[daa83b] | 1 | // $Id: deform.lib,v 1.33 2005-04-28 09:22:14 Singular Exp $ |
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[f0c6f4] | 2 | // author: Bernd Martin email: martin@math.tu-cottbus.de |
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[82716e] | 3 | //(bm, last modified 4/98) |
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[3d124a7] | 4 | /////////////////////////////////////////////////////////////////////////////// |
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[daa83b] | 5 | version="$Id: deform.lib,v 1.33 2005-04-28 09:22:14 Singular Exp $"; |
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[fd3fb7] | 6 | category="Singularities"; |
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[5480da] | 7 | info=" |
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[4ac997] | 8 | LIBRARY: deform.lib Miniversal Deformation of Singularities and Modules |
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| 9 | AUTHOR: Bernd Martin, email: martin@math.tu-cottbus.de |
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[f34c37c] | 10 | |
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| 11 | PROCEDURES: |
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[f0c6f4] | 12 | versal(Fo[,d,any]) miniversal deformation of isolated singularity Fo |
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| 13 | mod_versal(Mo,I,[,d,any]) miniversal deformation of module Mo modulo ideal I |
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[82716e] | 14 | lift_kbase(N,M); lifting N into standard kbase of M |
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[f0c6f4] | 15 | lift_rel_kb(N,M[,kbM,p]) relative lifting N into a kbase of M |
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[5480da] | 16 | "; |
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[4ac997] | 17 | |
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[f1201a] | 18 | LIB "inout.lib"; |
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| 19 | LIB "general.lib"; |
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| 20 | LIB "matrix.lib"; |
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| 21 | LIB "homolog.lib"; |
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[6f2edc] | 22 | LIB "sing.lib"; |
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[3d124a7] | 23 | /////////////////////////////////////////////////////////////////////////////// |
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[4ac997] | 24 | |
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[f1201a] | 25 | proc versal (ideal Fo,list #) |
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[d2b2a7] | 26 | "USAGE: versal(Fo[,d,any]); Fo=ideal, d=int, any=list |
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[11dddeb] | 27 | COMPUTE: miniversal deformation of Fo up to degree d (default d=100), |
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[6fe3a0] | 28 | RETURN: list L of 4 rings: |
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| 29 | L[1] extending the basering Po by new variables given by |
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| 30 | \"A,B,..\" (deformation parameters); the new variables precede |
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| 31 | the old ones, the ordering is the product of \"ls\" and \"ord(Po)\" @* |
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| 32 | L[2] = L[1]/Fo extending Qo=Po/Fo, @* |
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| 33 | L[3] = the embedding ring of the versal base space, @* |
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| 34 | L[4] = L[1]/Js extending L[3]/Js. @* |
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| 35 | In the ring L[1] the following matrices are stored: |
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[3fe3582] | 36 | @*Js = giving the versal base space (obstructions), |
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| 37 | @*Fs = giving the versal family of Fo, |
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| 38 | @*Rs = giving the lifting of Ro=syz(Fo). |
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| 39 | |
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[f1201a] | 40 | If d is defined (!=0), it computes up to degree d. |
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[3fe3582] | 41 | @*If 'any' is defined and any[1] is no string, interactive version. |
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[6fe3a0] | 42 | @*Otherwise 'any' is interpreted as a list of predefined strings: |
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| 43 | \"my\",\"param\",\"order\",\"out\": @* |
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| 44 | (\"my\" internal prefix, \"param\" is a letter (e.g. \"A\") for the |
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| 45 | name of the first parameter or (e.g. \"A(\") for index parameter |
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| 46 | variables, \"order\" ordering string for ring extension), \"out\" name |
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| 47 | of output file). |
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| 48 | NOTE: printlevel < 0 no additional output, |
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[82716e] | 49 | printlevel >=0,1,2,.. informs you, what is going on; |
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[f1201a] | 50 | this proc uses 'execute'. |
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| 51 | EXAMPLE:example versal; shows an example |
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[d2b2a7] | 52 | " |
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[3d124a7] | 53 | { |
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[f1201a] | 54 | //------- prepare ------------------------------------------------------------- |
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| 55 | string str,@param,@order,@my,@out,@degrees; |
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| 56 | int @d,d_max,@t1,t1',@t2,@colR,ok_ann,@smooth,@noObstr,@size,@j; |
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| 57 | int p = printlevel-voice+3; |
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| 58 | int time = timer; |
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| 59 | intvec @iv,@jv,@is_qh,@degr; |
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[82716e] | 60 | d_max = 100; |
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[f1201a] | 61 | @my = ""; @param="A"; @order="ds"; @out="no"; |
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| 62 | @size = size(#); |
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[f48213] | 63 | if( @size>0 ) { if (#[1]>0) { d_max = #[1];} } |
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[82716e] | 64 | if( @size>1 ) |
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| 65 | { if(typeof(#[2])!="string") |
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[f1201a] | 66 | { string @active; |
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| 67 | @my,@param,@order,@out = interact1(); |
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| 68 | } |
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| 69 | else |
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| 70 | { @my = #[2]; |
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| 71 | if (@size>2) {@param = #[3];} |
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| 72 | if (@size>3) {@order = #[4];} |
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| 73 | if (@size>4) {@out = #[5];} |
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| 74 | } |
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| 75 | } |
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| 76 | string myPx = @my+"Px"; |
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| 77 | string myQx = @my+"Qx"; |
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| 78 | string myOx = @my+"Ox"; |
