Generalized Multiplicities and Depth of Blowup Algebras Jonathan Monta˜ no Purdue University - University of Kansas Midwest Commutative Algebra Conference at Purdue f August 5, 2015 Jonathan Monta˜ no (PU-KU) Generalized Multiplicities August 5, 2015 1 / 36
Setting ( R , m , k ) Cohen-Macaulay (CM) local ring. | k | = ∞ , dim( R ) = d > 0. I an R -ideal. J ⊆ I is a minimal reduction of I , i.e., I n +1 = JI n for some n ∈ N and J is minimal with respect to inclusion. ℓ ( I ) is the analytic spread of I . Recall µ ( J ) = ℓ ( I ). r ( I ) = min { n | I n +1 = JI n , for some minimal reduction J } , the reduction number of I . Jonathan Monta˜ no (PU-KU) Generalized Multiplicities August 5, 2015 2 / 36
MULTIPLICITIES Jonathan Monta˜ no (PU-KU) Generalized Multiplicities August 5, 2015 3 / 36
Hilbert-Samuel multiplicity If I is m -primary, ( d − 1)! λ R ( I n / I n +1 ) e ( I ) = lim n d − 1 n →∞ d ! n d λ R ( R / I n ) = lim n →∞ is the Hilbert-Samuel multiplicity of I . If I is not m -primary, then λ R ( I n / I n +1 ) = ∞ , and λ R ( R / I n ) = ∞ . Jonathan Monta˜ no (PU-KU) Generalized Multiplicities August 5, 2015 4 / 36
Generalized multiplicities We use H 0 m ( M ) = 0 : M m ∞ , the 0 th -local cohomology of the R -module M , the largest finite length submodule of M . We obtain: � � ( d − 1)! H 0 m ( I n / I n +1 ) j ( I ) = lim λ R , n d − 1 n →∞ the j -muliplicity of I (Achilles-Maneresi, 1993). � � d ! H 0 m ( R / I n ) ε ( I ) = lim sup n d λ R , n →∞ the ε -muliplicity of I (Ulrich-Validashti, 2011). The limit exists when R is analytically unramified (Cutkosky, 2014). Jonathan Monta˜ no (PU-KU) Generalized Multiplicities August 5, 2015 5 / 36
Some properties j ( I ) ∈ Z � 0 . 1 ε ( I ) can be irrational (Cutkosky-H` a-Srinivasan-Theodorescu, 2005). 2 j ( I ) > 0 ⇔ ε ( I ) > 0 ⇔ ℓ ( I ) = d . 3 ε ( I ) � j ( I ) . 4 If I is m -primary ⇒ j ( I ) = ε ( I ) = e ( I ) . 5 Jonathan Monta˜ no (PU-KU) Generalized Multiplicities August 5, 2015 6 / 36
Applications j -multiplicity: Intersection theory (Achilles-Manaresi, 1993). Numerical criterion for integral dependence (Rees’ Theorem): If J ⊆ I , then I ⊆ J ⇔ j ( I p ) = j ( J p ) , ∀ p ∈ Spec R (Flenner-Manaresi, 2001) . Conditions for Cohen-Macaulayness of blowup algebras (Polini-Xie, 2013), (Mantero-Xie, 2014), (M, 2015). ε -multiplicity: Rees’ Theorem for ideals and modules (Ulrich-Validashti, 2011). Equisingularity Theory (Kleiman-Ulrich-Validashti). Jonathan Monta˜ no (PU-KU) Generalized Multiplicities August 5, 2015 7 / 36
COMPUTING GENERALIZED MULTIPLICITIES (with Jack Jeffries and Matteo Varbaro) Jonathan Monta˜ no (PU-KU) Generalized Multiplicities August 5, 2015 8 / 36
