On the Nef cones of blowups of the projective plane

TL;DR

This paper investigates the structure of the nef cone of blowups of the projective plane at multiple points, constructing fundamental domains under Cremona transformations for various numbers of points.

Contribution

It provides explicit descriptions of rational polyhedral fundamental domains of the nef cone for blowups of the projective plane at n points, especially for n=9 and n≥10.

Findings

Constructed a fundamental domain for n=9.

Described the nef cone's structure for n≥10.

Analyzed the Cremona action on the nef cone.

Abstract

In this paper, we study the Cremona action on the nef cone of $S_n$, the blowup of $\P^2$ in $n$ very general points, with $n\ge9$. We construct and describe a rational polyhedral fundamental domain of the nef cone for $n=9$ with respect to this action. In the case $n\ge10$, we give a rational polyhedral fundamental domain of the $K_{S_n}$-negative part of the nef cone with respect to the Cremona action.