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| 79 | string mySo = @my+"So"; |
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| 80 | Fo = simplify(Fo,10); |
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| 81 | @is_qh = qhweight(Fo); |
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| 82 | int @rowR= size(Fo); |
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| 83 | def Po = basering; |
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[82716e] | 84 | setring Po; |
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[f1201a] | 85 | poly X_s = product(maxideal(1)); |
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[0b59f5] | 86 | //------- reproduce T_12 ----------------------------------------------------- |
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| 87 | list Ls = T_12(Fo,1); |
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[f1201a] | 88 | matrix Ro = Ls[6]; // syz(i) |
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| 89 | matrix InfD = Ls[5]; // matrix of inf. deformations |
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| 90 | matrix PreO = Ls[7]; // representation of (Syz/Kos)* |
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| 91 | module PreO'= std(PreO); |
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[0b59f5] | 92 | module PreT = Ls[2]; // representation of modT_2 (sb) |
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[f1201a] | 93 | if(dim(PreT)==0) |
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| 94 | { |
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[0b59f5] | 95 | matrix kbT_2 = kbase(PreT); // kbase of T_2 |
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[f1201a] | 96 | } |
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| 97 | else |
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| 98 | { |
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[0b59f5] | 99 | matrix kbT_2 ; // kbase of T_2 : empty |
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[f1201a] | 100 | } |
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[0b59f5] | 101 | @t1 = Ls[3]; // vdim of T_1 |
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| 102 | @t2 = Ls[4]; // vdim of T_2 |
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[f1201a] | 103 | kill Ls; |
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[82716e] | 104 | t1' = @t1; |
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[6fe3a0] | 105 | if( @t1==0) { dbprint(p,"// rigid!"); return(list());} |
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[82716e] | 106 | if( @t2==0) { @smooth=1; dbprint(p,"// smooth base space");} |
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[0b59f5] | 107 | dbprint(p,"// ready: T_1 and T_2"); |
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[f1201a] | 108 | @colR = ncols(Ro); |
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| 109 | //----- test: quasi-homogeneous, choice of inf. def.-------------------------- |
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| 110 | @degrees = homog_test(@is_qh,matrix(Fo),InfD); |
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| 111 | @jv = 1..@t1; |
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[82716e] | 112 | if (@degrees!="") |
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[0b59f5] | 113 | { dbprint(p-1,"// T_1 is quasi-homogeneous represented with weight-vector", |
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[f1201a] | 114 | @degrees); |
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| 115 | } |
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| 116 | if (defined(@active)) |
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[82716e] | 117 | { "// matrix of infinitesimal deformations:";print(InfD); |
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[11dddeb] | 118 | "// weights of infinitesimal deformations ( empty ='not qhomog'):"; |
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[f1201a] | 119 | @degrees; |
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| 120 | matrix dummy; |
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| 121 | InfD,dummy,t1' = interact2(InfD,@jv);kill dummy; |
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[82716e] | 122 | } |
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[f1201a] | 123 | //---- create new rings and objects ------------------------------------------ |
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[6fe3a0] | 124 | list list_of_rings=get_rings(Fo,t1',1,@order,@param); |
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| 125 | def `myPx`= list_of_rings[1]; |
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| 126 | def `myQx`= list_of_rings[2]; |
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| 127 | def `myOx`= list_of_rings[3]; |
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| 128 | def `mySo`= list_of_rings[4]; |
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| 129 | kill list_of_rings; |
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| 130 | setring `myPx`; |
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[82716e] | 131 | @jv=0; @jv[t1']=0; @jv=@jv+1; @jv[nvars(basering)]=0; |
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[f1201a] | 132 | //weight-vector for calculating |
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| 133 | //rel-jet with resp to def-para |
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[82716e] | 134 | ideal Io = imap(Po,Fo); |
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[f1201a] | 135 | ideal J,m_J,tid; attrib(J,"isSB",1); |
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| 136 | matrix Fo = matrix(Io); //initial equations |
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| 137 | matrix homF = kohom(Fo,@colR); |
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| 138 | matrix Ro = imap(Po,Ro); |
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| 139 | matrix homR = transpose(Ro); |
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| 140 | matrix homFR= concat(homR,homF); |
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| 141 | module hom' = std(homFR); |
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[82716e] | 142 | matrix Js[1][@t2]; |
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| 143 | matrix F_R,Fs,Rs,Fn,Rn; |
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| 144 | export Js,Fs,Rs; |
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| 145 | matrix Mon[t1'][1]=maxideal(1); |
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[f1201a] | 146 | Fn = transpose(imap(Po,InfD)*Mon); //infinitesimal deformations |
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[82716e] | 147 | Fs = Fo + Fn; |
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[f1201a] | 148 | dbprint(p-1,"// infinitesimal deformation: Fs: ",Fs); |
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| 149 | Rn = (-1)*lift(Fo,Fs*Ro); //infinit. relations |
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| 150 | Rs = Ro + Rn; |
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| 151 | F_R = Fs*Rs; |
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| 152 | tid = 0 + ideal(F_R); |
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| 153 | if (tid[1]==0) {d_max=1;} //finished ? |
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[82716e] | 154 | setring `myOx`; |
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[f1201a] | 155 | matrix Fs,Rs,Cup,Cup',F_R,homFR,New,Rn,Fn; |
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| 156 | module hom'; |
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[82716e] | 157 | ideal null,tid; attrib(null,"isSB",1); |
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| 158 | setring `myQx`; |
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| 159 | poly X_s = imap(Po,X_s); |
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| 160 | matrix Cup,Cup',MASS; |