Computability Despite their importance, the generalized multiplicities are not easy to compute. The following formula expresses the j -multiplicity as the length of a module: � � R j ( I ) = λ R ( a 1 , a 2 , . . . , a d − 1 ) : I ∞ + ( a d ) for a 1 , a 2 , . . . , a d general elements in I . (Achilles-Manaresi 1993, Xie 2012) The ε -multiplicity has a better behavior than the j -multiplicity in some aspects, but it is harder to compute. Goal: Compute generalized multiplicities for large classes of ideals. Jonathan Monta˜ no (PU-KU) Generalized Multiplicities August 5, 2015 9 / 36
Teissier’s Theorem Let R = k [ x 1 , x 2 , . . . , x d ] and let I be a monomial ideal minimally generated by x v 1 , . . . , x v n , where x v = x v 1 1 · · · x v d if v = ( v 1 , . . . , v d ). d The Newton polyhedron of I is defined to be the following convex region: conv( I ) := conv( v 1 , . . . , v n ) + R d � 0 . We have x v ∈ I if and only if v ∈ conv( I ). Assume I is m -primary (i.e., I contains pure powers on each variable). Then � � R d covol( I ) := vol � 0 \ conv( I ) is finite. Theorem (Teissier, 1988) Let I be an m -primary monomial ideal, then e ( I ) = d ! covol( I ) . Jonathan Monta˜ no (PU-KU) Generalized Multiplicities August 5, 2015 10 / 36
Example The following picture corresponds to the ideal I = ( x 7 , x 2 y 2 , xy 5 , y 6 ). conv( I ) is the yellow region., and covol( I ) is the volume of the green region. e ( I ) = 2! covol( I ) = 26 . Jonathan Monta˜ no (PU-KU) Generalized Multiplicities August 5, 2015 11 / 36
The j -multiplicity of monomial ideals Let I be an arbitrary monomial ideal (not necessarily m -primary). If { P 1 , . . . , P b } are the bounded faces of dimension d − 1 of conv( I ), we call the region b � pyr( I ) = conv( P i , 0) , i =1 the pyramid of I . Theorem (Jeffries-M, 2013) � � j ( I ) = d ! vol pyr( I ) . Jonathan Monta˜ no (PU-KU) Generalized Multiplicities August 5, 2015 12 / 36
Example The following picture corresponds to the ideal I = ( xy 5 , x 2 y 3 , x 3 y 2 ). conv( I ) is the yellow region., and pyr( I ) is the green region. � � j ( I ) = 2!vol pyr( I ) = 6 Jonathan Monta˜ no (PU-KU) Generalized Multiplicities August 5, 2015 13 / 36
The ε -multiplicity of monomial ideals Let H i = { x ∈ R n | � x , b i � = c i } , with b i ∈ Q d , c i ∈ Q for i = 1 , . . . , w be the supporting hyperplanes of conv( I ) such that conv( I ) = H + 1 ∩ H + 2 ∩ · · · ∩ H + w . Assume that H 1 , . . . , H u , are the hyperplanes corresponding to unbounded facets and define out( I ) = ( H + 1 ∩ · · · ∩ H + u ) ∩ ( H − u +1 ∪ · · · ∪ H − w ) . Theorem (Jeffries-Monta˜ no, 2013) � � ε ( I ) = d ! vol out( I ) . Jonathan Monta˜ no (PU-KU) Generalized Multiplicities August 5, 2015 14 / 36
Example Let following picture corresponds to the ideal I = ( y 4 , x 2 y , xy 2 ). pyr( I ) is the green region and out( I ) is the portion of the green region that lies above the dotted line. � � � � j ( I ) = 2! vol pyr( I ) ε ( I ) = 2! vol out( I ) =7 =5 � � � � Notice covol( I ), vol pyr( I ) , and, vol out( I ) coincide when I is m -primary. Therefore, our theorems are generalizations of Teissier’s theorem. Jonathan Monta˜ no (PU-KU) Generalized Multiplicities August 5, 2015 15 / 36
