Points in \(\mathbb{P}^{1} \times \mathbb{P}^{1}\)

  • Elena Guardo
  • Adam Van Tuyl
Chapter
First Online:
Part of the SpringerBriefs in Mathematics book series (BRIEFSMATH)

Abstract

In this chapter we develop the basic properties of sets of points in \(\mathbb{P}^{1} \times \mathbb{P}^{1}\). We begin by describing the bihomogeneous ideal I(P) associated with a point \(P \in \mathbb{P}^{1} \times \mathbb{P}^{1}\). We also discuss the connection between sets of points X in \(\mathbb{P}^{1} \times \mathbb{P}^{1}\) and special configurations of lines in \(\mathbb{P}^{3}\). Because \(\mathbb{P}^{1} \times \mathbb{P}^{1}\) is isomorphic to the ruled quadric surface in \(\mathbb{P}^{3}\), we can visualize a collection of points X as sitting within a grid of lines. After describing how to represent a set of points X in \(\mathbb{P}^{1} \times \mathbb{P}^{1}\), we extract some combinatorial information from this representation.

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Copyright information

© The Authors 2015

Authors and Affiliations

  • Elena Guardo
    • 1
  • Adam Van Tuyl
    • 2
  1. 1.Dipartimento di Matematica e InformaticaUniversity of CataniaCataniaItaly
  2. 2.Department of Mathematics and StatisticsMcMaster UniversityHamiltonCanada

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