On the universal scheme of r-relative clusters of a family

ABSTRACT

We generalize the concept of r-point clusters of a scheme S to r-relative clusters of a B-scheme 𝒮, and define schemes Cl_r that naturally parametrize them. In the particular case that B = SpecK, the schemes Cl_r agree with Kleiman’s iterated blowups parametrizing r-point clusters. We show that there is an iterative construction of Cl_r + 1 from Cl_r, related but not equal in general to the blow up of ${Cl}_r{\times }_{{Cl}_{r-1}}{Cl}_r$ along the diagonal.

KEYWORDS:

• Birational morphism
• flat blowup
• Hilbert scheme
• infinitely near points
• iterated blowups
• universal family of sections

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 14E05
• 14D0
• 14D06

Acknowledgments

I thank my thesis advisor, Joaquim Roć, for his great support and I thank the referee for the patience and all the feedback.