1 | //////////////////////////////////////////////////////////////////////////////// |
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2 | version="Id"; |
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3 | |
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4 | category="Algebraic Geometry"; |
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5 | info=" |
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6 | LIBRARY: Schubert.lib Proceduces for Intersection Theory |
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7 | |
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8 | AUTHOR: Hiep Dang, email: hiep@mathematik.uni-kl.de |
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9 | |
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10 | OVERVIEW: |
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11 | |
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12 | We implement two new classes (variety and sheaf) and methods for computing |
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13 | with them. Here a variety is represented by the dimension and the Chow ring. |
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14 | A sheaf on a variety is represented by the Chern character. |
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15 | |
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16 | In particular, we implement the concrete varieties such as projective spaces |
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17 | , Grassmannians, and projective bundles. Finally, the most important thing |
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18 | is a method for computing the intersection numbers (degrees of 0-cycles). |
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19 | |
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20 | PROCEDURES: |
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21 | makeVariety(int,ideal) create a variety |
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22 | productVariety(variety,variety) product of two varieties |
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23 | ChowRing(variety) create the Chow ring of a variety |
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24 | dimension(variety) dimension of a variety |
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25 | relations(variety) relations of a variety |
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26 | Grassmannian(int,int) create a Grassmannian as a variety |
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27 | projectiveSpace(int) create a projective space as a variety |
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28 | projectiveBundle(sheaf) create a projective bundle as a variety |
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29 | integral(variety,poly) degree of a 0-cycle on a variety |
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30 | makeSheaf(variety,poly) create a sheaf |
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31 | ChernCharacter(sheaf) the Chern character of a sheaf |
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32 | totalChernClass(sheaf) the total Chern class of a sheaf |
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33 | ChernClass(sheaf,int) the k-th Chern class of a sheaf |
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34 | topChernClass(sheaf) the top Chern class of a sheaf |
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35 | totalSegreClass(sheaf) the total Segre class of a sheaf |
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36 | dualSheaf(sheaf) the dual of a sheaf |
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37 | tensorSheaf(sheaf,sheaf) the tensor of two sheaves |
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38 | symmetricPowerSheaf(sheaf,int) the k-th symmetric power of a sheaf |
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39 | minusSheaf(sheaf,sheaf) the quotient of two sheaves |
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40 | plusSheaf(sheaf,sheaf) the direct sum of two sheaves |
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41 | |
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42 | REFERENCES: |
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43 | |
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44 | Hiep Dang, Intersection theory with applications to the computation of |
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45 | Gromov-Witten invariants, Ph.D thesis, TU Kaiserslautern, 2013. |
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46 | |
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47 | Sheldon Katz and Stein A. Stromme, Schubert-A Maple package for intersection |
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48 | theory and enumerative geometry, 1992. |
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49 | |
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50 | Daniel R. Grayson, Michael E. Stillman, Stein A. Stromme, David Eisenbud and |
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51 | Charley Crissman, Schubert2-A Macaulay2 package for computation in |
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52 | intersection theory, 2010. |
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53 | |
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54 | KEYWORDS: Intersection Theory, Enumerative Geometry, Schubert Calculus |
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55 | "; |
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56 | |
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57 | //////////////////////////////////////////////////////////////////////////////// |
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58 | |
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59 | LIB "general.lib"; |
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60 | LIB "homolog.lib"; |
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61 | |
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62 | //////////////////////////////////////////////////////////////////////////////// |
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63 | /// create new classes: varieties and sheaves ////////////////////////////////// |
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64 | //////////////////////////////////////////////////////////////////////////////// |
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65 | |
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66 | proc mod_init() |
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67 | { |
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68 | newstruct("variety","int dimension, ring baseRing, ideal relations"); |
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69 | newstruct("sheaf","variety currentVariety, poly ChernCharacter"); |
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70 | |
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71 | system("install","variety","print",variety_print,1); |
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72 | system("install","variety","*",productVariety,2); |
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73 | system("install","sheaf","print",sheaf_print,1); |