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[f1201a] | 161 | ideal tid,null; attrib(null,"isSB",1); |
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[82716e] | 162 | ideal J,m_J; attrib(J,"isSB",1); |
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[f1201a] | 163 | attrib(m_J,"isSB",1); |
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[82716e] | 164 | matrix PreO = imap(Po,PreO); |
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[f1201a] | 165 | module PreO'= imap(Po,PreO'); attrib(PreO',"isSB",1); |
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| 166 | module PreT = imap(Po,PreT); attrib(PreT,"isSB",1); |
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[0b59f5] | 167 | matrix kbT_2 = imap(Po,kbT_2); |
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[f1201a] | 168 | matrix Mon = fetch(`myPx`,Mon); |
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| 169 | matrix F_R = fetch(`myPx`,F_R); |
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| 170 | matrix Js[1][@t2]; |
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[3d124a7] | 171 | //------- start the loop ------------------------------------------------------ |
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[f1201a] | 172 | for (@d=2;@d<=d_max;@d=@d+1) |
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[3d124a7] | 173 | { |
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[f1201a] | 174 | if( @t1==0) {break}; |
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[82716e] | 175 | dbprint(p,"// start computation in degree "+string(@d)+"."); |
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[f0c6f4] | 176 | dbprint(p-3,">>> TIME = "+string(timer-time)); |
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| 177 | dbprint(p-3,"==> memory = "+string(kmemory())+"k"); |
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[f1201a] | 178 | //------- compute obstruction-vector ----------------------------------------- |
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| 179 | if (@smooth) { @noObstr=1;} |
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| 180 | else |
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[82716e] | 181 | { Cup = jet(F_R,@d,@jv); |
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| 182 | Cup = matrix(reduce(ideal(Cup),m_J),@colR,1); |
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| 183 | Cup = jet(Cup,@d,@jv); |
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| 184 | } |
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[4ac997] | 185 | //------- express obstructions in kbase of T_2 ------------------------------- |
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[f1201a] | 186 | if ( @noObstr==0 ) |
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| 187 | { Cup' = reduce(Cup,PreO'); |
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| 188 | tid = simplify(ideal(Cup'),10); |
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| 189 | if(tid[1]!=0) |
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| 190 | { dbprint(p-4,"// *"); |
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| 191 | Cup=Cup-Cup'; |
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| 192 | } |
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| 193 | Cup = lift(PreO,Cup); |
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[0b59f5] | 194 | MASS = lift_rel_kb(Cup,PreT,kbT_2,X_s); |
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[82716e] | 195 | dbprint(p-3,"// next MASSEY-products:",MASS-jet(MASS,@d-1,@jv)); |
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[f1201a] | 196 | if (MASS==transpose(Js)) |
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[82716e] | 197 | { @noObstr=1;dbprint(p-1,"// no obstruction"); } |
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[f1201a] | 198 | else { @noObstr=0; } |
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[6f2edc] | 199 | } |
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[f1201a] | 200 | //------- obtain equations of base space -------------------------------------- |
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| 201 | if ( @noObstr==0 ) |
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| 202 | { Js = transpose(MASS); |
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| 203 | dbprint(p-2,"// next equation of base space:", |
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| 204 | simplify(ideal(Js),10)); |
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| 205 | setring `myPx`; |
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| 206 | Js = imap(`myQx`,Js); |
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[82716e] | 207 | degBound = @d+1; |
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[f1201a] | 208 | J = std(ideal(Js)); |
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| 209 | m_J = std(J*ideal(Mon)); |
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| 210 | degBound = 0; |
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| 211 | //--------------- obtain new base-ring ---------------------------------------- |
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[07329b2] | 212 | if(defined(`myOx`)) {kill `myOx`;} |
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[82716e] | 213 | qring `myOx` = J; |
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[f1201a] | 214 | matrix Fs,Rs,F_R,Cup,Cup',homFR,New,Rn,Fn; |
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| 215 | module hom'; |
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| 216 | ideal null,tid; attrib(null,"isSB",1); |
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[3d124a7] | 217 | } |
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[f1201a] | 218 | //---------------- lift equations F and relations R --------------------------- |
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| 219 | setring `myOx`; |
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[82716e] | 220 | Fs = fetch(`myPx`,Fs); |
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| 221 | Rs = fetch(`myPx`,Rs); |
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| 222 | F_R = Fs*Rs; |
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| 223 | F_R = matrix(reduce(ideal(F_R),null)); |
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[f1201a] | 224 | tid = 0 + ideal(F_R); |
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[82716e] | 225 | if (tid[1]==0) { dbprint(p-1,"// finished"); break;} |
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| 226 | Cup = (-1)*transpose(jet(F_R,@d,@jv)); |
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| 227 | homFR = fetch(`myPx`,homFR); |
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[f1201a] | 228 | hom' = fetch(`myPx`,hom'); attrib(hom',"isSB",1); |
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| 229 | Cup' = simplify(reduce(Cup,hom'),10); |
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| 230 | tid = simplify(ideal(Cup'),10); |
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| 231 | if (tid[1]!=0) |
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| 232 | { dbprint(p-4,"// #"); |
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| 233 | Cup=Cup-Cup'; |
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[6f2edc] | 234 | } |
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[f1201a] | 235 | New = lift(homFR,Cup); |
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| 236 | Rn = matrix(ideal(New[1+@rowR..nrows(New),1]),@rowR,@colR); |
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| 237 | Fn = matrix(ideal(New[1..@rowR,1]),1,@rowR); |
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| 238 | Fs = Fs+Fn; |
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| 239 | Rs = Rs+Rn; |
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| 240 | F_R = Fs*Rs; |
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[82716e] | 241 | tid = 0+reduce(ideal(F_R),null); |
