Weyl cycles on the blow-up of $\mathbb{P}^4$ at eight points

TL;DR

This paper introduces Weyl cycles on blown-up projective spaces, classifies all such cycles of codimension two in specific Mori Dream spaces, and proposes a generalized expected dimension for effective divisors.

Contribution

It defines Weyl cycles on blown-up projective spaces and classifies all codimension two Weyl cycles in certain Mori Dream spaces, extending the concept of expected dimension.

Findings

Classification of Weyl cycles in $X^3_7$ and $X^4_8$

Introduction of Weyl expected dimension for effective divisors

Generalization of linear and secant expected dimensions

Abstract

We define the Weyl cycles on $X^n_s$, the blown up projective space $\mathbb{P}^n$ in $s$ points in general position. In particular, we focus on the Mori Dream spaces $X^3_7$ and $X^{4}_{8}$, where we classify all the Weyl cycles of codimension two. We further introduce the Weyl expected dimension for the space of the global sections of any effective divisor that generalizes the linear expected dimension and the secant expected dimension.