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74 | system("install","sheaf","*",tensorSheaf,2); |
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75 | system("install","sheaf","+",plusSheaf,2); |
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76 | system("install","sheaf","-",minusSheaf,2); |
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77 | } |
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78 | |
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79 | //////////////////////////////////////////////////////////////////////////////// |
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80 | /// Auxilary Static Procedures in this Library ///////////////////////////////// |
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81 | //////////////////////////////////////////////////////////////////////////////// |
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82 | //////// - part //////////////////////////////////////////////////////////// |
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83 | //////// - parts //////////////////////////////////////////////////////////// |
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84 | //////// - logg //////////////////////////////////////////////////////////// |
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85 | //////// - expp //////////////////////////////////////////////////////////// |
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86 | //////// - adams //////////////////////////////////////////////////////////// |
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87 | //////// - wedges //////////////////////////////////////////////////////////// |
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88 | //////// - schur //////////////////////////////////////////////////////////// |
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89 | //////////////////////////////////////////////////////////////////////////////// |
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90 | |
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91 | proc part(poly f, int n) |
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92 | "USAGE: part(f,n); f poly, n int |
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93 | RETURN: poly |
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94 | PURPOSE: return the homogeneous component of degree n of the polynomial f. |
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95 | EXAMPLE: example part; shows examples |
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96 | " |
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97 | { |
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98 | int i; |
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99 | poly p; |
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100 | for (i=1;i<=size(f);i++) |
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101 | { |
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102 | if (deg(f[i])==n) {p=p+f[i];} |
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103 | } |
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104 | return (p); |
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105 | } |
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106 | example |
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107 | { |
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108 | "EXAMPLE:"; echo=2; |
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109 | ring r = 0,(x,y,z),wp(1,2,3); |
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110 | poly f = 1+x+x2+x3+x4+y+y2+y3+z+z2+xy+xz+yz+xyz; |
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111 | part(f,0); |
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112 | part(f,1); |
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113 | part(f,2); |
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114 | part(f,3); |
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115 | part(f,4); |
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116 | part(f,5); |
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117 | part(f,6); |
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118 | } |
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119 | |
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120 | proc parts(poly f, int i, int j) |
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121 | "USAGE: parts(f,i,j); f poly, i int, j int |
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122 | RETURN: poly |
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123 | PURPOSE: return a polynomial which is the sum of the homogeneous components |
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124 | of degree from i to j of the polynomial f. |
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125 | EXAMPLE: example parts; shows examples |
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126 | " |
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127 | { |
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128 | int k; |
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129 | poly p; |
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130 | for (k=i;k<=j;k++) |
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131 | { |
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132 | p=p+part(f,k); |
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133 | } |
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134 | return (p); |
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135 | } |
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136 | example |
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137 | { |
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138 | "EXAMPLE:"; echo=2; |
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139 | ring r = 0,(x,y,z),wp(1,2,3); |
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140 | poly f = 1+x+x2+x3+x4+y+y2+y3+z+z2+xy+xz+yz+xyz; |
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141 | parts(f,2,4); |
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142 | } |
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143 | proc logg(poly f, int n) |
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144 | "USAGE: logg(f,n); f poly, n int |
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145 | RETURN: poly |
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146 | PURPOSE: computing the Chern character from the total Chern class. |
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147 | EXAMPLE: example logg; shows examples |
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148 | " |
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149 | { |
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150 | poly p; |
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151 | int i,j,k,m; |
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152 | if (n==0) {p=0;} |
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153 | if (n==1) {p=part(f,1);} |
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154 | else |
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155 | { |
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156 | list l=-part(f,1); |