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[f1201a] | 242 | //---------------- fetch results into other rings ----------------------------- |
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| 243 | setring `myPx`; |
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| 244 | Fs = fetch(`myOx`,Fs); |
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| 245 | Rs = fetch(`myOx`,Rs); |
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| 246 | F_R = Fs*Rs; |
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| 247 | setring `myQx`; |
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| 248 | F_R = fetch(`myPx`,F_R); |
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| 249 | m_J = fetch(`myPx`,m_J); attrib(m_J,"isSB",1); |
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| 250 | J = fetch(`myPx`,J); attrib(J,"isSB",1); |
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[82716e] | 251 | Js = fetch(`myPx`,Js); |
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| 252 | tid = fetch(`myOx`,tid); |
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| 253 | if (tid[1]==0) { dbprint(p-1,"// finished");break;} |
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[6f2edc] | 254 | } |
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[f1201a] | 255 | //--------- end loop and final output ---------------------------------------- |
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| 256 | setring `myPx`; |
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| 257 | if (@out!="no") |
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| 258 | { string out = @out+"_"+string(@d); |
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[82716e] | 259 | "// writing file "+out+" with matrix Js, matrix Fs, matrix Rs ready |
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[f1201a] | 260 | for reading in rings "+myPx+" or "+myQx; |
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| 261 | write(out,"matrix Js[1][",@t2,"]=",Js,";matrix Fs[1][",@rowR,"]=",Fs, |
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| 262 | ";matrix Rs[",@rowR,"][",@colR,"]=",Rs,";"); |
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[82716e] | 263 | } |
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[f0c6f4] | 264 | dbprint(p-3,">>> TIME = "+string(timer-time)); |
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[f1201a] | 265 | if (@is_qh != 0) |
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| 266 | { @degr = qhweight(ideal(Js)); |
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| 267 | @degr = @degr[1..t1']; |
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| 268 | dbprint(p-1,"// quasi-homogeneous weights of miniversal base",@degr); |
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[82716e] | 269 | } |
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[f1201a] | 270 | dbprint(p-1, |
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| 271 | "// ___ Equations of miniversal base space ___",Js, |
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| 272 | "// ___ Equations of miniversal total space ___",Fs); |
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[6fe3a0] | 273 | dbprint(p,""," |
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| 274 | // 'versal' returned a list, say L, of four rings. In L[1] are stored: |
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| 275 | // as matrix Fs: Equations of total space of the miniversal deformation, |
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| 276 | // as matrix Js: Equations of miniversal base space, |
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| 277 | // as matrix Rs: syzygies of Fs mod Js. |
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| 278 | // To access these data, type |
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| 279 | def Px=L[1]; setring Px; print(Fs); print(Js); print(Rs); |
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| 280 | "); |
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| 281 | return(list(`myPx`,`myQx`,`mySo`,`myOx`)); |
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[3d124a7] | 282 | } |
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[6f2edc] | 283 | example |
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[3d124a7] | 284 | { "EXAMPLE:"; echo = 2; |
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[6f2edc] | 285 | int p = printlevel; |
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[f1201a] | 286 | printlevel = 0; |
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[6f2edc] | 287 | ring r1 = 0,(x,y,z,u,v),ds; |
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| 288 | matrix m[2][4] = x,y,z,u,y,z,u,v; |
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[82716e] | 289 | ideal Fo = minor(m,2); |
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[f1201a] | 290 | // cone over rational normal curve of degree 4 |
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[6fe3a0] | 291 | list L=versal(Fo); |
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| 292 | L; |
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| 293 | def Px=L[1]; |
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[f1201a] | 294 | setring Px; |
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[6f2edc] | 295 | // ___ Equations of miniversal base space ___: |
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[f1201a] | 296 | Js;""; |
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[6f2edc] | 297 | // ___ Equations of miniversal total space ___: |
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[f1201a] | 298 | Fs;""; |
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[6f2edc] | 299 | } |
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[3d124a7] | 300 | /////////////////////////////////////////////////////////////////////////////// |
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[4ac997] | 301 | |
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[f1201a] | 302 | proc mod_versal(matrix Mo, ideal I, list #) |
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[4ac997] | 303 | "USAGE: mod_versal(Mo,I[,d,any]); I=ideal, M=module, d=int, any =list |
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[11dddeb] | 304 | COMPUTE: miniversal deformation of coker(Mo) over Qo=Po/Io, Po=basering; |
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[6fe3a0] | 305 | RETURN: list L of 4 rings: |
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| 306 | L[1] extending the basering Po by new variables given by |
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| 307 | \"A,B,..\" (deformation parameters); the new variables precede |
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| 308 | the old ones, the ordering is the product of \"ls\" and \"ord(Po)\" @* |
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| 309 | L[2] = L[1]/Io extending Qo, @* |
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| 310 | L[3] = the embedding ring of the versal base space, @* |
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| 311 | L[4] = L[1]/(Io+Js) ring of the versal deformation of coker(Ms). @* |
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| 312 | In the ring L[1] the following matrices are stored: |
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[3fe3582] | 313 | @*Js = giving the versal base space (obstructions), |
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[6fe3a0] | 314 | @*Fs = giving the versal family of Mo, |
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| 315 | @*Rs = giving the lifting of syzygies Lo=syz(Mo). |
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[f1201a] | 316 | If d is defined (!=0), it computes up to degree d. |
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[3fe3582] | 317 | @*If 'any' is defined and any[1] is no string, interactive version. |
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[6fe3a0] | 318 | @*Otherwise 'any' is interpreted as a list of predefined strings: |
