EmbeddedProjectiveVariety ===> EmbeddedProjectiveVariety -- try to find an isomorphism between two projective varieties

Synopsis

Operator: ===>
Usage:

X ===> Y

Y <=== X
Inputs:
- X, an embedded projective variety
- Y, an embedded projective variety, projectively equivalent to X
Outputs:
- a multi-rational map, an isomorphism of the ambient spaces that sends X to Y (or an error if it fails)

Description

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 2.09717s (cpu); 1.34436s (thread); 0s (gc)

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 2.27083s (cpu); 1.47159s (thread); 0s (gc)

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[x ..x ])  <-- (K[x ..x ])
                 0   9            0   9

o14 = | 0                                                                    
      | 4699078x_0-1746876x_1-380897x_2+1422530x_3+4811841x_4-3896862x_5-3722
      | 27066x_0+3423234x_1+230399x_2+2782180x_3+309050x_4-1114049x_5+2286765
      | -3001071x_0+3626292x_1+3927398x_2-4508287x_3-1613351x_4-1776043x_5+12
      | -1128116x_0+2410838x_1+4011204x_2-1473177x_3+2441342x_4-4718496x_5+42
      -----------------------------------------------------------------------
                                               
      785x_6-3815668x_7-3313914x_8-4758195x_9  
      x_6-2212565x_7-74241x_8-860866x_9        
      19497x_6-2150772x_7+2179139x_8-692400x_9 
      95767x_6+1349994x_7-3469596x_8-4256627x_9
      -----------------------------------------------------------------------
      -4699078x_0+1746876x_1+380897x_2-1422530x_3-4811841x_4+3896862x_
      0                                                               
      -4775990x_0-1733951x_1+2685339x_2-101690x_3-1299785x_4-3383627x_
      -301724x_0+3056350x_1-4261084x_2+1869785x_3-1725095x_4+4002080x_
      2255774x_0+152589x_1-2805551x_2-411254x_3+2029789x_4-2013016x_5-
      -----------------------------------------------------------------------
      5+3722785x_6+3815668x_7+3313914x_8+4758195x_9
                                                   
      5+1688069x_6+4817905x_7-2628713x_8-4634439x_9
      5+3630364x_6+522185x_7+3993769x_8+117133x_9  
      421034x_6+4901792x_7-4988209x_8+494257x_9    
      -----------------------------------------------------------------------
      -27066x_0-3423234x_1-230399x_2-2782180x_3-309050x_4+1114049x_5-
      4775990x_0+1733951x_1-2685339x_2+101690x_3+1299785x_4+3383627x_
      0                                                              
      4824428x_0-260873x_1-3590724x_2-438108x_3+4564938x_4-837765x_5-
      1232609x_0+3795926x_1+3890630x_2-3831039x_3-1495247x_4-1456108x
      -----------------------------------------------------------------------
      2286765x_6+2212565x_7+74241x_8+860866x_9     
      5-1688069x_6-4817905x_7+2628713x_8+4634439x_9
                                                   
      4403511x_6+1089387x_7+1483485x_8-4660338x_9  
      _5+976817x_6-2292406x_7+4444574x_8-4380563x_9
      -----------------------------------------------------------------------
      3001071x_0-3626292x_1-3927398x_2+4508287x_3+1613351x_4+1776043x_
      301724x_0-3056350x_1+4261084x_2-1869785x_3+1725095x_4-4002080x_5
      -4824428x_0+260873x_1+3590724x_2+438108x_3-4564938x_4+837765x_5+
      0                                                               
      1260139x_0-2367455x_1+2074452x_2-1540641x_3-2096244x_4-604376x_5
      -----------------------------------------------------------------------
      5-1219497x_6+2150772x_7-2179139x_8+692400x_9
      -3630364x_6-522185x_7-3993769x_8-117133x_9  
      4403511x_6-1089387x_7-1483485x_8+4660338x_9 
                                                  
