Sequences of point blow-ups over perfect fields from a combinatorial point of view

TL;DR

This paper introduces a combinatorial framework for analyzing sequences of point blow-ups over perfect fields, establishing bijections between algebraic and combinatorial equivalence classes through graph-based objects.

Contribution

It develops a novel combinatorial approach using weighted directed graphs and intersection forms to classify sequences of point blow-ups over perfect fields.

Findings

Bijection between combinatorial equivalence classes of blow-up sequences and sequential morphisms.

Existence of a bijection between algebraic equivalence classes.

Framework facilitates understanding of blow-ups over different field extensions.

Abstract

We associate a combinatorial object to sequences of point blow-ups over perfect fields, the weighted directed graph, and another one to the composition of all blow-ups, which we call associated sequential morphisms, the $d-$ary intersection form. Then, in order to consider different fields extensions, we introduce the concepts of algebraically and combinatorially compatible partitions of the exceptional divisor for both sequences of point blow-ups and sequential morphisms, which lead us to define the corresponding algebraic and combinatorial equivalence classes. We prove that there exists a bijection between the respective combinatorial equivalence classes of sequences of point blow-ups and the associated sequential morphisms, and moreover, we also give a proof of the existence of a suitable bijection between the respective algebraic equivalence classes.