Determinantal ideals Consider the following matrix in m · n different variables { x i , j } , where 1 � i � m , 1 � j � n , and m � n . x 1 , 1 . . . x 1 , n . . . . A = . . x m , 1 . . . x m , n Let I t for t � m be the ideal of the polynomial ring R = k [ { x i , j } ] generated by all the t -minors of A . Jonathan Monta˜ no (PU-KU) Generalized Multiplicities August 5, 2015 16 / 36
Generalized multiplicities of determinantal ideals Theorem (Jeffries-M-Varbaro, 2015) Let ( mn − 1)! c = ( n − 1)!( n − 2)! · · · ( n − m )! · m !( m − 1)! · · · 1! . Then, (i) � � ( z j − z i ) 2 d ν ; ( z 1 · · · z m ) n − m j ( I t ) = ct 1 � i < j � m [0 , 1] m � z = t (ii) � � ( z j − z i ) 2 d z ; ( z 1 · · · z m ) n − m ε ( I t ) = cmn 1 � i < j � m [0 , 1] m max i { z i } + t − 1 � � z � t These integrals can be computed using the package NmzIntegrate of Normaliz. Jonathan Monta˜ no (PU-KU) Generalized Multiplicities August 5, 2015 17 / 36
Rational normal scrolls Consider positive integers a 1 � · · · � a r , and set N = � r i =1 a i + r − 1. The rational normal scroll associated to this sequence is the projective subvariety of P N , defined by the ideal I = I ( a 1 , . . . , a r ) ⊆ K [ { x i , j } 1 � i � r , 1 � j � a i +1 ] generated by the 2-minors of the matrix � x 1 , 1 � x 1 , 2 · · · x 1 , a 1 · · · x r , 1 x r , 2 · · · x r , a r . x 1 , 2 x 1 , 3 · · · x 1 , a 1 +1 · · · x r , 2 x r , 3 · · · x r , a r +1 Jonathan Monta˜ no (PU-KU) Generalized Multiplicities August 5, 2015 18 / 36
The j -multiplicity of rational normal scrolls Theorem (Jeffries-M-Varbaro, 2015) j ( I ( a 1 , . . . , a r )) = 0 if c < r + 3 , �� 2 c − 4 � � 2 c − 4 �� 2 · − if c = r + 3 , c − 2 c − 1 � c + r − 1 � � c + r − 1 � c − r − 1 � 2 · − ( c − r − 2) if c > r + 3 . c − j c − 1 j =2 Jonathan Monta˜ no (PU-KU) Generalized Multiplicities August 5, 2015 19 / 36
Example Example � x 1 � x 2 x 3 x 4 I (4) = I 2 x 2 x 3 x 4 x 5 Here c = 4 and r = 1, c = r + 3 therefore �� 4 � � 4 �� j ( I (4)) = 2 · − = 4 . 2 3 � x 1 � � � x 2 x 3 x 5 x 6 j ( I (3 , 2)) = j I 2 = 10 . x 2 x 3 x 4 x 6 x 7 These examples had been computed by Nishida-Ulrich in 2010 using residual intersection theory and some intricate computations. Jonathan Monta˜ no (PU-KU) Generalized Multiplicities August 5, 2015 20 / 36
Binomial ideals? Binomial ideals form another class of ideals with combinatorial structure. Problem: Compute the generalized multiplicities of binomial ideals. One may consider first ideals I defining numerical semigroup rings, i.e., k [[ x 1 , . . . , x d ]] / I ∼ = k [[ t a 1 , . . . , t a d ]] for some positive integers a 1 < · · · < a d . We know ℓ ( I ) = d if I is not a complete intersection (Cowsik-Nori, 1976). Hence j ( I ) � = 0. Nishida-Ulrich gave a explicit formula for j ( I ) in the case d = 3. Jonathan Monta˜ no (PU-KU) Generalized Multiplicities August 5, 2015 21 / 36
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