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

Synopsis

• Operator: ===>
• Usage:
X ===> Y
Y <=== X
• 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 6.0236s (cpu); 4.05431s (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 5.69123s (cpu); 3.92121s (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 11.0596s (cpu); 8.67077s (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 }