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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 2.6417 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 2.71749 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 : Z = projectiveVariety pfaffians(4,matrix pack(5,for i to 24 list random(1,ring PP^9)));

o13 : ProjectiveVariety, 6-dimensional subvariety of PP^9
i14 : ? Z

o14 = 6-dimensional subvariety of PP^9 cut out by 5 hypersurfaces of degree 2
i15 : G := projectiveVariety Grass(1,4,K)

o15 = GG(1,4) ⊂ PP^9

o15 : ProjectiveVariety, GG(1,4)
i16 : time h = Z ===> G
     -- used 5.38965 seconds

o16 = h

o16 : MultirationalMap (isomorphism from PP^9 to PP^9)
i17 : h||G

o17 = 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

o17 : MultirationalMap (rational map from Z to GG(1,4))
i18 : show oo

o18 = -- 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  - 1704465x0  + 3091309x0 x0  - 345843x0 x0  + 3033657x0 x0  + 2392839x0  + 4490358x0 x0  + 1990610x0 x0  - 3126908x0 x0  + 316416x0 x0  + 2849425x0  - 4286308x0 x0  - 4101708x0 x0  - 3961581x0 x0  - 1590650x0 x0  + 290760x0 x0  - 4424580x0  - 2695252x0 x0  + 399812x0 x0  - 3565813x0 x0  + 30234x0 x0  - 4560228x0 x0  + 1437063x0 x0  + 331402x0  - 85953x0 x0  - 1963777x0 x0  + 3845754x0 x0  + 3109335x0 x0  - 4737692x0 x0  + 2255895x0 x0  - 1314371x0 x0  + 3080756x0  + 4602897x0 x0  + 1765703x0 x0  + 3170176x0 x0  + 384366x0 x0  + 717488x0 x0  + 910345x0 x0  + 4841423x0 x0  - 1877932x0 x0  - 2204897x0  + 1724572x0 x0  - 4612505x0 x0  + 1161306x0 x0  + 4120997x0 x0  + 1518159x0 x0  - 1821133x0 x0  + 2552337x0 x0  + 1080518x0 x0  + 809049x0 x0  - 3881659x0 ,
                 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  + 3469833x0  + 1891088x0 x0  - 3034740x0 x0  - 3090216x0 x0  - 912199x0  + 4806644x0 x0  + 1077320x0 x0  - 3985426x0 x0  + 2473768x0 x0  + 813404x0  - 3365372x0 x0  + 2038834x0 x0  - 4212963x0 x0  - 1201602x0 x0  + 4073466x0 x0  - 1367136x0  + 3532393x0 x0  - 1432453x0 x0  - 2193242x0 x0  + 1766580x0 x0  - 2054119x0 x0  + 108836x0 x0  - 3161081x0  + 442509x0 x0  - 4711177x0 x0  - 1635182x0 x0  + 2597619x0 x0  + 3708832x0 x0  - 3245408x0 x0  + 3339670x0 x0  - 2261527x0  - 324753x0 x0  + 911676x0 x0  + 318418x0 x0  - 36840x0 x0  - 108970x0 x0  + 1132940x0 x0  + 3554022x0 x0  - 1535888x0 x0  - 4830398x0  + 3088469x0 x0  - 4906052x0 x0  - 3855544x0 x0  + 4633571x0 x0  + 3334193x0 x0  + 2900649x0 x0  + 2499964x0 x0  - 362675x0 x0  + 2341988x0 x0  - 3011543x0 ,
                 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  - 489614x0  - 3735863x0 x0  + 3709576x0 x0  - 2253326x0 x0  + 4633470x0  + 3464303x0 x0  - 2281714x0 x0  - 2968203x0 x0  - 509911x0 x0  - 523162x0  + 3434227x0 x0  - 4555125x0 x0  - 2752652x0 x0  + 2751055x0 x0  + 1930257x0 x0  + 4852251x0  - 84221x0 x0  - 2213642x0 x0  - 4220805x0 x0  + 3861276x0 x0  + 3920491x0 x0  - 2068378x0 x0  + 3745742x0  + 1331805x0 x0  + 1238070x0 x0  - 1290799x0 x0  - 3368120x0 x0  + 462570x0 x0  - 451917x0 x0  + 3434700x0 x0  - 4379051x0  - 407502x0 x0  - 339095x0 x0  + 4232832x0 x0  + 4011178x0 x0  + 1088725x0 x0  - 542937x0 x0  + 4864944x0 x0  + 1710513x0 x0  + 1160448x0  + 2286748x0 x0  + 237486x0 x0  + 2873763x0 x0  - 1674091x0 x0  - 947139x0 x0  + 1764302x0 x0  + 3773097x0 x0  + 2495007x0 x0  - 629519x0 x0  + 1590505x0 ,
                 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  - 3673168x0  - 4542731x0 x0  - 1764264x0 x0  - 190443x0 x0  - 2618756x0  - 3946344x0 x0  - 3788511x0 x0  - 3239779x0 x0  + 2567727x0 x0  - 3736026x0  + 1839628x0 x0  + 395659x0 x0  + 595204x0 x0  + 3695064x0 x0  - 3034356x0 x0  - 1601661x0  + 822622x0 x0  + 531426x0 x0  - 3275060x0 x0  + 1717070x0 x0  - 1049123x0 x0  - 2231183x0 x0  + 1764237x0  + 2626230x0 x0  + 3513565x0 x0  + 3676543x0 x0  - 549726x0 x0  + 3934492x0 x0  + 1643352x0 x0  + 248687x0 x0  + 2346227x0  - 239170x0 x0  + 1309256x0 x0  + 2229166x0 x0  - 3911968x0 x0  + 3443852x0 x0  - 1957572x0 x0  - 2258900x0 x0  - 2698176x0 x0  + 3484863x0  - 335580x0 x0  - 3405364x0 x0  - 3936196x0 x0  + 4127137x0 x0  - 3616296x0 x0  + 728732x0 x0  - 3342035x0 x0  - 4033783x0 x0  + 1946920x0 x0  - 593740x0 ,
                 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  - 1777807x0  - 819890x0 x0  + 1488402x0 x0  + 2208264x0 x0  - 1483087x0  - 2375566x0 x0  - 2444196x0 x0  + 1420876x0 x0  + 1365490x0 x0  - 2183902x0  - 4271115x0 x0  + 2004790x0 x0  + 625770x0 x0  - 3309635x0 x0  - 218120x0 x0  - 239750x0  - 2395097x0 x0  - 864368x0 x0  - 2290223x0 x0  - 4170830x0 x0  + 1496169x0 x0  + 2538167x0 x0  - 3994905x0  + 1758571x0 x0  - 1342155x0 x0  - 3423116x0 x0  - 3758980x0 x0  - 583134x0 x0  - 1906051x0 x0  - 1358181x0 x0  - 4085277x0  - 4434312x0 x0  - 1521374x0 x0  - 1378965x0 x0  + 1289475x0 x0  - 4667803x0 x0  + 2605667x0 x0  - 2577609x0 x0  - 4804495x0 x0  - 3505430x0  + 943001x0 x0  - 3746549x0 x0  - 1952754x0 x0  + 3618359x0 x0  + 3012654x0 x0  - 2347145x0 x0  + 4510972x0 x0  - 1971765x0 x0  - 3199426x0 x0  + 1681839x0
                 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[p   , p   , p   , p   , p   , p   , p   , p   , p   , p   ]) defined by
                                    0,1   0,2   1,2   0,3   1,3   2,3   0,4   1,4   2,4   3,4
              {
               p   p    - p   p    + p   p   ,
                2,3 1,4    1,3 2,4    1,2 3,4
               