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157 | for (j=2;j<=n;j++) |
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158 | { |
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159 | poly q; |
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160 | for (k=1;k<j;k++) |
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161 | { |
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162 | q=q+part(f,k)*l[j-k]; |
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163 | } |
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164 | q=-j*part(f,j)-q; |
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165 | l=insert(l,q,size(l)); |
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166 | kill q; |
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167 | } |
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168 | for (m=1;m<=n;m++) |
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169 | { |
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170 | p=p+1/factorial(m)*(-1)^m*l[m]; |
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171 | } |
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172 | } |
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173 | return (p); |
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174 | } |
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175 | example |
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176 | { |
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177 | "EXAMPLE:"; echo=2; |
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178 | ring r = 0,(x,y),wp(1,2); |
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179 | poly f = 1+x+y; |
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180 | logg(f,4); |
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181 | } |
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182 | |
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183 | proc expp(poly f, int n) |
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184 | "USAGE: expp(f,n); f poly, n int |
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185 | RETURN: poly |
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186 | PURPOSE: computing the total Chern class from the Chern character. |
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187 | EXAMPLE: example expp; shows examples |
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188 | " |
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189 | { |
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190 | poly p; |
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191 | int i,j,k; |
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192 | if (deg(f)==0) {p=1;} |
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193 | else |
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194 | { |
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195 | list l=1; |
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196 | for (i=1;i<=n;i++) |
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197 | { |
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198 | poly q; |
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199 | for (j=1;j<=i;j++) |
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200 | { |
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201 | q=q+factorial(j)*(-1)^(j-1)*l[i-j+1]*part(f,j)/i; |
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202 | } |
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203 | l=insert(l,q,size(l)); |
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204 | kill q; |
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205 | } |
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206 | for (k=1;k<=size(l);k++) |
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207 | { |
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208 | p=p+l[k]; |
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209 | } |
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210 | } |
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211 | return (p); |
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212 | } |
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213 | example |
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214 | { |
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215 | "EXAMPLE:"; echo=2; |
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216 | ring r = 0,(x),dp; |
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217 | poly f = 3+x; |
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218 | expp(f,3); |
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219 | } |
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220 | |
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221 | static proc adams(poly f, int n) |
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222 | { |
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223 | poly p; |
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224 | int i; |
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225 | for (i=0;i<=deg(f);i++) |
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226 | { |
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227 | p=p+n^i*part(f,i); |
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228 | } |
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229 | return (p); |
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230 | } |
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231 | |
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232 | static proc wedges(int n, poly f, int d) |
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233 | { |
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234 | int i,j; |
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235 | list l; |
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236 | if (n==0) {l=1;} |
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237 | if (n==1) {l=1,f;} |
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238 | else |
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239 | { |
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240 | l=1,f; |
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241 | for (i=2;i<=n;i++) |
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242 | { |
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243 | poly q; |
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244 | for (j=1;j<=i;j++) |
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245 | { |
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246 | q=q+((-1)^(i-j))*parts(l[j]*adams(f,i-j+1),0,d)/i; |
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247 | } |
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248 | l=insert(l,q,size(l)); |
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249 | kill q; |
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250 | } |
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251 | } |
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252 | return (l); |
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253 | } |
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254 | |
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255 | static proc schur(list p, poly f) |
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256 | { |