      -115065x_6-2900230x_7-1708776x_8+3939426x_9 
      -----------------------------------------------------------------------
      1128116x_0-2410838x_1-4011204x_2+1473177x_3-2441342x_4+4718496x_5
      -2255774x_0-152589x_1+2805551x_2+411254x_3-2029789x_4+2013016x_5+
      -1232609x_0-3795926x_1-3890630x_2+3831039x_3+1495247x_4+1456108x_
      -1260139x_0+2367455x_1-2074452x_2+1540641x_3+2096244x_4+604376x_5
      0                                                                
      -----------------------------------------------------------------------
      -4295767x_6-1349994x_7+3469596x_8+4256627x_9 |
      421034x_6-4901792x_7+4988209x_8-494257x_9    |
      5-976817x_6+2292406x_7-4444574x_8+4380563x_9 |
      +115065x_6+2900230x_7+1708776x_8-3939426x_9  |
                                                   |

                        5                5
o14 : Matrix (K[x ..x ])  <-- (K[x ..x ])
                 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 5.11979s (cpu); 3.40802s (thread); 0s (gc)

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[x , x , x , x , x , x , x , x , x , x ]) defined by
                                    0   1   2   3   4   5   6   7   8   9
              {
                              2                                                     2                                                                   2                                                                                2                                                                                              2                                                                                                            2                                                                                                                          2                                                                                                                                       2
               x x  + 1177025x  - 1095541x x  - 2449657x x  - 3243503x x  - 1061060x  - 2783252x x  - 3274449x x  + 3386662x x  + 4880814x x  + 3733032x  + 1640949x x  - 329156x x  - 2083947x x  - 2312431x x  - 1936834x x  + 1149240x  - 4060388x x  - 3611959x x  + 3508392x x  - 537860x x  - 4281865x x  - 4451400x x  + 2528307x  - 4230765x x  - 2408221x x  - 1286535x x  - 4336603x x  - 4220128x x  + 749283x x  + 3981741x x  + 3038844x  + 4740906x x  + 4843505x x  - 3886471x x  + 696812x x  - 4988346x x  + 2077593x x  - 2859698x x  - 1518929x x  - 3232058x  + 3121313x x  - 4756567x x  - 3127167x x  - 1627093x x  + 2611828x x  + 4968424x x  - 64090x x  + 2180367x x  + 2238599x x  + 2439061x ,
                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
               x x  + 4570442x  - 95455x x  + 4540866x x  - 3666920x x  - 2236818x  - 2988265x x  + 2811128x x  + 1843189x x  + 2468602x x  + 90348x  - 899258x x  - 4198922x x  + 2215739x x  - 1414418x x  - 4001772x x  + 1783295x  - 1769084x x  + 4558141x x  - 1508432x x  - 81167x x  - 625040x x  + 2937038x x  + 4090836x  + 3729984x x  + 4292910x x  - 1067772x x  - 2704646x x  - 4942689x x  - 3684702x x  + 2267021x x  + 1087738x  - 2739302x x  + 1562924x x  - 2847121x x  - 4121261x x  + 2174897x x  + 90757x x  - 2407863x x  - 2324974x x  - 174089x  + 4436267x x  + 487180x x  - 4151396x x  - 1383729x x  + 773635x x  + 3984741x x  + 3303811x x  - 4267053x x  - 4832450x x  - 4362709x ,
                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
               x  - 4983412x  + 4394308x x  + 2815203x x  - 331579x x  + 2764751x  - 2805962x x  + 668466x x  - 2554152x x  + 4279022x x  - 2965749x  - 1508144x x  - 1365547x x  + 4996438x x  - 3145140x x  + 1783510x x  + 2519033x  + 3099034x x  + 4073779x x  - 385562x x  + 2573406x x  + 3053509x x  - 1162366x x  + 4302327x  + 117773x x  + 1743374x x  + 2462036x x  - 4097671x x  - 1750639x x  - 418086x x  - 369814x x  - 1611145x  - 1066092x x  + 1462981x x  - 897233x x  - 4918316x x  - 442190x x  + 1847233x x  + 3694786x x  + 1873500x x  + 747368x  + 3942073x x  + 1611533x x  - 1063893x x  + 3564330x x  - 1700115x x  - 410720x x  + 2237535x x  + 316734x x  - 4299028x x  - 1057166x ,
                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
               x x  + 4945366x  + 122400x x  + 4613283x x  + 3607605x x  - 3797186x  + 880351x x  - 4741277x x  - 1205877x x  + 3486734x x  + 2898287x  + 3006671x x  + 3243145x x  - 1623964x x  + 4521923x x  + 4756659x x  - 4484352x  - 3236288x x  - 695220x x  + 970619x x  - 3098637x x  - 2898498x x  - 3885949x x  + 1379926x  + 452843x x  - 602246x x  - 2706941x x  - 290203x x  - 1724577x x  + 123395x x  + 583922x x  + 115646x  + 2172520x x  + 1125647x x  - 2817246x x  + 4180857x x  - 3227735x x  - 2538539x x  - 1811940x x  - 3865917x x  - 3160054x  - 738144x x  + 2508981x x  + 2624313x x  + 2088816x x  + 2135052x x  + 140567x x  - 3482168x x  + 3607342x x  - 205597x x  - 3307x ,
                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
               x  - 2084490x  + 3965062x x  + 3920717x x  - 748684x x  - 3601351x  - 4434745x x  + 2429080x x  + 4533987x x  - 3525867x x  + 316102x  - 989703x x  + 2718669x x  - 2734172x x  - 926135x x  + 1668848x x  - 3637865x  + 2187085x x  - 1706603x x  + 1210817x x  + 993302x x  - 3101990x x  - 1214647x x  + 4207879x  + 1139881x x  + 1251151x x  + 1168855x x  - 912010x x  - 3811666x x  - 3437312x x  + 1947189x x  + 4881584x  - 4659830x x  + 3854863x x  + 701591x x  + 3308028x x  + 2182166x x  + 3628590x x  + 1667603x x  - 1656132x x  - 4157810x  - 2253747x x  - 1789013x x  + 3891052x x  - 4072307x x  - 3350837x x  + 4382093x x  + 348517x x  - 3484961x x  + 1064900x x  - 2493096x
                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:
      {
       - 4955130x  - 2830792x  + 1376806x  + 4129359x  + 4511300x  - 4710574x  + 2905466x  - 4890882x  - 215730x  + 3759197x ,
                 0           1           2           3           4           5           6           7          8           9
       
