Depth of powers Matteo Varbaro (University of Genoa, Italy) - PowerPoint PPT Presentation

Depth of powers Matteo Varbaro (University of Genoa, Italy) 9/10/2015, Osnabr¨ uck, Germany Matteo Varbaro (University of Genoa, Italy) Depth of powers

Depth-function Let I be a homogeneous ideal of a polynomial ring S over a field K . Matteo Varbaro (University of Genoa, Italy) Depth of powers

Depth-function Let I be a homogeneous ideal of a polynomial ring S over a field K . The depth-function of I is the numerical function: φ I : N \ { 0 } − → N depth( S / I k ) �− → k Matteo Varbaro (University of Genoa, Italy) Depth of powers

Depth-function Let I be a homogeneous ideal of a polynomial ring S over a field K . The depth-function of I is the numerical function: φ I : N \ { 0 } − → N depth( S / I k ) �− → k Theorem (Brodmann, 1979) The depth-function is definitely constant. Matteo Varbaro (University of Genoa, Italy) Depth of powers

Depth-function Let I be a homogeneous ideal of a polynomial ring S over a field K . The depth-function of I is the numerical function: φ I : N \ { 0 } − → N depth( S / I k ) �− → k Theorem (Brodmann, 1979) The depth-function is definitely constant. Question What about the initial behavior of φ I ? Matteo Varbaro (University of Genoa, Italy) Depth of powers

Depth-function Let I be a homogeneous ideal of a polynomial ring S over a field K . The depth-function of I is the numerical function: φ I : N \ { 0 } − → N depth( S / I k ) �− → k Theorem (Brodmann, 1979) The depth-function is definitely constant. Question What about the initial behavior of φ I ? Matteo Varbaro (University of Genoa, Italy) Depth of powers

Depth-function: initial behavior At a first thought, probably one expects that the depth decreases when taking powers, that is: φ I (1) ≥ φ I (2) ≥ . . . ≥ φ I ( k ) ≥ φ I ( k + 1) ≥ . . . Matteo Varbaro (University of Genoa, Italy) Depth of powers

Depth-function: initial behavior At a first thought, probably one expects that the depth decreases when taking powers, that is: φ I (1) ≥ φ I (2) ≥ . . . ≥ φ I ( k ) ≥ φ I ( k + 1) ≥ . . . However, this is not true without any assumption on the ideal I : Matteo Varbaro (University of Genoa, Italy) Depth of powers

Depth-function: initial behavior At a first thought, probably one expects that the depth decreases when taking powers, that is: φ I (1) ≥ φ I (2) ≥ . . . ≥ φ I ( k ) ≥ φ I ( k + 1) ≥ . . . However, this is not true without any assumption on the ideal I : Theorem (Herzog-Hibi, 2005) For any bounded increasing numerical function φ : N > 0 → N , there exists a monomial ideal I such that φ I ( k ) = φ ( k ) ∀ k . Matteo Varbaro (University of Genoa, Italy) Depth of powers

Depth-function: initial behavior At a first thought, probably one expects that the depth decreases when taking powers, that is: φ I (1) ≥ φ I (2) ≥ . . . ≥ φ I ( k ) ≥ φ I ( k + 1) ≥ . . . However, this is not true without any assumption on the ideal I : Theorem (Herzog-Hibi, 2005) For any bounded increasing numerical function φ : N > 0 → N , there exists a monomial ideal I such that φ I ( k ) = φ ( k ) ∀ k . Theorem (Bandari-Herzog-Hibi, 2014) For any positive integer N , there exists a monomial ideal I such that φ I has N local maxima. Matteo Varbaro (University of Genoa, Italy) Depth of powers

Depth-function: initial behavior At a first thought, probably one expects that the depth decreases when taking powers, that is: φ I (1) ≥ φ I (2) ≥ . . . ≥ φ I ( k ) ≥ φ I ( k + 1) ≥ . . . However, this is not true without any assumption on the ideal I : Theorem (Herzog-Hibi, 2005) For any bounded increasing numerical function φ : N > 0 → N , there exists a monomial ideal I such that φ I ( k ) = φ ( k ) ∀ k . Theorem (Bandari-Herzog-Hibi, 2014) For any positive integer N , there exists a monomial ideal I such that φ I has N local maxima. The monomial ideals above are not square-free..... Matteo Varbaro (University of Genoa, Italy) Depth of powers

Depth-function: initial behavior If I is a square-free monomial ideal, then φ I (1) ≥ φ I ( k ) ∀ k > 1. Matteo Varbaro (University of Genoa, Italy) Depth of powers

Depth-function: initial behavior If I is a square-free monomial ideal, then φ I (1) ≥ φ I ( k ) ∀ k > 1. Question If I is a square-free monomial ideal, is φ I decreasing? Matteo Varbaro (University of Genoa, Italy) Depth of powers

Depth-function: initial behavior If I is a square-free monomial ideal, then φ I (1) ≥ φ I ( k ) ∀ k > 1. Question If I is a square-free monomial ideal, is φ I decreasing? Analogously, any projective scheme X smooth over C admits an embedding such that φ I X (1) ≥ φ I X ( k ) ∀ k > 1 ( ). Matteo Varbaro (University of Genoa, Italy) Depth of powers

