Abstract
In the previous chapters we focused on sets of reduced points X in \(\mathbb{P}^{1} \times \mathbb{P}^{1}\), that is, \(I(X) = \sqrt{I(X)}\). We now relax this condition to study “sets of fat points”. Roughly speaking, given a set of reduced points X, we assign to each \(P_{i} \in X\) a positive integer m i , called its multiplicity, and we consider the ideal \(I(Z) =\bigcap _{ i=1}^{s}I(P_{i})^{m_{i}}\). We can then ask similar questions for the set of fat point Z defined by I(Z): when is Z arithmetically Cohen-Macaulay? If Z is ACM, what is its Hilbert function H Z ? What is the bigraded minimal free resolution of I(Z)? We answer these questions in this chapter.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
M. Baczyńska, M. Dumnicki, A. Habura, G. Malara, P. Pokora, T. Szemberg, J. Szpond, H. Tutaj-Gasińska, Points fattening on \(\mathbb{P}^{1} \times \mathbb{P}^{1}\) and symbolic powers of bi-homogeneous ideals. J. Pure Appl. Algebra 218(8), 1555–1562 (2014)
A. Gimigliano, Our thin knowledge of fat points, in The Curves Seminar at Queen’s, Vol. VI (Kingston, ON, 1989). Queen’s Papers in Pure and Applied Mathematics, vol. 83 (Queen’s University, Kingston, 1989), Exp. No. B, 50 pp.
E. Guardo, Schemi di “Fat Points”. PhD Thesis, Università di Messina (2000)
E. Guardo, Fat point schemes on a smooth quadric. J. Pure Appl. Algebra 162(2–3), 183–208 (2001)
E. Guardo, A survey on fat points on a smooth quadric, in Algebraic Structures and Their Representations. Contemporary Mathematics, vol. 376 (American Mathematical Society, Providence, 2005), pp. 61–87
E. Guardo, A. Van Tuyl, Fat Points in \(\mathbb{P}^{1} \times \mathbb{P}^{1}\) and their Hilbert functions. Can. J. Math. 56(4), 716–741 (2004)
E. Guardo, A. Van Tuyl, Separators of fat points in \(\mathbb{P}^{n} \times \mathbb{P}^{m}\). J. Pure Appl. Algebra 215(8), 1990–1998 (2011)
E. Guardo, A. Van Tuyl, Separators of arithmetically Cohen-Macaulay fatpoints in \(\mathbb{P}^{1} \times \mathbb{P}^{1}\). J. Commut. Algebra 4(2), 255–268 (2012)
E. Guardo, B. Harbourne, A. Van Tuyl, Symbolic powers versus regular powers of ideals of general points in \(\mathbb{P}^{1} \times \mathbb{P}^{1}\). Can. J. Math. 65(4), 823–842 (2013)
H.T. Hà, A. Van Tuyl, The regularity of points in multi–projective spaces. J. Pure Appl. Algebra 187(1–3), 153–167 (2004)
B. Harbourne, Problems and progress: a survey on fat points in \(\mathbb{P}^{2}\), in Zero-Dimensional Schemes and Applications (Naples, 2000) Queen’s Papers in Pure and Applied Mathematics, vol. 123 (Queen’s University, Kingston, 2002), pp. 85–132
B. Hassett, Introduction to Algebraic Geometry (Cambridge University Press, Cambridge, 2007)
D. Maclagan, G.G. Smith, Multigraded Castelnuovo-Mumford regularity. J. Reine Angew. Math. 571, 179–212 (2004)
E. Miller, B. Sturmfels, Combinatorial Commutative Algebra. Graduate Texts in Mathematics, vol. 227 (Springer, New York, 2005)
I. Peeva, Graded Syzygies. Algebra and Applications, vol. 14 (Springer, London, 2011)
J. Sidman, A. Van Tuyl, Multigraded regularity: syzygies and fat points. Beitr. Algebra Geom. 47(1), 67–87 (2006)
R.H. Villarreal, Monomial Algebras. Monographs and Textbooks in Pure and Applied Mathematics, vol. 238 (Marcel Dekker, New York, 2001)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2015 The Authors
About this chapter
Cite this chapter
Guardo, E., Van Tuyl, A. (2015). Fat points in \(\mathbb{P}^{1} \times \mathbb{P}^{1}\) . In: Arithmetically Cohen-Macaulay Sets of Points in P^1 x P^1. SpringerBriefs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-24166-1_6
Download citation
DOI: https://doi.org/10.1007/978-3-319-24166-1_6
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-24164-7
Online ISBN: 978-3-319-24166-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)