       - 2350531x  + 647071x  - 2010797x  - 1679088x  - 1605354x  + 245004x  - 1952375x  + 4032072x  + 4274163x  - 3460346x ,
                 0          1           2           3           4          5           6           7           8           9
       
       - 2105465x  + 2212889x  + 2560271x  + 546146x  + 1007583x  - 3570671x  + 2251582x  - 1355141x  - 188875x  + 87464x ,
                 0           1           2          3           4           5           6           7          8         9
       
       613134x  + 3423965x  - 2992038x  - 1116142x  - 1947368x  - 1631649x  + 1335061x  + 4143364x  + 259754x  + 3440927x ,
              0           1           2           3           4           5           6           7          8           9
       
       - 853563x  + 2955972x  + 1107025x  + 4901955x  - 4405095x  + 768881x  - 121812x  + 1770730x  + 558569x  + 3067493x ,
                0           1           2           3           4          5          6           7          8           9
       
       - 1429409x  + 4258621x  - 134232x  - 2023131x  - 4008192x  + 2021415x  + 257400x  + 3674927x  + 4256589x  + 784860x ,
                 0           1          2           3           4           5          6           7           8          9
       
       668204x  + 314424x  - 3430507x  - 2810996x  + 3883757x  + 1562229x  - 3244720x  + 1895569x  + 3350914x  + 1627251x ,
              0          1           2           3           4           5           6           7           8           9
       
       409606x  - 1564530x  + 4411376x  - 3966210x  + 2237618x  + 474772x  - 2750700x  + 1022834x  - 98736x  + 2004957x ,
              0           1           2           3           4          5           6           7         8           9
       
       - 4005338x  + 4473865x  + 514745x  + 2120182x  + 2345402x  - 417906x  + 2386551x  - 754907x  - 2731424x  + 4928807x ,
                 0           1          2           3           4          5           6          7           8           9
       
       - 4867678x  + 3796787x  - 1874516x  + 4528024x  - 1592517x  + 4922242x  - 408054x  - 4897063x  + 4835457x  - 3641859x
                 0           1           2           3           4           5          6           7           8           9
      }

Ways to use this method: