Knowledge (XXG)

Hyperplane section

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and its refinements. Because the dimension drops by one in taking hyperplane sections, the process is potentially an inductive method for understanding varieties of higher dimension. A basic tool for that is the
258: 251: 191: 282: 244: 183: 155: 277: 178: 36: 96: 143: 115: 100: 88: 151: 111: 224: 150:) − 1. What more can be said is addressed by a collection of results known collectively as 123: 187: 228: 173: 160: 29: 201: 197: 119: 76: 271: 139: 216: 92: 17: 44: 154:. The topology of hyperplane sections is studied in the topic of the 106:
From a geometrical point of view, the most interesting case is when
232: 252: 8: 186:, vol. 52, New York: Springer-Verlag, 134:, a subvariety not lying completely in any 259: 245: 67:that satisfy the single linear condition 50:. In other words, we look at the subset 7: 213: 211: 231:. You can help Knowledge (XXG) by 14: 215: 138:, the hyperplane sections are 1: 184:Graduate Texts in Mathematics 114:; for more general cases, in 156:Lefschetz hyperplane theorem 299: 210: 126:, assuming therefore that 283:Algebraic geometry stubs 118:, some analogue of the 97:homogeneous coordinates 227:–related article is a 144:irreducible components 146:all of dimension dim( 116:mathematical analysis 101:scalar multiplication 89:dual projective space 112:algebraic subvariety 87:can range over the 278:Algebraic geometry 225:algebraic geometry 179:Algebraic Geometry 124:algebraic geometry 59:of those elements 22:hyperplane section 240: 239: 193:978-0-387-90244-9 174:Hartshorne, Robin 152:Bertini's theorem 290: 261: 254: 247: 219: 212: 204: 161:Lefschetz pencil 30:projective space 298: 297: 293: 292: 291: 289: 288: 287: 268: 267: 266: 265: 208: 194: 172: 169: 120:Radon transform 77:linear subspace 58: 12: 11: 5: 296: 294: 286: 285: 280: 270: 269: 264: 263: 256: 249: 241: 238: 237: 220: 206: 205: 192: 168: 165: 140:algebraic sets 54: 13: 10: 9: 6: 4: 3: 2: 295: 284: 281: 279: 276: 275: 273: 262: 257: 255: 250: 248: 243: 242: 236: 234: 230: 226: 221: 218: 214: 209: 203: 199: 195: 189: 185: 181: 180: 175: 171: 170: 166: 164: 162: 157: 153: 149: 145: 141: 137: 133: 129: 125: 121: 117: 113: 109: 104: 102: 98: 94: 90: 86: 82: 78: 74: 71:= 0 defining 70: 66: 62: 57: 53: 49: 46: 42: 38: 34: 31: 27: 23: 19: 233:expanding it 222: 207: 177: 147: 135: 131: 127: 122:applies. In 107: 105: 93:linear forms 91:of non-zero 84: 80: 72: 68: 64: 60: 55: 51: 47: 40: 37:intersection 32: 25: 24:of a subset 21: 15: 18:mathematics 272:Categories 167:References 45:hyperplane 43:with some 176:(1977), 99:, up to 202:0463157 95:in the 79:. Here 35:is the 200:  190:  110:is an 223:This 142:with 75:as a 229:stub 188:ISBN 20:, a 130:is 83:or 63:of 39:of 28:of 16:In 274:: 198:MR 196:, 182:, 163:. 103:. 260:e 253:t 246:v 235:. 148:V 136:H 132:V 128:X 108:X 85:H 81:L 73:H 69:L 65:X 61:x 56:H 52:X 48:H 41:X 33:P 26:X

Index

mathematics
projective space
intersection
hyperplane
linear subspace
dual projective space
linear forms
homogeneous coordinates
scalar multiplication
algebraic subvariety
mathematical analysis
Radon transform
algebraic geometry
algebraic sets
irreducible components
Bertini's theorem
Lefschetz hyperplane theorem
Lefschetz pencil
Hartshorne, Robin
Algebraic Geometry
Graduate Texts in Mathematics
ISBN
978-0-387-90244-9
MR
0463157
Stub icon
algebraic geometry
stub
expanding it
v

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