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| 319 | \"my\",\"param\",\"order\",\"out\": @* |
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| 320 | (\"my\" internal prefix, \"param\" is a letter (e.g. \"A\") for the |
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| 321 | name of the first parameter or (e.g. \"A(\") for index parameter |
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| 322 | variables, \"order\" ordering string for ring extension), \"out\" name |
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| 323 | of output file). |
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| 324 | NOTE: printlevel < 0 no additional output, |
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[82716e] | 325 | printlevel >=0,1,2,.. informs you, what is going on, |
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[f1201a] | 326 | this proc uses 'execute'. |
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| 327 | EXAMPLE:example mod_versal; shows an example |
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[d2b2a7] | 328 | " |
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[3d124a7] | 329 | { |
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[f1201a] | 330 | //------- prepare ------------------------------------------------------------- |
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[11dddeb] | 331 | intvec save_opt=option(get); |
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| 332 | option(cancelunit); |
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[f1201a] | 333 | string str,@param,@order,@my,@out,@degrees; |
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| 334 | int @d,d_max,f0,f1,f2,e1,e1',e2,ok_ann,@smooth,@noObstr,@size,@j; |
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| 335 | int p = printlevel-voice+3; |
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| 336 | int time = timer; |
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| 337 | intvec @iv,@jv,@is_qh,@degr; |
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[82716e] | 338 | d_max = 100; |
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[f1201a] | 339 | @my = ""; @param="A"; @order="ds"; @out="no"; |
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| 340 | @size = size(#); |
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| 341 | if( @size>0 ) { d_max = #[1]; } |
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[82716e] | 342 | if( @size>1 ) |
---|
| 343 | { if(typeof(#[2])!="string") |
---|
[f1201a] | 344 | { string @active; |
---|
| 345 | @my,@param,@order,@out = interact1(); |
---|
| 346 | } |
---|
| 347 | else |
---|
| 348 | { @my = #[2]; |
---|
| 349 | if (@size>2) {@param = #[3];} |
---|
| 350 | if (@size>3) {@order = #[4];} |
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| 351 | if (@size>4) {@out = #[5];} |
---|
| 352 | } |
---|
[82716e] | 353 | } |
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[f1201a] | 354 | string myPx = @my+"Px"; |
---|
| 355 | string myQx = @my+"Qx"; |
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| 356 | string myOx = @my+"Ox"; |
---|
| 357 | string mySo = @my+"So"; |
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| 358 | @is_qh = qhweight(I); |
---|
| 359 | def Po = basering; |
---|
| 360 | setring Po; |
---|
| 361 | poly X_s = product(maxideal(1)); |
---|
| 362 | //-------- compute Ext's ------------------------------------------------------ |
---|
| 363 | I = std(I); |
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[82716e] | 364 | qring Qo = I; |
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[f1201a] | 365 | matrix Mo = fetch(Po,Mo); |
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[82716e] | 366 | list Lo = compute_ext(Mo,p); |
---|
[f1201a] | 367 | f0,f1,f2,e1,e2,ok_ann=Lo[1]; |
---|
| 368 | matrix Ls,kb1,lift1 = Lo[2],Lo[3],Lo[4]; |
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| 369 | matrix kb2,C',D' = Lo[5][2],Lo[5][3],Lo[5][5]; |
---|
| 370 | module ex2,Co,Do = Lo[5][1],Lo[5][4],Lo[5][6]; |
---|
| 371 | kill Lo; |
---|
| 372 | dbprint(p,"// ready: Ext1 and Ext2"); |
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| 373 | //----- test: quasi-homogeneous, choice of inf. def.-------------------------- |
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[82716e] | 374 | @degrees = homog_test(@is_qh,Mo,kb1); |
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[f1201a] | 375 | e1' = e1; @jv = 1..e1; |
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[82716e] | 376 | if (@degrees != "") |
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[f1201a] | 377 | { dbprint(p-1,"// Ext1 is quasi-homogeneous represented: "+@degrees); |
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| 378 | } |
---|
| 379 | if (defined(@active)) |
---|
| 380 | { "// kbase of Ext1:"; |
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[82716e] | 381 | print(kb1); |
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[f1201a] | 382 | "// weights of kbase of Ext1 ( empty = 'not qhomog')";@degrees; |
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| 383 | kb1,lift1,e1' = interact2(kb1,@jv,lift1); |
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[82716e] | 384 | } |
---|
[f1201a] | 385 | //-------- get new rings and objects ------------------------------------------ |
---|
| 386 | setring Po; |
---|
[6fe3a0] | 387 | list list_of_rings=get_rings(I,e1',0,@order,@param); |
---|
| 388 | def `myPx`= list_of_rings[1]; |
---|
| 389 | def `myQx`= list_of_rings[2]; |
---|
| 390 | def `myOx`= list_of_rings[3]; |
---|
| 391 | def `mySo`= list_of_rings[4]; |
---|
| 392 | kill list_of_rings; |
---|
[f1201a] | 393 | setring `myPx`; |
---|
| 394 | ideal J,m_J; |
---|
| 395 | ideal I_J = imap(Po,I); |
---|
| 396 | ideal Io = I_J; |
---|
| 397 | matrix Mon[e1'][1] = maxideal(1); |
---|
[82716e] | 398 | matrix Ms = imap(Qo,Mo); |
---|
| 399 | matrix Ls = imap(Qo,Ls); |
---|
| 400 | matrix Js[1][e2]; |
---|
[f1201a] | 401 | setring `myQx`; |
---|
| 402 | ideal J,I_J,tet,null; attrib(null,"isSB",1); |
---|
| 403 | ideal m_J = fetch(`myPx`,m_J); attrib(m_J,"isSB",1); |
---|
| 404 | @jv=0; @jv[e1] = 0; @jv = @jv+1; @jv[nvars(`myPx`)] = 0; |
---|
[82716e] | 405 | matrix Ms = imap(Qo,Mo); export(Ms); |
---|
[f1201a] | 406 | matrix Ls = imap(Qo,Ls); export(Ls); |
---|
| 407 | matrix Js[e2][1]; export(Js); |
---|
[82716e] | 408 | matrix MASS; |
---|
[f1201a] | 409 | matrix Mon = fetch(`myPx`,Mon); |
---|
| 410 | matrix Mn,Ln,ML,Cup,Cup',Lift; |
---|
| 411 | matrix C' = imap(Qo,C'); |
---|
| 412 | module Co = imap(Qo,Co); attrib(Co,"isSB",1); |
---|
| 413 | module ex2 = imap(Qo,ex2); attrib(ex2,"isSB",1); |
---|
| 414 | matrix D' = imap(Qo,D'); |
---|
| 415 | module Do = imap(Qo,Do); attrib(Do,"isSB",1); |
---|
[82716e] | 416 | matrix kb2 = imap(Qo,kb2); |
---|
[f1201a] | 417 | matrix kb1 = imap(Qo,kb1); |
---|
| 418 | matrix lift1= imap(Qo,lift1); |
---|
| 419 | poly X_s = imap(Po,X_s); |
---|
[82716e] | 420 | intvec intv = e1',e1,f0,f1,f2; |
---|
| 421 | Ms,Ls= get_inf_def(Ms,Ls,kb1,lift1,X_s); |
---|
[f1201a] | 422 | kill kb1,lift1; |
---|
| 423 | dbprint(p-1,"// infinitesimal extension",Ms); |
---|
| 424 | //----------- start the loop -------------------------------------------------- |
---|
| 425 | for (@d=2;@d<=d_max;@d=@d+1) |
---|
[82716e] | 426 | { |
---|
[f0c6f4] | 427 | dbprint(p-3,">>> time = "+string(timer-time)); |
---|
| 428 | dbprint(p-3,"==> memory = "+string(memory(0)/1000)+ |
---|
[f1201a] | 429 | ", allocated = "+string(memory(1)/1000)); |
---|
[82716e] | 430 | dbprint(p,"// start deg = "+string(@d)); |
---|
[f1201a] | 431 | //-------- get obstruction ---------------------------------------------------- |
---|
| 432 | Cup = matrix(ideal(Ms*Ls),f0*f2,1); |
---|
| 433 | Cup = jet(Cup,@d,@jv); |
---|
| 434 | Cup = reduce(ideal(Cup),m_J); |
---|
| 435 | Cup = jet(Cup,@d,@jv); |
---|
| 436 | //-------- express obstruction in kbase --------------------------------------- |
---|
