next | previous | forward | backward | up | top | index | toc | Macaulay2 website
MultiprojectiveVarieties :: EmbeddedProjectiveVariety ===> EmbeddedProjectiveVariety

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

Synopsis

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 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
      }

See also