4. Example: the fat point of multiplicity four

Kähler Differentials for Fat Point Schemes in ℙ1×ℙ1

Journal of Commutative Algebra ◽

10.1216/jca.2021.13.179 ◽

2021 ◽

Vol 13 (2) ◽

Author(s):

Elena Guardo ◽

Martin Kreuzer ◽

Tran N. K. Linh ◽

Le Ngoc Long

Keyword(s):

Fat Point ◽

Kähler Differentials

Download Full-text

Fat point embeddings in affine space

Journal of Pure and Applied Algebra ◽

10.1016/j.jpaa.2007.10.013 ◽

2008 ◽

Vol 212 (7) ◽

pp. 1583-1593

Author(s):

Jean-Philippe Furter

Keyword(s):

Affine Space ◽

Fat Point

Download Full-text

Fat Points in ℙ1× ℙ1and Their Hilbert Functions

Canadian Journal of Mathematics ◽

10.4153/cjm-2004-033-0 ◽

2004 ◽

Vol 56 (4) ◽

pp. 716-741 ◽

Cited By ~ 15

Author(s):

Elena Guardo ◽

Adam Van Tuyl

Keyword(s):

Finite Number ◽

Hilbert Function ◽

Free Resolution ◽

Hilbert Functions ◽

Minimal Free Resolution ◽

Fat Points ◽

Fat Point ◽

Point Scheme

AbstractWe study the Hilbert functions of fat points in ℙ1× ℙ1. IfZ⊆ ℙ1× ℙ1is an arbitrary fat point scheme, then it can be shown that for everyiandjthe values of the Hilbert functionHZ(l,j) andHZ(i,l) eventually become constant forl≫ 0. We show how to determine these eventual values by using only the multiplicities of the points, and the relative positions of the points in ℙ1× ℙ1. This enables us to compute all but a finite number values ofHZwithout using the coordinates of points. We also characterize the ACM fat point schemes using our description of the eventual behaviour. In fact, in the case thatZ⊆ ℙ1× ℙ1is ACM, then the entire Hilbert function and its minimal free resolution depend solely on knowing the eventual values of the Hilbert function.

Download Full-text