| 437 | Cup' = reduce(Cup,Do); |
---|
| 438 | tet = simplify(ideal(Cup'),10); |
---|
[82716e] | 439 | if (tet[1]!=0) |
---|
[f1201a] | 440 | { dbprint(p-4,"// *"); |
---|
| 441 | Cup = Cup-Cup'; |
---|
| 442 | } |
---|
| 443 | Cup = lift(D',Cup); |
---|
| 444 | if (ok_ann) |
---|
| 445 | { MASS = lift_rel_kb(Cup,ex2,kb2,X_s);} |
---|
| 446 | else |
---|
[82716e] | 447 | { MASS = reduce(Cup,ex2);} |
---|
[f1201a] | 448 | dbprint(p-3,"// next MATRIC-MASSEY-products", |
---|
| 449 | MASS-jet(MASS,@d-1,@jv)); |
---|
| 450 | if ( MASS==transpose(Js)) |
---|
| 451 | { @noObstr = 1;dbprint(p-1,"//no obstruction"); } |
---|
[82716e] | 452 | else { @noObstr = 0; } |
---|
[f1201a] | 453 | //-------- obtain equations of base space ------------------------------------- |
---|
| 454 | if (@noObstr == 0) |
---|
| 455 | { Js = MASS; |
---|
| 456 | dbprint(p-2,"// next equation of base space:",simplify(ideal(Js),10)); |
---|
| 457 | setring `myPx`; |
---|
| 458 | Js = imap(`myQx`,Js); |
---|
| 459 | degBound=@d+1; |
---|
| 460 | J = std(ideal(Js)); |
---|
| 461 | m_J = std(ideal(Mon)*J); |
---|
| 462 | degBound=0; |
---|
| 463 | I_J = Io,J; attrib(I_J,"isSB",1); |
---|
[82716e] | 464 | //-------- obtain new base ring ----------------------------------------------- |
---|
[07329b2] | 465 | if (defined(`myOx`)) {kill `myOx`;} |
---|
[82716e] | 466 | qring `myOx` = I_J; |
---|
[f1201a] | 467 | ideal null,tet; attrib(null,"isSB",1); |
---|
| 468 | matrix Ms = imap(`myQx`,Ms); |
---|
| 469 | matrix Ls = imap(`myQx`,Ls); |
---|
| 470 | matrix Mn,Ln,ML,Cup,Cup',Lift; |
---|
[82716e] | 471 | matrix C' = imap(Qo,C'); |
---|
[f1201a] | 472 | module Co = imap(Qo,Co); attrib(Co,"isSB",1); |
---|
| 473 | module ex2 = imap(Qo,ex2); attrib(ex2,"isSB",1); |
---|
| 474 | matrix kb2 = imap(Qo,kb2); |
---|
| 475 | poly X_s = imap(Po,X_s); |
---|
[82716e] | 476 | } |
---|
[f1201a] | 477 | //-------- get lifts ---------------------------------------------------------- |
---|
| 478 | setring `myOx`; |
---|
| 479 | ML = matrix(reduce(ideal(Ms*Ls),null),f0,f2); |
---|
| 480 | Cup = matrix(ideal(ML),f0*f2,1); |
---|
| 481 | Cup = jet(Cup,@d,@jv); |
---|
| 482 | Cup'= reduce(Cup,Co); |
---|
[82716e] | 483 | tet = simplify(ideal(Cup'),10); |
---|
| 484 | if (tet[1]!=0) |
---|
[f1201a] | 485 | { dbprint(p-4,"// #"); |
---|
| 486 | Cup = Cup-Cup'; |
---|
| 487 | } |
---|
[82716e] | 488 | Lift = lift(C',Cup); |
---|
[f1201a] | 489 | Mn = matrix(ideal(Lift),f0,f1); |
---|
| 490 | Ln = matrix(ideal(Lift[f0*f1+1..nrows(Lift),1]),f1,f2); |
---|
| 491 | Ms = Ms-Mn; |
---|
| 492 | Ls = Ls-Ln; |
---|
| 493 | dbprint(p-3,"// next extension of Mo",Mn); |
---|
| 494 | dbprint(p-3,"// next extension of syz(Mo)",Ln); |
---|
| 495 | ML = reduce(ideal(Ms*Ls),null); |
---|
[82716e] | 496 | //--------- test: finished ---------------------------------------------------- |
---|
[f1201a] | 497 | tet = simplify(ideal(ML),10); |
---|
| 498 | if (tet[1]==0) { dbprint(p-1,"// finished in degree ",@d);} |
---|
| 499 | //---------fetch results into Qx and Px --------------------------------------- |
---|
| 500 | setring `myPx`; |
---|
| 501 | Ms = fetch(`myOx`,Ms); |
---|
[82716e] | 502 | Ls = fetch(`myOx`,Ls); |
---|
[f1201a] | 503 | setring `myQx`; |
---|
| 504 | Ms = fetch(`myOx`,Ms); |
---|
[82716e] | 505 | Ls = fetch(`myOx`,Ls); |
---|
[f1201a] | 506 | ML = Ms*Ls; |
---|
[82716e] | 507 | ML = matrix(reduce(ideal(ML),null),f0,f2); |
---|
[f1201a] | 508 | tet = imap(`myOx`,tet); |
---|
| 509 | if (tet[1]==0) { break;} |
---|
[82716e] | 510 | } |
---|
| 511 | //------- end of loop, final output ------------------------------------------- |
---|
[f1201a] | 512 | if (@out != "no") |
---|
| 513 | { string out = @out+"_"+string(@d); |
---|
[82716e] | 514 | "// writing file '"+out+"' with matrix Js, matrix Ms, matrix Ls |
---|
[f1201a] | 515 | ready for reading in rings "+myPx+" or "+myQx; |
---|
| 516 | write(out,"matrix Js[1][",e2,"]=",Js,";matrix Ms[",f0,"][",f1,"]=",Ms, |
---|
| 517 | ";matrix Ls[",f1,"][",f2,"]=",Ls,";"); |
---|
| 518 | } |
---|
[f0c6f4] | 519 | dbprint(p-3,">>> TIME = "+string(timer-time)); |
---|
[f1201a] | 520 | if (@is_qh != 0) |
---|
| 521 | { @degr = qhweight(ideal(Js)); |
---|
| 522 | @degr = @degr[1..e1']; |
---|
| 523 | dbprint(p-1,"// quasi-homogeneous weights of miniversal base",@degr); |
---|
[82716e] | 524 | } |
---|
[6fe3a0] | 525 | dbprint(p," |
---|
| 526 | // 'mod_versal' returned a list, say L, of four rings. In L[2] are stored: |
---|
| 527 | // as matrix Ms: presentation matrix of the deformed module, |
---|
| 528 | // as matrix Ls: lifted syzygies, |
---|
| 529 | // as matrix Js: Equations of total space of miniversal deformation |
---|
| 530 | // To access these data, type |
---|
| 531 | def Qx=L[2]; setring Qx; print(Ms); print(Ls); print(Js); |
---|
| 532 | "); |
---|
[11dddeb] | 533 | option(set,save_opt); |
---|
[6fe3a0] | 534 | return(list(`myPx`,`myQx`,`mySo`,`myOx`)); |
---|
[3d124a7] | 535 | } |
---|
[6f2edc] | 536 | example |
---|
[3d124a7] | 537 | { "EXAMPLE:"; echo = 2; |
---|
[f1201a] | 538 | int p = printlevel; |
---|
| 539 | printlevel = 1; |
---|
| 540 | ring Ro = 0,(x,y),wp(3,4); |
---|
| 541 | ideal Io = x4+y3; |
---|
| 542 | matrix Mo[2][2] = x2,y,-y2,x2; |
---|
[6fe3a0] | 543 | list L = mod_versal(Mo,Io); |
---|
| 544 | def Qx=L[2]; setring Qx; |
---|
| 545 | print(Ms); |
---|
| 546 | print(Ls); |
---|
| 547 | print(Js); |
---|
[f1201a] | 548 | printlevel = p; |
---|
[07329b2] | 549 | if (defined(Px)) {kill Px,Qx,So;} |
---|
[3d124a7] | 550 | } |
---|
| 551 | /////////////////////////////////////////////////////////////////////////////// |
---|
[f1201a] | 552 | proc kill_rings(list #) |
---|
[d2b2a7] | 553 | "USAGE: kill_rings([string]); |
---|
[3fe3582] | 554 | RETURN: nothing, but kills exported rings generated by procedures |
---|
| 555 | 'versal' and 'mod_versal' with optional prefix 'string' |
---|
[6fe3a0] | 556 | NOTE: obsolete |
---|
[d2b2a7] | 557 | " |
---|
[3d124a7] | 558 | { |
---|
[f1201a] | 559 | string my,br; |
---|
| 560 | if (size(#)>0) { my = #[1];} |
---|
| 561 | string na=nameof(basering); |
---|
| 562 | br = my+"Qx"; |
---|
| 563 | if (defined(`br`)) { kill `br`;} |
---|
| 564 | br = my+"Px"; |
---|
| 565 | if (defined(`br`)) { kill `br`;} |
---|
| 566 | br = my+"So"; |
---|
| 567 | if (defined(`br`)) { kill `br`;} |
---|
| 568 | br = my+"Ox"; |
---|
| 569 | if (defined(`br`)) { kill `br`;} |
---|
| 570 | br = my+"Sx"; |
---|
| 571 | if (defined(`br`)) { kill `br`} |
---|
[917fb5] | 572 | if(system("with","Namespaces")) |
---|
| 573 | { |
---|
[c67136] | 574 | br = my+"Qx"; |
---|
| 575 | if (defined(Top::`br`)) { kill Top::`br`;} |
---|
| 576 | br = my+"Ox"; |
---|
| 577 | if (defined(Top::`br`)) { kill Top::`br`;} |
---|
| 578 | br = my+"Px"; |
---|
| 579 | if (defined(Ring::`br`)) { kill Ring::`br`;} |
---|
| 580 | br = my+"So"; |
---|
| 581 | if (defined(Ring::`br`)) { kill Ring::`br`;} |
---|
| 582 | } |
---|
[f1201a] | 583 | if (defined(basering)==0) |
---|
| 584 | { "// choose new basering?"; |
---|
[c67136] | 585 | if(system("with","Namespaces")) { listvar(Top,ring); } |
---|
| 586 | else { listvar(ring); } |
---|
[f1201a] | 587 | } |
---|
| 588 | return(); |
---|
[3d124a7] | 589 | } |
---|
| 590 | /////////////////////////////////////////////////////////////////////////////// |
---|
[f1201a] | 591 | proc compute_ext(matrix Mo,int p) |
---|
[d2b2a7] | 592 | " |
---|
[f1201a] | 593 | Sub-procedure: obtain Ext1 and Ext2 and other objects used by mod_versal |
---|
[d2b2a7] | 594 | " |
---|
[82716e] | 595 | { |
---|
[f1201a] | 596 | int l,f0,f1,f2,f3,e1,e2,ok_ann; |
---|
| 597 | module Co,Do,ima,ex1,ex2; |
---|
[82716e] | 598 | matrix M0,M1,M2,ker,kb1,lift1,kb2,A,B,C,D; |
---|
[f1201a] | 599 | //------- resM --------------------------------------------------------------- |
---|
[3939bc] | 600 | list resM = nres(Mo,3); |
---|
[f1201a] | 601 | M0 = resM[1]; |
---|
| 602 | M1 = resM[2]; |
---|
| 603 | M2 = resM[3]; kill resM; |
---|
