Generic deformations of matroid ideals Alexandru Constantinescu - PowerPoint PPT Presentation

Generic deformations of matroid ideals Alexandru Constantinescu (joint work with Thomas Kahle and Matteo Varbaro) Universit´ e de Neuchˆ atel, Switzerland 1

A matroid is a simplicial complex ∆ on [ n ], such that ∀ F , G ∈ ∆ with | F | < | G | , ∃ v ∈ G \ F , such that F ∪ { v } ∈ ∆ . 2

A matroid is a simplicial complex ∆ on [ n ], such that ∀ F , G ∈ ∆ with | F | < | G | , ∃ v ∈ G \ F , such that F ∪ { v } ∈ ∆ . 3

A matroid is a simplicial complex ∆ on [ n ], such that ∀ F , G ∈ ∆ with | F | < | G | , ∃ v ∈ G \ F , such that F ∪ { v } ∈ ∆ . 4

A matroid is a simplicial complex ∆ on [ n ], such that ∀ F , G ∈ ∆ with | F | < | G | , ∃ v ∈ G \ F , such that F ∪ { v } ∈ ∆ . [ n ] 5

A matroid is a simplicial complex ∆ on [ n ], such that ∀ F , G ∈ ∆ with | F | < | G | , ∃ v ∈ G \ F , such that F ∪ { v } ∈ ∆ . [ n ] x 1 , . . . , x n 6

A matroid is a simplicial complex ∆ on [ n ], such that ∀ F , G ∈ ∆ with | F | < | G | , ∃ v ∈ G \ F , such that F ∪ { v } ∈ ∆ . 2 [ n ] x 1 , . . . , x n 7

A matroid is a simplicial complex ∆ on [ n ], such that ∀ F , G ∈ ∆ with | F | < | G | , ∃ v ∈ G \ F , such that F ∪ { v } ∈ ∆ . 2 [ n ] S = K [ x 1 , . . . , x n ] 8

A matroid is a simplicial complex ∆ on [ n ], such that ∀ F , G ∈ ∆ with | F | < | G | , ∃ v ∈ G \ F , such that F ∪ { v } ∈ ∆ . 2 [ n ] ⊇ ∆ matroid complex S = K [ x 1 , . . . , x n ] 9

A matroid is a simplicial complex ∆ on [ n ], such that ∀ F , G ∈ ∆ with | F | < | G | , ∃ v ∈ G \ F , such that F ∪ { v } ∈ ∆ . 2 [ n ] ⊇ ∆ matroid complex S = K [ x 1 , . . . , x n ] ⊇ I ∆ monomial ideal 10

A matroid is a simplicial complex ∆ on [ n ], such that ∀ F , G ∈ ∆ with | F | < | G | , ∃ v ∈ G \ F , such that F ∪ { v } ∈ ∆ . 2 [ n ] ⊇ ∆ − → ( h 0 , . . . , h s ) h -vector of ∆: S = K [ x 1 , . . . , x n ] ⊇ I ∆ monomial ideal 11

A matroid is a simplicial complex ∆ on [ n ], such that ∀ F , G ∈ ∆ with | F | < | G | , ∃ v ∈ G \ F , such that F ∪ { v } ∈ ∆ . 2 [ n ] ⊇ ∆ − → ( h 0 , . . . , h s ) h -vector of ∆: h 0 + h 1 t + ··· + h s t s S = K [ x 1 , . . . , x n ] ⊇ I ∆ − → Hilbert series of S / I ∆ : (1 − t ) d 12

A matroid is a simplicial complex ∆ on [ n ], such that ∀ F , G ∈ ∆ with | F | < | G | , ∃ v ∈ G \ F , such that F ∪ { v } ∈ ∆ . 2 [ n ] ⊇ ∆ − → ( h 0 , . . . , h s ) h -vector of ∆: h 0 + h 1 t + ··· + h s t s S = K [ x 1 , . . . , x n ] ⊇ I ∆ − → Hilbert series of S / I ∆ : (1 − t ) d 13

