Canadian Mathematical Society
www.cms.math.ca
|          |    |           |    |
|----------|----|-----------|----|
|          | |  |           | |  |
| Site map | |  | CMS store |    |

location:            Publications → journals → CJM
Abstract view

# Fat Points in $\mathbb{P}^1 \times \mathbb{P}^1$ and Their Hilbert Functions

Read article
[PDF: 281KB]
http://dx.doi.org/10.4153/CJM-2004-033-0
Canad. J. Math. 56(2004), 716-741
Published:2004-08-01
Printed: Aug 2004
• Elena Guardo
• Adam Van Tuyl
Features coming soon:
Citations   (via CrossRef) Tools: Search Google Scholar:

Format:  HTML  LaTeX  MathJax  PDF  PostScript

## Abstract

We study the Hilbert functions of fat points in $\popo$. If $Z \subseteq \popo$ is an arbitrary fat point scheme, then it can be shown that for every $i$ and $j$ the values of the Hilbert function $_{Z}(l,j)$ and $H_{Z}(i,l)$ eventually become constant for $l \gg 0$. We show how to determine these eventual values by using only the multiplicities of the points, and the relative positions of the points in $\popo$. This enables us to compute all but a finite number values of $H_{Z}$ without using the coordinates of points. We also characterize the ACM fat point schemes sing our description of the eventual behaviour. In fact, n the case that $Z \subseteq \popo$ is ACM, then the entire Hilbert function and its minimal free resolution depend solely on knowing the eventual values of the Hilbert function.

Keywords:  Hilbert function, points, fat points, Cohen-Macaulay, multi-projective space

MSC Classifications:  13D40 - Hilbert-Samuel and Hilbert-Kunz functions; Poincare series 13D02 - Syzygies, resolutions, complexes 13H10 - Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) [See also 14M05] 14A15 - Schemes and morphisms
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 11, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.6914311051368713, "perplexity_flag": "middle"}