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257 | int i,j; |
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258 | int n = size(p); |
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259 | matrix M[n][n]; |
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260 | for (i=1;i<=n;i++) |
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261 | { |
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262 | for (j=1;j<=n;j++) |
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263 | { |
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264 | M[i,j] = part(f,p[i]+j-i); |
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265 | } |
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266 | } |
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267 | return (det(M)); |
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268 | } |
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269 | |
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270 | //////////////////////////////////////////////////////////////////////////////// |
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271 | //////// Procedures concerned with varieties /////////////////////////////////// |
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272 | //////////////////////////////////////////////////////////////////////////////// |
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273 | |
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274 | proc variety_print(variety V) |
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275 | { |
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276 | "A variety of dimension", V.dimension; |
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277 | } |
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278 | |
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279 | proc makeVariety(int d, ideal i) |
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280 | { |
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281 | def R = basering; |
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282 | variety V; |
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283 | V.dimension = d; |
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284 | V.baseRing = R; |
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285 | V.relations = i; |
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286 | return(V); |
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287 | } |
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288 | |
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289 | proc dimension(variety V) |
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290 | { |
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291 | return (V.dimension); |
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292 | } |
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293 | |
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294 | proc ChowRing(variety V) |
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295 | { |
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296 | def R = V.baseRing; |
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297 | setring R; |
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298 | ideal rels = V.relations; |
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299 | qring CR = std(rels); |
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300 | return (CR); |
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301 | } |
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302 | |
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303 | proc relations(variety V) |
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304 | { |
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305 | def R = V.baseRing; |
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306 | setring R; |
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307 | ideal i = V.relations; |
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308 | return (i); |
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309 | } |
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310 | |
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311 | //////////////////////////////////////////////////////////////////////////////// |
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312 | |
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313 | proc Grassmannian(int k, int n, list #) |
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314 | "USAGE: Grassmannian(k,n); k int, n int |
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315 | RETURN: variety |
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316 | PURPOSE: create a variety as a Grassmannian G(k,n). This variety has |
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317 | diemnsion k(n-k) and its Chow ring is the quotient ring of a |
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318 | polynomial ring in n-k variables q(1),...,q(n-k) which are the |
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319 | Chern classes of the tautological quotient bundle on the |
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320 | Grassmannian G(k,n), modulo the ideal generated by n-k polynomials |
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321 | which come from the Giambelli formula. The monomial ordering of this |
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322 | ring is 'wp' with vector (1..k,1..n-k). Moreover, we export the |
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323 | Chern characters of the tautological subbundle and quotient bundle |
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324 | on G(k,n) (say 'subBundle' and 'quotientBundle'). |
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325 | EXAMPLE: example Grassmannian; shows examples |
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326 | " |
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327 | { |
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328 | string q; |
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329 | if (size(#)==0) {q = "q";} |
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330 | else |
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331 | { |
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332 | if (typeof(#[1]) == "string") {q = #[1];} |
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333 | else {Error("invalid optional argument");} |
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334 | } |
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335 | variety G; |
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336 | G.dimension = k*(n-k); |
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337 | execute("ring r = 0,("+q+"(1..n-k)),wp(1..n-k);"); |
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338 | setring r; |
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339 | G.baseRing = r; |
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340 | int i,j; |
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341 | poly v = 1; |
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342 | poly u = 1; |
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343 | for (j=1;j<=n-k;j++) {v=v+q(j);} |
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344 | list l; |
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345 | for (i=1;i<=k;i++) |
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346 | { |
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347 | l=insert(l,1,size(l)); |
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348 | u=u+(-1)^i*schur(l,v); |
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349 | } |
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350 | l=insert(l,1,size(l)); |
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351 | ideal rels = schur(l,v); |
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352 | int h = k+2; |
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353 | while (h<=n) |
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354 | { |
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355 | l=insert(l,1,size(l)); |
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356 | rels = rels,schur(l,v); |
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357 | h++; |
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358 | } |
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359 | G.relations = rels; |
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360 | int d = k*(n-k); |
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361 | poly subBundle = reduce(logg(u,d)+k,std(rels)); |
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362 | poly quotientBundle = reduce(logg(v,d)+n-k,std(rels)); |
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363 | export (subBundle,quotientBundle); |
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364 | kill u,v,d,l,rels; |
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365 | return (G); |
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366 | } |
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367 | example |
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368 | { |
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369 | "EXAMPLE:"; echo=2; |
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370 | def G24 = Grassmannian(2,4); |
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371 | G24; |
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372 | def r = G24.baseRing; |
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373 | subBundle; |
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374 | quotientBundle; |
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375 | G24.dimension; |
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376 | G24.relations; |
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377 | ChowRing(G24); |
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378 | } |
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379 | |
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380 | proc projectiveSpace(int n, list #) |
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381 | "USAGE: projectiveSpace(n); n int |
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382 | RETURN: variety |
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383 | PURPOSE: create a variety as a projective space of dimension n. The Chow ring |
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384 | is the quotient ring in one variable h modulo the ideal generated by |
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385 | h^(n+1). |
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386 | EXAMPLE: example projectiveSpace; shows examples |
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387 | " |
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388 | { |
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389 | string h; |
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390 | if (size(#)==0) {h = "h";} |
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391 | else |
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392 | { |
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393 | if (typeof(#[1]) == "string") {h = #[1];} |
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394 | else {Error("invalid optional argument");} |
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395 | } |
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396 | variety P; |
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397 | P.dimension = n; |
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398 | execute("ring r = 0, ("+h+"), wp(1);"); |
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399 | setring r; |
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400 | P.baseRing = r; |
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401 | ideal rels = var(1)^(n+1); |
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402 | P.relations = rels; |
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403 | poly u = 1; |
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404 | poly v = 1 + var(1); |
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405 | list l; |
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406 | int i; |
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407 | for (i=1;i<=n;i++) |
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408 | { |
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409 | l=insert(l,1,size(l)); |
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410 | u=u+(-1)^i*schur(l,v); |
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411 | } |
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412 | poly subBundle = reduce(logg(u,n)+n,std(rels)); |
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413 | poly quotientBundle = reduce(logg(v,n)+1,std(rels)); |
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414 | export(subBundle,quotientBundle); |
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415 | kill rels,u,v,l; |
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416 | return (P); |
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417 | } |
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418 | example |
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419 | { |
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420 | "EXAMPLE:"; echo=2; |
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421 | def P = projectiveSpace(3); |
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422 | P; |
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423 | P.dimension; |
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424 | def r = P.baseRing; |
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425 | setring r; |
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426 | P.relations; |
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427 | ChowRing(P); |
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428 | } |
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429 | |
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430 | proc projectiveBundle(sheaf S, list #) |
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431 | "USAGE: projectiveBundle(S); S sheaf |
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432 | RETURN: variety |
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433 | PURPOSE: create a variety which we work on. |
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434 | EXAMPLE: example projectiveBundle; shows examples |
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435 | " |
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436 | { |
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437 | string z; |
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438 | if (size(#)==0) {z = "z";} |
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439 | else |
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440 | { |
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441 | if (typeof(#[1]) == "string") {z = #[1];} |
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442 | else {Error("invalid optional argument");} |
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443 | } |
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444 | variety A; |
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445 | def B = S.currentVariety; |
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446 | def R = B.baseRing; |
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447 | setring R; |
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448 | ideal rels = B.relations; |
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449 | int r = rankSheaf(S); |
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450 | A.dimension = r - 1 + B.dimension; |
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451 | poly c = totalChernClass(S); |
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452 | execute("ring P = 0, ("+z+"), wp(1);"); |
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453 | def CR = P + R; |
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454 | setring CR; |
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455 | A.baseRing = CR; |
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456 | poly c = imap(R,c); |
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457 | ideal rels = imap(R,rels); |
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458 | poly g = var(1)^r; |
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459 | int i; |
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460 | for (i=1;i<=r;i++) {g=g+var(1)^(r-i)*part(c,i);} |
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461 | A.relations = rels,g; |
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462 | poly u = 1 + var(1); |
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463 | poly f = logg(u,A.dimension)+1; |
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464 | poly QuotientBundle = reduce(f,std(A.relations)); |
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465 | export (QuotientBundle); |
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466 | kill f,rels; |
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467 | return (A); |
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468 | } |
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469 | example |
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470 | { |
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471 | "EXAMPLE:"; echo=2; |
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472 | def G35 = Grassmannian(3,5); |
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473 | def R = G35.baseRing; |
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474 | setring R; |
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475 | def S = makeSheaf(G35, subBundle); |
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476 | def B = symmetricPowerSheaf(dualSheaf(S),2); |
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477 | def PB = projectiveBundle(B); |
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478 | PB; |
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479 | def P = PB.baseRing; |
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480 | setring P; |
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481 | QuotientBundle; |
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482 | ChowRing(PB); |
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483 | } |
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484 | |
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485 | proc productVariety(variety U, variety V) |
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486 | { |
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487 | //def br = basering; |
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488 | def ur = U.baseRing; setring ur; |
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489 | ideal ii1 = U.relations; |
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490 | def vr = V.baseRing; setring vr; |
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491 | ideal ii2 = V.relations; |
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492 | variety W; |
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493 | W.dimension = U.dimension + V.dimension; |
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494 | def temp = ringtensor(ur,vr); |
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495 | setring temp; |
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496 | W.baseRing = temp; |
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497 | ideal i1 = imap(ur,ii1); |
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498 | ideal i2 = imap(vr,ii2); |
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499 | W.relations = i1 + i2; |
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500 | //setring br; |
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501 | return (W); |
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502 | } |
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503 | |
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504 | //////////////////////////////////////////////////////////////////////////////// |
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505 | |
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506 | proc integral(variety V, poly f) |
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507 | "USAGE: integral(V,f); V variety, f poly |
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508 | RETURN: int |
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509 | PURPOSE: compute the intersection numbers. |
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510 | EXAMPLE: example integral; shows examples |
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511 | " |
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512 | { |
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513 | def R = V.baseRing; |
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514 | setring R; |
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515 | ideal rels = V.relations; |
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516 | return (leadcoef(reduce(f,std(rels)))); |
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517 | } |
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518 | example |
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519 | { |
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520 | "EXAMPLE:"; echo=2; |
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521 | def G24 = Grassmannian(2,4); |
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522 | def R = G24.baseRing; |