Depth-function: initial behavior If I is a square-free monomial ideal, then φ I (1) ≥ φ I ( k ) ∀ k > 1. Question If I is a square-free monomial ideal, is φ I decreasing? Analogously, any projective scheme X smooth over C admits an embedding such that φ I X (1) ≥ φ I X ( k ) ∀ k > 1 ( ). Question If Proj( S / I ) is smooth over C , is φ I decreasing? Matteo Varbaro (University of Genoa, Italy) Depth of powers

Depth-function: initial behavior If I is a square-free monomial ideal, then φ I (1) ≥ φ I ( k ) ∀ k > 1. Question If I is a square-free monomial ideal, is φ I decreasing? Analogously, any projective scheme X smooth over C admits an embedding such that φ I X (1) ≥ φ I X ( k ) ∀ k > 1 ( ). Question If Proj( S / I ) is smooth over C , is φ I decreasing? Matteo Varbaro (University of Genoa, Italy) Depth of powers

Useful tools The Rees ring of I ⊆ S = K [ x 1 , . . . , x n ] is the S -algebra: � I k . R ( I ) = k ≥ 0 Matteo Varbaro (University of Genoa, Italy) Depth of powers

Useful tools The Rees ring of I ⊆ S = K [ x 1 , . . . , x n ] is the S -algebra: � I k . R ( I ) = k ≥ 0 If m = S + , then H i k ≥ 0 H i m ( I k ). m R ( I ) ( R ( I )) = � Matteo Varbaro (University of Genoa, Italy) Depth of powers

Useful tools The Rees ring of I ⊆ S = K [ x 1 , . . . , x n ] is the S -algebra: � I k . R ( I ) = k ≥ 0 If m = S + , then H i k ≥ 0 H i m ( I k ). So we see that: m R ( I ) ( R ( I )) = � k { depth( I k ) } . grade( m R ( I ) , R ( I )) = min Matteo Varbaro (University of Genoa, Italy) Depth of powers

Useful tools The Rees ring of I ⊆ S = K [ x 1 , . . . , x n ] is the S -algebra: � I k . R ( I ) = k ≥ 0 If m = S + , then H i k ≥ 0 H i m ( I k ). So we see that: m R ( I ) ( R ( I )) = � k { depth( I k ) } . grade( m R ( I ) , R ( I )) = min So height( m R ( I )) ≥ min k { depth( S / I k ) } + 1, with equality if R ( I ) is Cohen-Macaulay. Matteo Varbaro (University of Genoa, Italy) Depth of powers

Useful tools The Rees ring of I ⊆ S = K [ x 1 , . . . , x n ] is the S -algebra: � I k . R ( I ) = k ≥ 0 If m = S + , then H i k ≥ 0 H i m ( I k ). So we see that: m R ( I ) ( R ( I )) = � k { depth( I k ) } . grade( m R ( I ) , R ( I )) = min So height( m R ( I )) ≥ min k { depth( S / I k ) } + 1, with equality if R ( I ) is Cohen-Macaulay. Now, let us remind that the fiber cone of I is the K -algebra: F ( I ) = R ( I ) / m R ( I ) . Matteo Varbaro (University of Genoa, Italy) Depth of powers

Useful tools (For instance, if I is generated by polynomials f 1 , . . . , f r of the same degree, then F ( I ) = K [ f 1 , . . . , f r ].) Matteo Varbaro (University of Genoa, Italy) Depth of powers

Useful tools (For instance, if I is generated by polynomials f 1 , . . . , f r of the same degree, then F ( I ) = K [ f 1 , . . . , f r ].) Therefore, dim( F ( I )) = dim( R ( I )) − height( m R ( I )) k { depth( S / I k ) } − 1 ≤ n + 1 − min k { depth( S / I k ) } , = n − min Matteo Varbaro (University of Genoa, Italy) Depth of powers

Useful tools (For instance, if I is generated by polynomials f 1 , . . . , f r of the same degree, then F ( I ) = K [ f 1 , . . . , f r ].) Therefore, dim( F ( I )) = dim( R ( I )) − height( m R ( I )) k { depth( S / I k ) } − 1 ≤ n + 1 − min k { depth( S / I k ) } , = n − min with equality if R ( I ) is Cohen-Macaulay (these results are due to Burch and to Eisenbud-Huneke). Matteo Varbaro (University of Genoa, Italy) Depth of powers

Useful tools (For instance, if I is generated by polynomials f 1 , . . . , f r of the same degree, then F ( I ) = K [ f 1 , . . . , f r ].) Therefore, dim( F ( I )) = dim( R ( I )) − height( m R ( I )) k { depth( S / I k ) } − 1 ≤ n + 1 − min k { depth( S / I k ) } , = n − min with equality if R ( I ) is Cohen-Macaulay (these results are due to Burch and to Eisenbud-Huneke). So, it is evident that the study of depth-functions is closely related to the study of blow-up algebras. Matteo Varbaro (University of Genoa, Italy) Depth of powers

Constant depth-functions In this talk, I want to inquire on ideals having constant depth-function. Matteo Varbaro (University of Genoa, Italy) Depth of powers

Constant depth-functions In this talk, I want to inquire on ideals having constant depth-function. Most of what I’ll say, is part of a joint work with Le Dinh Nam. Matteo Varbaro (University of Genoa, Italy) Depth of powers