Cremona Orbits in $\mathbb{P}^4$ and Applications

TL;DR

This paper studies the geometry of the Mori dream space obtained by blowing up eight points in projective 4-space, developing techniques to analyze Cremona orbits and applying these to understand the dimension of global sections of certain divisors.

Contribution

It introduces explicit computational methods for Weyl orbits of linear cycles in four dimensions and applies them to problems in algebraic geometry related to effective divisors.

Findings

Developed techniques for determining Weyl orbits in $P^4$

Computed the Chow ring of the Cremona transformation resolution

Applied results to the dimension of global sections of divisors with base points

Abstract

This article is motivated by the authors interest in the geometry of the Mori dream space $\mathbb{P}^4$ blown up in $8$ general points. In this article we develop the necessary technique for determining Weyl orbits of linear cycles for the four-dimensional case, by explicit computations in the Chow ring of the resolution of the standard Cremona transformation. In particular, we close this paper with applications to the question of the dimension of the space global sections of effective divisors having at most $8$ base points.