On Nash Blowing-Up
Heisuke Hironaka · Birkhäuser Boston eBooks · 1983
Let X be an algebraic variety, reduced and equidimensional, over the base field k of characteristic zero. Let us consider a sequence of transformations % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 % qacaWGybWdamaaBaaaleaapeGaaGimaaWdaeqaaOWdbiabg2da9iaa % dIfadaGduaWcpaqaa8qacqaHdpWCpaWaaSbaaWqaa8qacaaIXaaapa % qabaaal8qabeGccaGLqgcacaWGybWdamaaBaaaleaapeGaaGymaaWd % aeqaaOWdbmaaoqbal8aabaWdbiabeo8aZ9aadaWgaaadbaWdbiaaik % daa8aabeaaaSWdbeqakiaawcziaiaadIfapaWaaSbaaSqaa8qacaaI % YaaapaqabaGcpeGaeyiKHWQaeS47IWeaaa!4B25! $$ {X_0} = X\xleftarrow{{{\sigma _1}}}{X_1}\xleftarrow{{{\sigma _2}}}{X_2} \leftarrow \cdots $$ where % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 % qacqaHdpWCpaWaaSbaaSqaa8qacaWGPbaapaqabaGcpeGaaiOoaiaa % dIfapaWaaSbaaSqaa8qacaWGPbaapaqabaGcpeGaeyOKH4Qaamiwa8 % aadaWgaaWcbaWdbiaadMgacqGHsislcaaIXaaapaqabaaaaa!41F0! $$ {\sigma _i}:{X_i} \to {X_{i - 1}} $$ for each % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaiabgw % MiZkaaigdaaaa!3963! $$ i \geqslant 1 $$ is