 1 3 2 3 3 0 4 1 4 2 4 3 4 4 0 5 1 5 2 5 3 5 4 5 5 0 6 1 6 2 6 3 6 4 6 5 6 6 0 7 1 7 2 7 3 7 4 7 5 7 6 7 7 0 8 1 8 2 8 3 8 4 8 5 8 6 8 7 8 8 0 9 1 9 2 9 3 9 4 9 5 9 6 9 7 9 8 9 9
}
target: subvariety of Proj(K[x , x , x , x , x , x , x , x , x , x ]) defined by
0,1 0,2 1,2 0,3 1,3 2,3 0,4 1,4 2,4 3,4
{
x x - x x + x x ,
2,3 1,4 1,3 2,4 1,2 3,4
x x - x x + x x ,
2,3 0,4 0,3 2,4 0,2 3,4
x x - x x + x x ,
1,3 0,4 0,3 1,4 0,1 3,4
x x - x x + x x ,
1,2 0,4 0,2 1,4 0,1 2,4
x x - x x + x x
1,2 0,3 0,2 1,3 0,1 2,3
}
-- rational map 1/1 --
map 1/1, one of its representatives:
{
636063x0 + 1910149x0 + 3469156x0 + 1782944x0 + 1681964x0 - 2267546x0 + 4403821x0 - 4541128x0 - 4187947x0 - 2404965x0 ,
0 1 2 3 4 5 6 7 8 9
- 949439x0 + 3012986x0 + 3641878x0 - 3305612x0 + 495381x0 - 323026x0 - 2045984x0 + 3305654x0 - 3092516x0 - 3990680x0 ,
0 1 2 3 4 5 6 7 8 9
- 1503238x0 - 380567x0 - 2542707x0 - 3235971x0 + 1420604x0 + 2472548x0 + 4662742x0 - 3436356x0 - 1377206x0 - 1850827x0 ,
0 1 2 3 4 5 6 7 8 9
3161208x0 - 295908x0 + 4659338x0 - 3299384x0 - 1366852x0 + 4490584x0 - 3672140x0 - 397925x0 - 4109903x0 - 2645351x0 ,
0 1 2 3 4 5 6 7 8 9
- 4845468x0 + 298140x0 + 492319x0 + 1871395x0 + 3419462x0 - 2736262x0 - 1695206x0 - 2773655x0 + 4763600x0 + 1207008x0 ,
0 1 2 3 4 5 6 7 8 9
310790x0 - 2652464x0 - 1111166x0 + 656374x0 + 4083837x0 + 1294467x0 + 2345982x0 - 1487466x0 + 1010936x0 - 2926035x0 ,
0 1 2 3 4 5 6 7 8 9
- 4153718x0 - 3952331x0 - 519801x0 + 3475384x0 - 3951007x0 - 3183733x0 - 485341x0 - 76783x0 - 1134223x0 + 2116163x0 ,
0 1 2 3 4 5 6 7 8 9
2729508x0 - 4494994x0 + 230951x0 + 4160430x0 - 2268160x0 + 4275431x0 - 3493822x0 + 3220931x0 - 4891262x0 - 2371118x0 ,
0 1 2 3 4 5 6 7 8 9
- 786935x0 + 916099x0 - 929197x0 + 3275195x0 + 1353045x0 - 158323x0 - 3848537x0 - 4681844x0 - 4697057x0 + 1685092x0 ,
0 1 2 3 4 5 6 7 8 9
3200839x0 - 1324382x0 + 2124226x0 - 1706941x0 + 2264598x0 - 113322x0 + 2004009x0 + 2730665x0 - 1393874x0 - 802548x0
0 1 2 3 4 5 6 7 8 9
}
|