On the Hilbert Function of General Fat Points in $\mathbb{P}^{1}\times \mathbb{P}^{1}$

2020 ◽  
Vol 69 (3) ◽  
pp. 601-632
Author(s):  
Enrico Carlini ◽  
Maria Virginia Catalisano ◽  
Alessandro Oneto
Keyword(s):  
Hilbert Function ◽  
Fat Points

Fat Points in ℙ1× ℙ1and Their Hilbert Functions

2004 ◽  
Vol 56 (4) ◽  
pp. 716-741 ◽  
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.

scholarly journals On the Hilbert Function of Fat Points on a Rational Normal Cubic

1996 ◽  
Vol 183 (1) ◽  
pp. 245-265 ◽  
Author(s):  
Maria Virginia Catalisano ◽  
Alessandro Gimigliano
Keyword(s):  
Hilbert Function ◽  
Fat Points

scholarly journals On the Hilbert Function of General Fat Points in $\mathbb{P}^{1}\times \mathbb{P}^{1}$

2020 ◽  
Author(s):  
Enrico Carlini ◽  
Maria Virginia Catalisano ◽  
Alessandro Oneto
Keyword(s):  
Hilbert Function ◽  
Fat Points

Fat points, partial intersections and Hamming distance

2019 ◽  
Vol 19 (04) ◽  
pp. 2050071 ◽  
Author(s):  
Susan M. Cooper ◽  
Elena Guardo
Keyword(s):  
Complete Intersection ◽  
Linear Code ◽  
Hilbert Function ◽  
Hamming Distance ◽  
Special Linear ◽  
Fat Points ◽  
Fat Point ◽  
Minimum Hamming Distance ◽  
Point Set ◽  
Set Of Points

We use two main techniques, namely, residuation and separators of points, to show that the Hilbert function of a certain fat point set supported on a grid complete intersection is the same as the Hilbert function of a reduced set of points called a partial intersection. As an application, we answer a question of Tohǎneanu and Van Tuyl which relates the minimum Hamming distance of a special linear code and the minimum socle degree of the associated fat point set.

scholarly journals On Segre's bound for fat points inPn

2016 ◽  
Vol 220 (6) ◽  
pp. 2307-2323 ◽  
Author(s):  
Edoardo Ballico ◽  
Olivia Dumitrescu ◽  
Elisa Postinghel
Keyword(s):  
Fat Points
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