| 604 | f0 = nrows(M0); |
---|
| 605 | f1 = ncols(M0); |
---|
| 606 | f2 = ncols(M1); |
---|
| 607 | f3 = ncols(M2); |
---|
| 608 | //------ compute Ext^2 ------------------------------------------------------ |
---|
[82716e] | 609 | B = kohom(M0,f3); |
---|
[f1201a] | 610 | A = kontrahom(M2,f0); |
---|
[82716e] | 611 | D = modulo(A,B); |
---|
| 612 | Do = std(D); |
---|
[f1201a] | 613 | ima = kohom(M0,f2),kontrahom(M1,f0); |
---|
| 614 | ex2 = modulo(D,ima); |
---|
| 615 | ex2 = std(ex2); |
---|
| 616 | e2 = vdim(ex2); |
---|
| 617 | kb2 = kbase(ex2); |
---|
| 618 | dbprint(p,"// vdim (Ext^2) = "+string(e2)); |
---|
| 619 | //------ test: max = Ann(Ext2) ----------------------------------------------- |
---|
| 620 | for (l=1;l<=e2;l=l+1) |
---|
| 621 | { ok_ann = ok_ann+ord(kb2[l]); |
---|
| 622 | } |
---|
| 623 | if (ok_ann==0) |
---|
[82716e] | 624 | { e2 =nrows(ex2); |
---|
[f1201a] | 625 | dbprint(p,"// Ann(Ext2) is maximal"); |
---|
| 626 | } |
---|
| 627 | //------ compute Ext^1 ------------------------------------------------------- |
---|
[82716e] | 628 | B = kohom(M0,f2); |
---|
[f1201a] | 629 | A = kontrahom(M1,f0); |
---|
| 630 | ker = modulo(A,B); |
---|
[82716e] | 631 | ima = kohom(M0,f1),kontrahom(M0,f0); |
---|
[f1201a] | 632 | ex1 = modulo(ker,ima); |
---|
| 633 | ex1 = std(ex1); |
---|
| 634 | e1 = vdim(ex1); |
---|
| 635 | dbprint(p,"// vdim (Ext^1) = "+string(e1)); |
---|
| 636 | kb1 = kbase(ex1); |
---|
| 637 | kb1 = ker*kb1; |
---|
| 638 | C = concat(A,B); |
---|
| 639 | Co = std(C); |
---|
| 640 | //------ compute the liftings of Ext^1 --------------------------------------- |
---|
| 641 | lift1 = A*kb1; |
---|
[82716e] | 642 | lift1 = lift(B,lift1); |
---|
[f1201a] | 643 | intvec iv = f0,f1,f2,e1,e2,ok_ann; |
---|
| 644 | list L' = ex2,kb2,C,Co,D,Do; |
---|
| 645 | return(iv,M1,kb1,lift1,L'); |
---|
[3d124a7] | 646 | } |
---|
[4ac997] | 647 | /////////////////////////////////////////////////////////////////////////////// |
---|
[6fe3a0] | 648 | static proc get_rings(ideal Io,int e1,int switch, list #) |
---|
[d2b2a7] | 649 | " |
---|
[6fe3a0] | 650 | Sub-procedure: creating ring-extensions, returned as a list of 4 rings |
---|
[d2b2a7] | 651 | " |
---|
[82716e] | 652 | { |
---|
| 653 | def Po = basering; |
---|
[f1201a] | 654 | string my; |
---|
| 655 | string my_ord = "ds"; |
---|
[82716e] | 656 | string my_var = "A"; |
---|
[6fe3a0] | 657 | if (size(#)>1) |
---|
[3d124a7] | 658 | { |
---|
[6fe3a0] | 659 | my_ord = #[1]; |
---|
| 660 | my_var = #[2]; |
---|
[3d124a7] | 661 | } |
---|
[6fe3a0] | 662 | def my_Px=extendring(e1,my_var,my_ord); |
---|
| 663 | setring my_Px; |
---|
| 664 | ideal Io = imap(Po,Io); |
---|
| 665 | attrib(Io,"isSB",1); |
---|
| 666 | qring my_Qx = Io; |
---|
[f1201a] | 667 | if (switch) |
---|
[3d124a7] | 668 | { |
---|
[6fe3a0] | 669 | setring my_Px; |
---|
| 670 | qring my_Ox = std(ideal(0)); |
---|
[3d124a7] | 671 | } |
---|
[f1201a] | 672 | else |
---|
[6f2edc] | 673 | { |
---|
[6fe3a0] | 674 | def my_Ox = my_Qx; |
---|
[3d124a7] | 675 | } |
---|
[6fe3a0] | 676 | def my_So=defring(charstr(Po),e1,my_var,my_ord); |
---|
| 677 | setring my_So; |
---|
| 678 | list erg=list(my_Px,my_Qx,my_Ox,my_So); |
---|
| 679 | return(erg); |
---|
[3d124a7] | 680 | } |
---|
[4ac997] | 681 | /////////////////////////////////////////////////////////////////////////////// |
---|
[d2b2a7] | 682 | proc get_inf_def(list #) |
---|
| 683 | " |
---|
[82716e] | 684 | Sub-procedure: compute infinitesimal family of a module and its syzygies |
---|
[f1201a] | 685 | from a kbase of Ext1 and its lifts |
---|
[d2b2a7] | 686 | " |
---|
[f1201a] | 687 | { |
---|
| 688 | matrix Ms = #[1]; |
---|
| 689 | matrix Ls = #[2]; |
---|
| 690 | matrix kb1 = #[3]; |
---|
| 691 | matrix li1 = #[4]; |
---|
| 692 | int e1,f0,f1,f2; |
---|
| 693 | poly X_s = #[5]; |
---|
| 694 | e1 = ncols(kb1); |
---|
| 695 | f0 = nrows(Ms); |
---|
| 696 | f1 = nrows(Ls); |
---|
| 697 | f2 = ncols(Ls); |
---|
| 698 | int l; |
---|
| 699 | for (l=1;l<=e1;l=l+1) |
---|
| 700 | { |
---|
| 701 | Ms = Ms + var(l)*matrix(ideal(kb1[l]),f0,f1); |
---|
| 702 | Ls = Ls - var(l)*matrix(ideal(li1[l]),f1,f2); |
---|
| 703 | } |
---|
| 704 | return(Ms,Ls); |
---|
[82716e] | 705 | } |
---|
[f1201a] | 706 | ////////////////////////////////////////////////////////////////////////////// |
---|
| 707 | proc lift_rel_kb (module N, module M, list #) |
---|
[4ac997] | 708 | "USAGE: lift_rel_kb(N,M[,kbaseM,p]); |
---|
[3fe3582] | 709 | ASSUME: [p a monomial ] or the product of all variables |
---|
[b9b906] | 710 | N, M modules of same rank, M depending only on variables not in p |
---|
[4ac997] | 711 | and vdim(M) is finite in this ring, |
---|
[8675b0] | 712 | [ kbaseM the kbase of M in the subring given by variables not in p ] @* |
---|
| 713 | warning: these assumptions are not checked by the procedure |
---|
| 714 | RETURN: matrix A, whose j-th columns present the coeff's of N[j] in kbaseM, |
---|
[3fe3582] | 715 | i.e. kbaseM*A = reduce(N,std(M)) |
---|
| 716 | EXAMPLE: example lift_rel_kb; shows examples |
---|
[d2b2a7] | 717 | " |
---|
[f1201a] | 718 | { |
---|
| 719 | poly p = product(maxideal(1)); |
---|
| 720 | M = std(M); |
---|
[82716e] | 721 | matrix A; |
---|
[f1201a] | 722 | if (size(#)>0) { p=#[2]; module kbaseM=#[1];} |
---|
[82716e] | 723 | else |
---|
[f1201a] | 724 | { if (vdim(M)<=0) { "// vdim(M) not finite";return(A);} |
---|
| 725 | module kbaseM = kbase(M); |
---|
| 726 | } |
---|
| 727 | N = reduce(N,M); |
---|
| 728 | if (simplify(N,10)[1]==[0]) {return(A);} |
---|
| 729 | A = coeffs(N,kbaseM,p); |
---|
| 730 | return(A); |
---|
[82716e] | 731 | } |
---|
[3d124a7] | 732 | example |
---|
[f1201a] | 733 | { |
---|
[3fe3582] | 734 | "EXAMPLE:"; echo=2; |
---|
[f1201a] | 735 | ring r=0,(A,B,x,y),dp; |
---|
| 736 | module M = [x2,xy],[xy,y3],[y2],[0,x]; |
---|
| 737 | module kbaseM = [1],[x],[xy],[y],[0,1],[0,y],[0,y2]; |
---|
| 738 | poly f=xy; |
---|
| 739 | module N = [AB,BBy],[A3xy+x4,AB*(1+y2)]; |
---|
| 740 | matrix A = lift_rel_kb(N,M,kbaseM,f); |
---|
| 741 | print(A); |
---|
| 742 | "TEST:"; |
---|
| 743 | print(matrix(kbaseM)*A-matrix(reduce(N,std(M)))); |
---|
[82716e] | 744 | } |
---|
[f0c6f4] | 745 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 746 | proc lift_kbase (N, M) |
---|
[d2b2a7] | 747 | "USAGE: lift_kbase(N,M); N,M=poly/ideal/vector/module |
---|
[f0c6f4] | 748 | RETURN: matrix A, coefficient matrix expressing N as linear combination of |
---|
| 749 | k-basis of M. Let the k-basis have k elements and size(N)=c columns. |
---|
| 750 | Then A satisfies: |
---|
| 751 | matrix(reduce(N,std(M)),k,c) = matrix(kbase(std(M)))*A |
---|
| 752 | ASSUME: dim(M)=0 and the monomial ordering is a well ordering or the last |
---|
| 753 | block of the ordering is c or C |
---|
| 754 | EXAMPLE: example lift_kbase; shows an example |
---|
[d2b2a7] | 755 | " |
---|
[f0c6f4] | 756 | { |
---|
| 757 | return(lift_rel_kb(N,M)); |
---|
| 758 | } |
---|
| 759 | example |
---|
| 760 | {"EXAMPLE:"; echo=2; |
---|
| 761 | ring R=0,(x,y),ds; |
---|
| 762 | module M=[x2,xy],[y2,xy],[0,xx],[0,yy]; |
---|
| 763 | module N=[x3+xy,x],[x,x+y2]; |
---|
| 764 | print(M); |
---|
| 765 | module kb=kbase(std(M)); |
---|
| 766 | print(kb); |
---|
| 767 | print(N); |
---|
| 768 | matrix A=lift_kbase(N,M); |
---|
| 769 | print(A); |
---|
| 770 | matrix(reduce(N,std(M)),nrows(kb),ncols(A)) - matrix(kbase(std(M)))*A; |
---|
| 771 | } |
---|
| 772 | |
---|
| 773 | |
---|
[f1201a] | 774 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 775 | proc interact1 () |
---|
[d2b2a7] | 776 | " |
---|
[82716e] | 777 | Sub_procedure: asking for and reading your input-strings |
---|
[d2b2a7] | 778 | " |
---|
[f1201a] | 779 | { |
---|
| 780 | string my = "@"; |
---|
| 781 | string str,out,my_ord,my_var; |
---|
| 782 | my_ord = "ds"; |
---|
[82716e] | 783 | my_var = "A"; |
---|
[f1201a] | 784 | "INPUT: name of output-file (ENTER = no output, do not use \"my\"!)"; |
---|
[82716e] | 785 | str = read(""); |