               p   p    - p   p    + p   p   ,
                2,3 0,4    0,3 2,4    0,2 3,4
               
               p   p    - p   p    + p   p   ,
                1,3 0,4    0,3 1,4    0,1 3,4
               
               p   p    - p   p    + p   p   ,
                1,2 0,4    0,2 1,4    0,1 2,4
               
               p   p    - p   p    + p   p
                1,2 0,3    0,2 1,3    0,1 2,3
              }
      -- rational map 1/1 -- 
      map 1/1, one of its representatives:
      {
       - 3859043x0  + 2739124x0  - 3791482x0  - 2481358x0  - 1350948x0  - 2848334x0  - 1962136x0  + 2085469x0  + 3228074x0  - 2995844x0 ,
                  0            1            2            3            4            5            6            7            8            9
       
       2163490x0  + 4049720x0  + 1934174x0  - 4262754x0  - 3459672x0  - 4971809x0  + 4128180x0  - 742887x0  + 1109668x0  - 2939177x0 ,
                0            1            2            3            4            5            6           7            8            9
       
       - 2705209x0  + 2375473x0  + 646547x0  + 2953755x0  + 3046788x0  - 1379975x0  - 4394589x0  + 844025x0  - 3672616x0  - 770907x0 ,
                  0            1           2            3            4            5            6           7            8           9
       
       3225774x0  - 1010026x0  - 4493340x0  + 2592046x0  - 3307303x0  - 1264286x0  + 3954538x0  - 3907777x0  + 2427570x0  - 2639628x0 ,
                0            1            2            3            4            5            6            7            8            9
       
       - 3238044x0  - 2280456x0  - 1690478x0  + 1009298x0  - 1216691x0  + 904259x0  + 2624748x0  - 3711397x0  + 2074039x0  + 2513227x0 ,
                  0            1            2            3            4           5            6            7            8            9
       
       - 4619922x0  + 536904x0  - 421667x0  + 4882755x0  + 1899459x0  - 2333124x0  + 784763x0  - 417246x0  - 1275833x0  + 3105312x0 ,
                  0           1           2            3            4            5           6           7            8            9
       
       805149x0  + 369172x0  + 397519x0  + 4522825x0  + 101664x0  - 4804567x0  - 991011x0  - 4385463x0  + 1214478x0  + 1667067x0 ,
               0           1           2            3           4            5           6            7            8            9
       
       2874492x0  - 2124563x0  + 2351827x0  - 2053155x0  + 1613498x0  + 587805x0  - 1167881x0  + 3653243x0  - 4869241x0  + 1116973x0 ,
                0            1            2            3            4           5            6            7            8            9
       
       3244467x0  + 2466739x0  + 3171858x0  + 1875247x0  + 2617875x0  - 2879193x0  + 4407255x0  - 1173409x0  + 284839x0  - 4686890x0 ,
                0            1            2            3            4            5            6            7           8            9
       
       2440857x0  - 1918965x0  + 1019117x0  + 381109x0  - 4470561x0  + 2314152x0  - 718555x0  + 3396036x0  - 1023810x0  + 3601598x0
                0            1            2           3            4            5           6            7            8            9
      }

See also