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523 | setring R; |
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524 | integral(G24,q(1)^4); |
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525 | } |
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526 | |
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527 | proc SchubertClass(list p) |
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528 | "USAGE: SchubertClass(p); p list |
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529 | RETURN: poly |
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530 | PURPOSE: compute the Schubert class on a Grassmannian. |
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531 | EXAMPLE: example SchubertClass; shows examples |
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532 | " |
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533 | { |
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534 | def R = basering; |
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535 | setring R; |
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536 | poly f = 1; |
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537 | if (size(p) == 0) {return (f);} |
---|
538 | int i; |
---|
539 | for (i=1;i<=nvars(R);i++) |
---|
540 | { |
---|
541 | f = f + var(i); |
---|
542 | } |
---|
543 | return (schur(p,f)); |
---|
544 | } |
---|
545 | example |
---|
546 | { |
---|
547 | "EXAMPLE:"; echo=2; |
---|
548 | def G = Grassmannian(2,4); |
---|
549 | def r = G.baseRing; |
---|
550 | setring r; |
---|
551 | list p = 1,1; |
---|
552 | SchubertClass(p); |
---|
553 | } |
---|
554 | |
---|
555 | //////////////////////////////////////////////////////////////////////////////// |
---|
556 | |
---|
557 | proc dualPartition(int k, int n, list p) |
---|
558 | { |
---|
559 | while (size(p) < k) |
---|
560 | { |
---|
561 | p = insert(p,0,size(p)); |
---|
562 | } |
---|
563 | int i; |
---|
564 | list l; |
---|
565 | for (i=1;i<=size(p);i++) |
---|
566 | { |
---|
567 | l[i] = n-k-p[size(p)-i+1]; |
---|
568 | } |
---|
569 | return (l); |
---|
570 | } |
---|
571 | |
---|
572 | //////////////////////////////////////////////////////////////////////////////// |
---|
573 | |
---|
574 | //////////////////////////////////////////////////////////////////////////////// |
---|
575 | ////////// Procedures concerned with sheaves /////////////////////////////////// |
---|
576 | //////////////////////////////////////////////////////////////////////////////// |
---|
577 | |
---|
578 | //////////////////////////////////////////////////////////////////////////////// |
---|
579 | |
---|
580 | proc sheaf_print(sheaf S) |
---|
581 | { |
---|
582 | def V = S.currentVariety; |
---|
583 | def R = V.baseRing; |
---|
584 | setring R; |
---|
585 | poly f = S.ChernCharacter; |
---|
586 | "A sheaf of rank ", int(part(f,0)); |
---|
587 | } |
---|
588 | |
---|
589 | proc makeSheaf(variety V, poly ch) |
---|
590 | { |
---|
591 | def R = basering; |
---|
592 | sheaf S; |
---|
593 | S.currentVariety = V; |
---|
594 | S.ChernCharacter = ch; |
---|
595 | return(S); |
---|
596 | } |
---|
597 | |
---|
598 | proc currentVariety(sheaf S) |
---|
599 | { |
---|
600 | return (S.currentVariety); |
---|
601 | } |
---|
602 | |
---|
603 | proc ChernCharacter(sheaf S) |
---|
604 | { |
---|
605 | return (S.ChernCharacter) |
---|
606 | } |
---|
607 | |
---|
608 | proc rankSheaf(sheaf S) |
---|
609 | "USAGE: rankSheaf(S); S sheaf |
---|
610 | RETURN: int |
---|
611 | INPUT: S -- a sheaf |
---|
612 | OUTPUT: an integer which is the rank of the sheaf S. |
---|
613 | EXAMPLE: example rankSheaf(S); shows examples |
---|
614 | " |
---|
615 | { |
---|
616 | def V = S.currentVariety; |
---|
617 | def R = V.baseRing; |
---|
618 | setring R; |
---|
619 | poly f = S.ChernCharacter; |
---|
620 | return (int(part(f,0))); |
---|
621 | } |
---|
622 | example |
---|
623 | { |
---|
624 | "EXAMPLE:"; echo=2; |
---|
625 | def G24 = Grassmannian(2,4); |
---|
626 | def R = G24.baseRing; |
---|
627 | setring R; |
---|
628 | def S = makeSheaf(G24,subBundle); |
---|
629 | rankSheaf(S); |
---|
630 | } |
---|
631 | |
---|
632 | proc totalChernClass(sheaf S) |
---|
633 | "USAGE: totalChernClass(S); f sheaf |
---|
634 | RETURN: poly |
---|
635 | INPUT: S -- a sheaf |
---|
636 | OUTPUT: a polynomial which is the total Chern class of the sheaf S |
---|
637 | EXAMPLE: example totalChernClass(S); shows examples |
---|
638 | " |
---|
639 | { |
---|
640 | def V = S.currentVariety; |
---|
641 | int d = V.dimension; |
---|
642 | def R = V.baseRing; |
---|
643 | setring R; |
---|
644 | poly ch = S.ChernCharacter; |
---|
645 | poly f = expp(ch,d); |
---|
646 | ideal rels = std(V.relations); |
---|
647 | return (reduce(f,rels)); |
---|
648 | } |
---|
649 | |
---|
650 | proc ChernClass(sheaf S, int i) |
---|
651 | { |
---|
652 | return (part(totalChernClass(S),i)); |
---|
653 | } |
---|
654 | |
---|
655 | proc topChernClass(sheaf S) |
---|
656 | "USAGE: topChernClass(S); S sheaf |
---|
657 | RETURN: poly |
---|
658 | INPUT: S -- a sheaf |
---|
659 | OUTPUT: the top Chern class of the sheaf S |
---|
660 | EXAMPLE: example topChernClass(S); shows examples |
---|
661 | " |
---|
662 | { |
---|
663 | return (ChernClass(S,rankSheaf(S))); |
---|
664 | } |
---|
665 | example |
---|
666 | { |
---|
667 | "EXAMPLE:"; echo=2; |
---|
668 | def G24 = Grassmannian(2,4); |
---|
669 | def R = G24.baseRing; |
---|
670 | setring R; |
---|
671 | def S = makeSheaf(G24,quotientBundle); |
---|
672 | def B = symmetricPowerSheaf(S,3); |
---|
673 | topChernClass(B); |
---|
674 | } |
---|
675 | |
---|
676 | proc totalSegreClass(sheaf S) |
---|
677 | "USAGE: totalSegreClass(S); S sheaf |
---|
678 | RETURN: poly |
---|
679 | INPUT: S -- a sheaf |
---|
680 | OUTPUT: a polynomial which is the total Segre class of the sheaf S. |
---|
681 | EXAMPLE: example totalSegreClass(S); shows examples |
---|
682 | " |
---|
683 | { |
---|
684 | //def D = dualSheaf(S); |
---|
685 | def V = S.currentVariety; |
---|
686 | def R = V.baseRing; |
---|
687 | setring R; |
---|
688 | poly f = totalChernClass(S); |
---|
689 | poly g; |
---|
690 | int d = V.dimension; |
---|
691 | ideal rels = std(V.relations); |
---|
692 | if (f == 1) {return (1);} |
---|
693 | else |
---|
694 | { |
---|
695 | poly t,h; |
---|
696 | int i,j; |
---|
697 | for (i=0;i<=d;i++) {t = t + (1-f)^i;} |
---|
698 | for (j=0;j<=d;j++) {h = h + part(t,j);} |
---|
699 | return (reduce(h,rels)); |
---|
700 | } |
---|
701 | } |
---|
702 | example |
---|
703 | { |
---|
704 | "EXAMPLE:"; echo=2; |
---|
705 | def G24 = Grassmannian(2,4); |
---|
706 | def R = G24.baseRing; |
---|
707 | setring R; |
---|
708 | def S = makeSheaf(G24,subBundle); |
---|
709 | totalSegreClass(S); |
---|
710 | } |
---|
711 | |
---|
712 | proc dualSheaf(sheaf S) |
---|
713 | "USAGE: dualSheaf(S); S sheaf |
---|
714 | RETURN: sheaf |
---|
715 | INPUT: S -- a sheaf |
---|
716 | OUTPUT: the dual of the sheaf S |
---|
717 | EXAMPLE: example dualSheaf(S); shows examples |
---|
718 | " |
---|
719 | { |
---|
720 | def V = S.currentVariety; |
---|
721 | int d = V.dimension; |
---|
722 | def R = V.baseRing; |
---|
723 | setring R; |
---|
724 | poly ch = S.ChernCharacter; |
---|
725 | poly f = adams(ch,-1); |
---|
