: - tag lMnt d ! - din R ECI ) easy ecm ) d- Here in H ) Take I - - PDF document

A Lech - Mumford constant ( work in progress with Ilya Smirnov ) Noetherian all rings are commutative , Throughout . , 1 with multiplicative identity . Lech 's inequality 4960 ) . m ) - local IER m - primary CR CA ) E d ! ECR ) LCRII ) ECI ) Then : - tag lMnt d ! - din R ECI ) easy ecm ) d- Here in H ) ① Take I - - M Example - i = elk ) is sharp when LHS * ) RHS =D ! elk ) del in G ) " ② 1=8 Take as > o = nd et ) LHS e CRI nd et ) RHS - when sharp ( p ) B - regular R -

ftp.mfd?I#jf(d--d.mR CR . mk local caulk ) Definition B the Lech - Mumford constant of R ) - Lech 's inequality ⇐ Guck ) seem Remade , ⑥ if de I GMCRKECR ) ② { " by considering 1=8 Caulk ) at n > so ② - Gauck ) ③ Caulk ) - Gm CR ) to denote e. CR ) Mumford use " of R " flat multiplicity - th called the o - tilt ) - EIR Eti further defined eick ) - - - - tilt ) = Cay CRAG - Zim ( Mumford 1977 ) LR . mt local Then e CR ) Z Caulk ) zcuuckft.lt/zCuuCRfti.tiHz---z/#-- so

- local is called food 1977 ) Definite . m ) ( R - • seLbe if = I GMCRATA ) and the sap • stable if semis table in the definition of GMC RETI ) is not attained Suppose Theorem C Mumford 1977 ) k= projective . - - ample . L ) If on X is L scheme Cx - . O x. a asymptoticakychowsemi-stable.TK en • semi stable for all KEN is Chow point Teak ) correspond charo * normal . prog . Q - Goemteh " ) to CX . L # . D= asymptotically theorem ( od aka 12 ) CX . x - log canonical Chow semis table . Then Ox x EX F

- X log resolution logcanoniewl Y → Speck ) - , . coeff of exceptional in Kya E - I is g- CR . m ) = normal - Smirnov 20 ) theorem ( Ma . ( essentially finite type ) Charo Q - Goren stein . then F- log canned then ① If E- semi stable , R= isolated singularity Cun CR ) =/ ② and R= canonical then

⇒ .ft 1=1 CR .mI= regular ⇒ ¥2 , ⑨ Example RB semi stable ( in fact , stable by Lech ' ) 1960 hypersurface of dim d CK . m )=kf × oij"XdI ⑦ - semi stable ⇒ degf Edit R - • CMS ] → deg fed CLMIR ) - I • CM - semis table ⇐ dim R -4 R ② - or R -=kC¥ [ Mumford ) Regular HI or p > 5 ③ CM char O dim R -2 - pg =p ADE or As . Do ( MS ) ⇐ E- regular CLMLRH . : k% ) E- semi stable incomplete list of candidates [ Mumford Ms ] . Shah . . Z ) kcxey x¥z