A matroid is a simplicial complex ∆ on [ n ], such that ∀ F , G ∈ ∆ with | F | < | G | , ∃ v ∈ G \ F , such that F ∪ { v } ∈ ∆ . 2 [ n ] ⊇ ∆ − → ( h 0 , . . . , h s ) h -vector of ∆: h 0 + h 1 t + ··· + h s t s S = K [ x 1 , . . . , x n ] ⊇ I ∆ − → Hilbert series of S / I ∆ : (1 − t ) d Conjecture (Stanley ’77) The h-vector of a matroid complex is a pure O-sequence. 14

A matroid is a simplicial complex ∆ on [ n ], such that ∀ F , G ∈ ∆ with | F | < | G | , ∃ v ∈ G \ F , such that F ∪ { v } ∈ ∆ . 2[ n ] ⊇ ∆ − → ( h 0 , . . . , hs ) h -vector of ∆: h 0+ h 1 t + ··· + hs ts S = K [ x 1 , . . . , xn ] ⊇ I ∆ − → Hilbert series of S / I ∆: (1 − t ) d Conjecture (Stanley ’77) The h-vector of a matroid complex is a pure O-sequence. 15

A matroid is a simplicial complex ∆ on [ n ], such that ∀ F , G ∈ ∆ with | F | < | G | , ∃ v ∈ G \ F , such that F ∪ { v } ∈ ∆ . 2[ n ] ⊇ ∆ − → ( h 0 , . . . , hs ) h -vector of ∆: h 0+ h 1 t + ··· + hs ts S = K [ x 1 , . . . , xn ] ⊇ I ∆ − → Hilbert series of S / I ∆: (1 − t ) d Conjecture (Stanley ’77) The h-vector of a matroid complex is a pure O-sequence. 16

A matroid is a simplicial complex ∆ on [ n ], such that ∀ F , G ∈ ∆ with | F | < | G | , ∃ v ∈ G \ F , such that F ∪ { v } ∈ ∆ . 2[ n ] ⊇ ∆ − → ( h 0 , . . . , hs ) h -vector of ∆: h 0+ h 1 t + ··· + hs ts S = K [ x 1 , . . . , xn ] ⊇ I ∆ − → Hilbert series of S / I ∆: (1 − t ) d Conjecture (Stanley ’77) The h-vector of a matroid complex is a pure O-sequence. A pure O-sequence is the Hilbert function of some artinian , monomial , level algebra. 17

A matroid is a simplicial complex ∆ on [ n ], such that ∀ F , G ∈ ∆ with | F | < | G | , ∃ v ∈ G \ F , such that F ∪ { v } ∈ ∆ . 2[ n ] ⊇ ∆ − → ( h 0 , . . . , hs ) h -vector of ∆: h 0+ h 1 t + ··· + hs ts S = K [ x 1 , . . . , xn ] ⊇ I ∆ − → Hilbert series of S / I ∆: (1 − t ) d Conjecture (Stanley ’77) The h-vector of a matroid complex is a pure O-sequence. A pure O-sequence is the Hilbert function of some artinian , monomial , level algebra. 18

A matroid is a simplicial complex ∆ on [ n ], such that ∀ F , G ∈ ∆ with | F | < | G | , ∃ v ∈ G \ F , such that F ∪ { v } ∈ ∆ . 2[ n ] ⊇ ∆ − → ( h 0 , . . . , hs ) h -vector of ∆: h 0+ h 1 t + ··· + hs ts S = K [ x 1 , . . . , xn ] ⊇ I ∆ − → Hilbert series of S / I ∆: (1 − t ) d Conjecture (Stanley ’77) The h-vector of a matroid complex is a pure O-sequence. A pure O-sequence is the Hilbert function of some artinian , monomial , level algebra. A graded, CM 1 algebra S / I is level if: 0 − → S ( − a ) β p − → · · · − → F 0 − → S / I − → 0 1 CM= Cohen Macaulay 19