726 | sheaf D; |
---|
727 | D.currentVariety = V; |
---|
728 | D.ChernCharacter = f; |
---|
729 | return (D); |
---|
730 | } |
---|
731 | example |
---|
732 | { |
---|
733 | "EXAMPLE:"; echo=2; |
---|
734 | def G24 = Grassmannian(2,4); |
---|
735 | def R = G24.baseRing; |
---|
736 | setring R; |
---|
737 | def S = makeSheaf(G24,subBundle); |
---|
738 | dualSheaf(S); |
---|
739 | } |
---|
740 | |
---|
741 | proc tensorSheaf(sheaf A, sheaf B) |
---|
742 | "USAGE: tensorSheaf(A,B); A sheaf, B sheaf |
---|
743 | RETURN: sheaf |
---|
744 | INPUT: A, B -- two sheaves |
---|
745 | OUTPUT: the tensor of two sheaves A and B |
---|
746 | EXAMPLE: example tensorSheaf(A,B); shows examples |
---|
747 | " |
---|
748 | { |
---|
749 | sheaf S; |
---|
750 | def V1 = A.currentVariety; |
---|
751 | def V2 = B.currentVariety; |
---|
752 | def R1 = V1.baseRing; |
---|
753 | setring R1; |
---|
754 | poly c1 = A.ChernCharacter; |
---|
755 | def R2 = V2.baseRing; |
---|
756 | setring R2; |
---|
757 | poly c2 = B.ChernCharacter; |
---|
758 | if (nvars(R1) < nvars(R2)) |
---|
759 | { |
---|
760 | S.currentVariety = V2; |
---|
761 | poly c = imap(R1,c1); |
---|
762 | poly f = parts(c*c2,0,V2.dimension); |
---|
763 | S.ChernCharacter = f; |
---|
764 | return (S); |
---|
765 | } |
---|
766 | else |
---|
767 | { |
---|
768 | setring R1; |
---|
769 | S.currentVariety = V1; |
---|
770 | poly c = imap(R2,c2); |
---|
771 | poly f = parts(c1*c,0,V1.dimension); |
---|
772 | S.ChernCharacter = f; |
---|
773 | return (S); |
---|
774 | } |
---|
775 | } |
---|
776 | example |
---|
777 | { |
---|
778 | "EXAMPLE:"; echo=2; |
---|
779 | def G34 = Grassmannian(3,4); |
---|
780 | def R = G34.baseRing; |
---|
781 | setring R; |
---|
782 | def S = makeSheaf(G34,subBundle); |
---|
783 | def Q = makeSheaf(G34, quotientBundle); |
---|
784 | S*Q; |
---|
785 | } |
---|
786 | |
---|
787 | proc symmetricPowerSheaf(sheaf S, int n) |
---|
788 | "USAGE: symmetricPowerSheaf(S,n); S sheaf, n int |
---|
789 | RETURN: sheaf |
---|
790 | INPUT: S -- a sheaf |
---|
791 | n -- an integer |
---|
792 | OUTPUT: the n-th symmetric power of the sheaf S |
---|
793 | EXAMPLE: example symmetricPowerSheaf(S,n); shows examples |
---|
794 | " |
---|
795 | { |
---|
796 | def V = S.currentVariety; |
---|
797 | def R = V.baseRing; |
---|
798 | setring R; |
---|
799 | int r = rankSheaf(S); |
---|
800 | int d = V.dimension; |
---|
801 | int i,j,m; |
---|
802 | poly f = S.ChernCharacter; |
---|
803 | poly result; |
---|
804 | list s,w; |
---|
805 | if (n==0) {result=1;} |
---|
806 | if (n==1) {result=f;} |
---|
807 | else |
---|
808 | { |
---|
809 | s = 1,f; |
---|
810 | w = wedges(n,f,d); |
---|
811 | for (i=2;i<=n;i++) |
---|
812 | { |
---|
813 | if (i<=r) {m=i;} |
---|
814 | else {m=r;} |
---|
815 | poly q; |
---|
816 | for (j=1;j<=m;j++) |
---|
817 | { |
---|
818 | q = q + ((-1)^(j+1))*parts(w[j+1]*s[i-j+1],0,d); |
---|
819 | } |
---|
820 | s = insert(s,q,size(s)); |
---|
821 | kill q; |
---|
822 | } |
---|
823 | result = s[n+1]; |
---|
824 | } |
---|
825 | sheaf A; |
---|
826 | A.currentVariety = V; |
---|
827 | A.ChernCharacter = result; |
---|
828 | return (A); |
---|
829 | } |
---|
830 | example |
---|
831 | { |
---|
832 | "EXAMPLE:"; echo=2; |
---|
833 | def G24 = Grassmannian(2,4); |
---|
834 | def R = G24.baseRing; |
---|
835 | setring R; |
---|
836 | def S = makeSheaf(G24,quotientBundle); |
---|
837 | def B = symmetricPowerSheaf(S,3); |
---|
838 | B; |
---|
839 | } |
---|
840 | |
---|
841 | proc minusSheaf(sheaf A, sheaf B) |
---|
842 | { |
---|
843 | sheaf S; |
---|
844 | def V1 = A.currentVariety; |
---|
845 | def V2 = B.currentVariety; |
---|
846 | def R1 = V1.baseRing; |
---|
847 | setring R1; |
---|
848 | poly c1 = A.ChernCharacter; |
---|
849 | def R2 = V2.baseRing; |
---|
850 | setring R2; |
---|
851 | poly c2 = B.ChernCharacter; |
---|
852 | if (nvars(R1) < nvars(R2)) |
---|
853 | { |
---|
854 | S.currentVariety = V2; |
---|
855 | poly c = imap(R1,c1); |
---|
856 | S.ChernCharacter = c - c2; |
---|
857 | return (S); |
---|
858 | } |
---|
859 | else |
---|
860 | { |
---|
861 | setring R1; |
---|
862 | S.currentVariety = V1; |
---|
863 | poly c = imap(R2,c2); |
---|
864 | S.ChernCharacter = c1 - c; |
---|
865 | return (S); |
---|
866 | } |
---|
867 | } |
---|
868 | |
---|
869 | proc plusSheaf(sheaf A, sheaf B) |
---|
870 | { |
---|
871 | sheaf S; |
---|
872 | def V1 = A.currentVariety; |
---|
873 | def V2 = B.currentVariety; |
---|
874 | def R1 = V1.baseRing; |
---|
875 | setring R1; |
---|
876 | poly c1 = A.ChernCharacter; |
---|
877 | def R2 = V2.baseRing; |
---|
878 | setring R2; |
---|
879 | poly c2 = B.ChernCharacter; |
---|
880 | if (nvars(R1) < nvars(R2)) |
---|
881 | { |
---|
882 | S.currentVariety = V2; |
---|
883 | poly c = imap(R1,c1); |
---|
884 | S.ChernCharacter = c + c2; |
---|
885 | return (S); |
---|
886 | } |
---|
887 | else |
---|
888 | { |
---|
889 | setring R1; |
---|
890 | S.currentVariety = V1; |
---|
891 | poly c = imap(R2,c2); |
---|
892 | S.ChernCharacter = c1 + c; |
---|
893 | return (S); |
---|
894 | } |
---|
895 | } |
---|
896 | |
---|
897 | //////////////////////////////////////////////////////////////////////////////// |
---|
898 | |
---|
899 | proc geometricMultiplicity(ideal I, ideal J) |
---|
900 | "USAGE: geometricMultiplicity(I,J); I,J = ideals |
---|
901 | RETURN: int |
---|
902 | INPUT: I, J -- two ideals |
---|
903 | OTPUT: an integer which is the intersection multiplicity of two subvarieties |
---|
904 | defined by the ideals I, J at the origin. |
---|
905 | EXAMPLE: example geometricMultiplicity(I,J); shows examples |
---|
906 | " |
---|
907 | { |
---|
908 | def R = basering; |
---|
909 | setring R; |
---|
910 | ideal K = I + J; |
---|
911 | int v = vdim(std(K)); |
---|
912 | return (v); |
---|
913 | } |
---|
914 | example |
---|
915 | { |
---|
916 | "EXAMPLE:"; echo=2; |
---|
917 | ring r = 0, (x,y,z,w), ds; |
---|
918 | ideal I = xz,xw,yz,yw; |
---|
919 | ideal J = x-z,y-w; |
---|
920 | geometricMultiplicity(I,J); |
---|
921 | } |
---|
922 | |
---|
923 | //////////////////////////////////////////////////////////////////////////////// |
---|
924 | |
---|
925 | proc serreMultiplicity(ideal I, ideal J) |
---|
926 | "USAGE: serreMultiplicity(I,J); I,J = ideals |
---|
927 | RETURN: int |
---|
928 | INPUT: I, J -- two ideals |
---|
929 | OUTPUT: the intersection multiplicity (defined by J. P. Serre) |
---|
930 | of two subvarieties defined by the ideals I, J at the origin. |
---|
931 | EXAMPLE: example serreMultiplicity(I,J); shows examples |
---|
932 | " |
---|
933 | { |
---|
934 | def R = basering; |
---|
935 | setring R; |
---|
936 | int i = 0; |
---|
937 | int s = 0; |
---|
938 | module m = std(Tor(i,I,J)); |
---|
939 | int t = sum(hilb(m,2)); |
---|
940 | kill m; |
---|
941 | while (t != 0) |
---|
942 | { |
---|
943 | s = s + ((-1)^i)*t; |
---|
944 | i++; |
---|
945 | module m = std(Tor(i,I,J)); |
---|
946 | t = sum(hilb(m,2)); |
---|
947 | kill m; |
---|
948 | } |
---|
949 | return (s); |
---|
950 | } |
---|
951 | example |
---|
952 | { |
---|
953 | "EXAMPLE:"; echo=2; |
---|
954 | ring r = 0, (x,y,z,w), ds; |
---|
955 | ideal I = xz,xw,yz,yw; |
---|
956 | ideal J = x-z,y-w; |
---|
957 | serreMultiplicity(I,J); |
---|
958 | } |
---|