Hilbert Polynomials of Kähler Differential Modules for Fat Point Schemes

Kreuzer, Martin; Linh, Tran N. K.; Long, Le Ngoc

doi:10.1007/s40306-021-00432-3

Cited by 2 publications

(2 citation statements)

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“…This provides a positive answer to a question posed in [7]. See also [8] for a different proof of this result.…”

Section: Kähler Differentials Of Fat Point Schemesmentioning

confidence: 61%

“…. , x n ], the homogeneous coordinate ring R = P/I X , its Hilbert function HF X (i) = dim K (R i ), and the canonical module ω R of R. In several previous works, the authors introduced and started using the Kähler differential module Ω 1 R/K of R and even the entire Kähler differential algebra Ω • R/K = Λ R (Ω 1 R/K ) in order to advance this topic (see [4], [7], and [8]).…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Differential theory of zero-dimensional schemes

Kreuzer¹,

Linh²,

Long³

2023

Preprint

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For a 0-dimensional scheme X in P n over a perfect field K, we first embed the homogeneous coordinate ring R into its truncated integral closure R. Then we use the corresponding map from the module of Kähler differentialsto find a formula for the Hilbert polynomial HP(Ω 1 R/K ) and a sharp bound for the regularity index ri(Ω 1 R/K ). Additionally, we extend this to formulas for the Hilbert polynomials HP(Ω m R/K ) and bounds for the regularity indices of the higher modules of Kähler differentials. Next we derive a new characterization of a weakly curvilinear scheme X which can be checked without computing a primary decomposition of its vanishing ideal I X . Moreover, we prove precise formulas for the Hilbert polynomial of Ω m R/K of a fat point scheme X, extending and settling previous partial results and conjectures. Finally, we characterize uniformity conditions on X using the Hilbert functions of the Kähler differential modules of X and its subschemes.

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“…This provides a positive answer to a question posed in [7]. See also [8] for a different proof of this result.…”

Section: Kähler Differentials Of Fat Point Schemesmentioning

confidence: 61%