A matroid is a simplicial complex ∆ on [ n ], such that ∀ F , G ∈ ∆ with | F | < | G | , ∃ v ∈ G \ F , such that F ∪ { v } ∈ ∆ . 2[ n ] ⊇ ∆ − → ( h 0 , . . . , hs ) h -vector of ∆: h 0+ h 1 t + ··· + hs ts S = K [ x 1 , . . . , xn ] ⊇ I ∆ − → Hilbert series of S / I ∆: (1 − t ) d Conjecture (Stanley ’77) The h-vector of a matroid complex is a pure O-sequence. A pure O-sequence is the Hilbert function of some artinian , monomial , level algebra. Stanley-Reisner ring S / I ∆ A graded, CM 1 algebra S / I is level if: 0 − → S ( − a ) β p − → · · · − → F 0 − → S / I − → 0 1 CM= Cohen Macaulay 20

A matroid is a simplicial complex ∆ on [ n ], such that ∀ F , G ∈ ∆ with | F | < | G | , ∃ v ∈ G \ F , such that F ∪ { v } ∈ ∆ . 2[ n ] ⊇ ∆ − → ( h 0 , . . . , hs ) h -vector of ∆: h 0+ h 1 t + ··· + hs ts S = K [ x 1 , . . . , xn ] ⊇ I ∆ − → Hilbert series of S / I ∆: (1 − t ) d Conjecture (Stanley ’77) The h-vector of a matroid complex is a pure O-sequence. A pure O-sequence is the Hilbert function of some artinian , monomial , level algebra. Stanley-Reisner ring S / I ∆ A graded, CM 1 algebra S / I is level if: 0 − → S ( − a ) β p − → · · · − → F 0 − → S / I − → 0 1 CM= Cohen Macaulay 21

A matroid is a simplicial complex ∆ on [ n ], such that ∀ F , G ∈ ∆ with | F | < | G | , ∃ v ∈ G \ F , such that F ∪ { v } ∈ ∆ . 2[ n ] ⊇ ∆ − → ( h 0 , . . . , hs ) h -vector of ∆: h 0+ h 1 t + ··· + hs ts S = K [ x 1 , . . . , xn ] ⊇ I ∆ − → Hilbert series of S / I ∆: (1 − t ) d Conjecture (Stanley ’77) The h-vector of a matroid complex is a pure O-sequence. A pure O-sequence is the Hilbert function of some artinian , monomial , level algebra. Stanley-Reisner ring S / I ∆ A graded, CM 1 algebra S / I is level if: 0 − → S ( − a ) β p − → · · · − → F 0 − → S / I − → 0 1 CM= Cohen Macaulay 22

A matroid is a simplicial complex ∆ on [ n ], such that ∀ F , G ∈ ∆ with | F | < | G | , ∃ v ∈ G \ F , such that F ∪ { v } ∈ ∆ . 2[ n ] ⊇ ∆ − → ( h 0 , . . . , hs ) h -vector of ∆: h 0+ h 1 t + ··· + hs ts S = K [ x 1 , . . . , xn ] ⊇ I ∆ − → Hilbert series of S / I ∆: (1 − t ) d Conjecture (Stanley ’77) The h-vector of a matroid complex is a pure O-sequence. A pure O-sequence is the Hilbert function of some artinian , monomial , level algebra. Stanley-Reisner ring S / I ∆ A graded, CM 1 algebra S / I is level if: 0 − → S ( − a ) β p − → · · · − → F 0 − → S / I − → 0 1 CM= Cohen Macaulay 23

A matroid is a simplicial complex ∆ on [ n ], such that ∀ F , G ∈ ∆ with | F | < | G | , ∃ v ∈ G \ F , such that F ∪ { v } ∈ ∆ . 2[ n ] ⊇ ∆ − → ( h 0 , . . . , hs ) h -vector of ∆: h 0+ h 1 t + ··· + hs ts S = K [ x 1 , . . . , xn ] ⊇ I ∆ − → Hilbert series of S / I ∆: (1 − t ) d Conjecture (Stanley ’77) The h-vector of a matroid complex is a pure O-sequence. A pure O-sequence is the Hilbert function of some artinian , monomial , level algebra. Stanley-Reisner ring S / I ∆ Artinian reduction S / ( I ∆ + ( ℓ i )) A graded, CM 1 algebra S / I is level if: 0 − → S ( − a ) β p − → · · · − → F 0 − → S / I − → 0 1 CM= Cohen Macaulay 24