Coxeter theory for curves on blowups of $\mathbb{P}^r$

TL;DR

This paper explores the geometry of rational curves on blowups of projective space using Coxeter groups, providing criteria to identify special curves called $(i)$-Weyl lines, especially in three dimensions.

Contribution

It introduces a Coxeter-theoretic framework for classifying rational curves on blowups of projective space and establishes criteria for identifying $(i)$-Weyl lines, with sharp results in three dimensions.

Findings

Coxeter groups help classify rational curves on blowups.

Numerical criteria determine when an $(i)$-curve is an $(i)$-Weyl line.

A Noether-type inequality is proved for $r=3$.

Abstract

We investigate the study of smooth irreducible rational curves in $Y_s^r$, a general blowup of $\mathbb{P}^r$ at $s$ general points, whose normal bundle splits as a direct sum of line bundles all of degree $i$, for $i \in \{-1,0,1\}$: we call these $(i)$-curves. We systematically exploit the theory of Coxeter groups applied to the Chow space of curves in $Y_s^r$, which provides us with a useful bilinear form that helps to expose properties of $(i)$-curves. We are particularly interested in the orbits of lines (through $1-i$ points) under the Weyl group of standard Cremona transformations (all of which are $(i)$-curves): we call these $(i)$-Weyl lines. We prove various theorems related to understanding when an $(i)$-curve is an $(i)$-Weyl line, via numerical criteria expressed in terms of the bilinear form. We obtain stronger results for $r=3$, where we prove a Noether-type inequality that gives a sharp criterion.