On $(i)$-Curves in Blowups of $\mathbb{P}^r$

TL;DR

This paper investigates special classes of curves in blown-up projective spaces, establishing finiteness conditions, defining invariants, and analyzing extremal rays, with implications for Mori Dream Spaces and curve counting.

Contribution

It introduces new notions of $(0)$- and $(1)$-curves, characterizes their finiteness, and links these to Mori Dream Spaces using bilinear forms and Weyl invariants.

Findings

Finiteness of certain curves characterizes Mori Dream Spaces.

Defined a bilinear form and a Weyl-invariant class for curve analysis.

Reproved that $F^2 leq 0$ implies the space is not a Mori Dream Space.

Abstract

In this paper we study $(i)$-curves with $i\in \{-1, 0, 1\}$ in the blown up projective space $\mathbb{P}^r$ in general points. The notion of $(-1)$-curves was analyzed in the early days of mirror symmetry by Kontsevich with the motivation of counting curves on a Calabi-Yau threefold. In dimension two, Nagata studied planar $(-1)$-curves in order to construct counterexample to Hilbert's 14th problem. We introduce the notion of classes of $(0)$- and $(1)$-curves in $\mathbb{P}^r$ with $s$ points blown up and we prove that their number is finite if and only if the space is a Mori Dream Space. We further introduce a bilinear form on a space of curves, and a unique symmetric Weyl-invariant class, $F$, (that we will refer to as the anticanonical curve class). For Mori Dream Spaces we prove that $(-1)$-curves can be defined arithmetically by the linear and quadratic invariants determined by the bilinear form. Moreover, $(0)$- and $(1)$-Weyl lines give the extremal rays for the cone of movable curves in $\mathbb{P}^r$ with $r+3$ points blown up. As an application, we use the technique of movable curves to reprove that if $F^2\leq 0$ then $Y$ is not a Mori Dream Space and we propose to apply this technique to other spaces.