---|
| 786 | if (size(str)>1) |
---|
[f1201a] | 787 | { out = str[1..size(str)-1];} |
---|
| 788 | else |
---|
| 789 | { out = "no";} |
---|
| 790 | "INPUT: prefix-string of ring-extension (ENTER = '@')"; |
---|
[82716e] | 791 | str = read(""); |
---|
| 792 | if ( size(str) > 1 ) |
---|
| 793 | { my = str[1..size(str)-1]; } |
---|
| 794 | "INPUT:parameter-string |
---|
[f1201a] | 795 | (give a letter corresponding to first new variable followed by the next letters, |
---|
| 796 | or 'T(' - a letter + '(' - getting a string of indexed variables) |
---|
| 797 | (ENTER = A) :"; |
---|
[82716e] | 798 | str = read(""); |
---|
[f1201a] | 799 | if (size(str)>1) { my_var=str[1..size(str)-1]; } |
---|
| 800 | "INPUT:order-string (local or weighted!) (ENTER = ds) :"; |
---|
[82716e] | 801 | str = read(""); |
---|
| 802 | if (size(str)>1) { my_ord=str[1..size(str)-1]; } |
---|
[f1201a] | 803 | if( find(my_ord,"s")+find(my_ord,"w") == 0 ) |
---|
| 804 | { "// ordering must be an local! changed into 'ds'"; |
---|
| 805 | my_ord = "ds"; |
---|
| 806 | } |
---|
| 807 | return(my,my_var,my_ord,out); |
---|
[3d124a7] | 808 | } |
---|
| 809 | /////////////////////////////////////////////////////////////////////////////// |
---|
[f1201a] | 810 | proc interact2 (matrix A, intvec col_vec, list #) |
---|
[d2b2a7] | 811 | " |
---|
[f1201a] | 812 | Sub-procedure: asking for and reading your input |
---|
[d2b2a7] | 813 | " |
---|
[3d124a7] | 814 | { |
---|
[f1201a] | 815 | module B,C; |
---|
| 816 | matrix D; |
---|
| 817 | int flag; |
---|
| 818 | if (size(#)>0) { D=#[1];flag=1;} |
---|
| 819 | int t1 = ncols(A); |
---|
| 820 | ">>Do you want all deformations? (ENTER=yes)"; |
---|
| 821 | string str = read(""); |
---|
[11dddeb] | 822 | if ((size(str)>1) and (str<>"yes")) |
---|
| 823 | { ">> Choose columns of the matrix"; |
---|
| 824 | ">> (Enter = all columns)"; |
---|
| 825 | "INPUT (number of columns to use as integer-list 'i_1,i_2,.. ,i_t' ):"; |
---|
[f1201a] | 826 | string columnes = read(""); |
---|
[11dddeb] | 827 | |
---|
| 828 | // improved: CL |
---|
| 829 | // ========================================================== |
---|
| 830 | // old: if (size(columnes)<2) {columnes=string(col_vec);} |
---|
| 831 | // t1 = size(columnes)/2; |
---|
| 832 | // new: |
---|
| 833 | if (columnes=="") |
---|
| 834 | { |
---|
| 835 | intvec vvvv=1..ncols(A); |
---|
| 836 | } |
---|
| 837 | else |
---|
| 838 | { |
---|
| 839 | execute("intvec vvvv="+columnes); |
---|
| 840 | } |
---|
| 841 | t1=size(vvvv); |
---|
| 842 | // ========================================================== |
---|
| 843 | |
---|
[f1201a] | 844 | int l,l1; |
---|
| 845 | for (l=1;l<=t1;l=l+1) |
---|
[3d124a7] | 846 | { |
---|
[11dddeb] | 847 | // old: execute("l1= "+columnes[2*l-1]+";"); |
---|
| 848 | l1=vvvv[l]; |
---|
[f1201a] | 849 | B[l] = A[l1]; |
---|
[82716e] | 850 | if(flag) { C[l]=D[l1];} |
---|
[3d124a7] | 851 | } |
---|
[f1201a] | 852 | A = matrix(B,nrows(A),size(B)); |
---|
| 853 | D = matrix(C,nrows(D),size(C)); |
---|
[6f2edc] | 854 | } |
---|
[f1201a] | 855 | return(A,D,t1); |
---|
[6f2edc] | 856 | } |
---|
[f1201a] | 857 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 858 | proc negative_part(intvec iv) |
---|
[d2b2a7] | 859 | " |
---|
[82716e] | 860 | RETURNS intvec of indices of jv having negative entries (or iv, if non) |
---|
[d2b2a7] | 861 | " |
---|
[f1201a] | 862 | { |
---|
| 863 | intvec jv; |
---|
| 864 | int l,k; |
---|
| 865 | for (l=1;l<=size(iv);l=l+1) |
---|
[82716e] | 866 | { if (iv[l]<0) |
---|
[f1201a] | 867 | { k = k+1; |
---|
| 868 | jv[k]=l; |
---|
| 869 | } |
---|
| 870 | } |
---|
[75089b] | 871 | if (jv==0) {jv=1; dbprint(printlevel-1,"// empty negative part, return all ");} |
---|
[f1201a] | 872 | return(jv); |
---|
[3d124a7] | 873 | } |
---|
| 874 | /////////////////////////////////////////////////////////////////////////////// |
---|
[f1201a] | 875 | proc find_ord(matrix A, intvec w_vec) |
---|
[d2b2a7] | 876 | " |
---|
[f1201a] | 877 | Sub-proc: return martix ord(a_ij) with respect to weight_vec, or |
---|
| 878 | 0 if A non-qh |
---|
[d2b2a7] | 879 | " |
---|
[f1201a] | 880 | { |
---|
| 881 | int @r = nrows(A); |
---|
| 882 | int @c = ncols(A); |
---|
| 883 | int i,j; |
---|
| 884 | string ord_str = "wp("+string(w_vec)+")"; |
---|
| 885 | def br = basering; |
---|
[daa83b] | 886 | def nr=changeord(ord_str); |
---|
| 887 | setring nr; |
---|
[f1201a] | 888 | matrix A = imap(br,A); |
---|
| 889 | intmat degA[@r][@c]; |
---|
[82716e] | 890 | if (homog(ideal(A))) |
---|
[f1201a] | 891 | { for (i=1;i<=@r;i=i+1) |
---|
| 892 | { for(j=1;j<=@c;j=j+1) |
---|
| 893 | { degA[i,j]=ord(A[i,j]); } |
---|
| 894 | } |
---|
| 895 | } |
---|
| 896 | setring br; |
---|
[13af82] | 897 | if(system("with","Namespaces")) { kill Top::nr; } |
---|
| 898 | if (defined(nr)) { kill nr; } |
---|
[f1201a] | 899 | return(degA); |
---|
| 900 | } |
---|
[4ac997] | 901 | /////////////////////////////////////////////////////////////////////////////// |
---|
[f1201a] | 902 | proc homog_test(intvec w_vec, matrix Mo, matrix A) |
---|
[d2b2a7] | 903 | " |
---|
[11dddeb] | 904 | Sub proc: return relative weight string of columns of A with respect |
---|
[82716e] | 905 | to the given w_vec and to Mo, or \"\" if not qh |
---|
[f1201a] | 906 | NOTE: * means weight is not determined |
---|
[d2b2a7] | 907 | " |
---|
[f1201a] | 908 | { |
---|
| 909 | int k,l; |
---|
| 910 | intvec tv; |
---|
| 911 | string @nv; |
---|
| 912 | int @r = nrows(A); |
---|
| 913 | int @c = ncols(A); |
---|
[82716e] | 914 | A = concat(matrix(ideal(Mo),@r,1),A); |
---|
| 915 | intmat a = find_ord(A,w_vec); |
---|
[f1201a] | 916 | intmat b[@r][@c]; |
---|
| 917 | for (l=1;l<=@c;l=l+1) |
---|
[82716e] | 918 | { |
---|
[f1201a] | 919 | for (k=1;k<=@r;k=k+1) |
---|
[82716e] | 920 | { if (A[k,l+1]!=0) |
---|
[f1201a] | 921 | { b[k,l] = a[k,l+1]-a[k,1];} |
---|
| 922 | } |
---|
| 923 | tv = 0; |
---|
| 924 | for (k=1;k<=@r;k=k+1) |
---|
[82716e] | 925 | { if (A[k,l+1]*A[k,1]!=0) |
---|
[f1201a] | 926 | {tv = tv,b[k,l];} |
---|
| 927 | } |
---|
| 928 | if (size(tv)>1) |
---|
[82716e] | 929 | { k = tv[2]; |
---|
[11dddeb] | 930 | tv = tv[2..size(tv)]; |
---|
| 931 | tv = tv -k; |
---|
[82716e] | 932 | if (tv==0) { @nv = @nv+string(-k)+",";} |
---|
[f1201a] | 933 | else {return("");} |
---|
| 934 | } |
---|
| 935 | else { @nv = @nv+"*,";} |
---|
| 936 | } |
---|
| 937 | @nv = @nv[1..size(@nv)-1]; |
---|
| 938 | return(@nv); |
---|
| 939 | } |
---|
[4ac997] | 940 | /////////////////////////////////////////////////////////////////////////////// |
---|
[f1201a] | 941 | proc homog_t(intvec d_vec, matrix Fo, matrix A) |
---|
[d2b2a7] | 942 | " |
---|
[82716e] | 943 | Sub-procedure: Computing relative (with respect to flatten(Fo)) weight_vec |
---|
[11dddeb] | 944 | of columns of A (return zero if Fo or A not qh) |
---|
[d2b2a7] | 945 | " |
---|
[f1201a] | 946 | { |
---|
| 947 | Fo = matrix(Fo,nrows(A),1); |
---|
| 948 | A = concat(Fo,A); |
---|
| 949 | A = transpose(A); |
---|
| 950 | def br = basering; |
---|
| 951 | string o_str = "wp("+string(d_vec)+")"; |
---|
[daa83b] | 952 | def nr=changeord(o_str); |
---|
| 953 | setring nr; |
---|
[f1201a] | 954 | module A = fetch(br,A); |
---|
| 955 | intvec dv; |
---|
| 956 | int l = homog(A) ; |
---|
[c67136] | 957 | if (l==0) { |
---|
| 958 | setring br; |
---|
[13af82] | 959 | if(system("with","Namespaces")) { kill Top::nr; } |
---|
| 960 | if (defined(nr)) { kill nr; } |
---|
[c67136] | 961 | return(l); |
---|
| 962 | } |
---|
[f1201a] | 963 | dv = attrib(A,"isHomog"); |
---|
| 964 | l = dv[1]; |
---|
| 965 | dv = dv[2..size(dv)]; |
---|
| 966 | dv = dv-l; |
---|
[82716e] | 967 | setring br; |
---|
[13af82] | 968 | if(system("with","Namespaces")) { kill Top::nr; } |
---|
| 969 | if (defined(nr)) { kill nr; } |
---|
[f1201a] | 970 | return(dv); |
---|
| 971 | } |
---|
[4ac997] | 972 | /////////////////////////////////////////////////////////////////////////////// |
---|