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

## Synopsis

• Operator: ===>
• Usage:
X ===> Y
• Inputs:
• Y, , projectively equivalent to X
• Outputs:
• , 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 2.6417 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 }