# Mathematics Research Institute

## Sequences of point blow-ups from a combinatorial point of view

Fecha: 02/07/2021 09:45

Abstract:
Fixed a perfect field $k$, we will focus on the study of sequential morphisms, that is morphism which can be expressed, in at least one way, as a composition of a sequence of blow-ups of smooth $d$-dimensional projective varieties over $K$, with $K$ and algebraic extension of $k$ such that $K\subset\overline{k}$. The motivation is that this kind of morphisms play a central role on singularity theory due to the existence of embedded resolutions of singular projective varieties. More concretely, we are interested in finding the geometrical information encoded by the numerical data given by the multilinear intersection form on divisors with exceptional support. In order to consider different fields $K$, with $k\subset K\subset\overline{k}$, we will define the notion of compatible partition of the exceptional divisor $E$. Our main tool will be the existence of regular and projective divisorial contractions under certain hypothesis. After defining combinatorial and algebraic equivalence for both sequences of point blow-ups and their associated sequential morphism, we will revisit the key concept of final divisor and characterize it by some numerical criteria.