Cox rings of blow-ups of multiprojective spaces

TL;DR

This paper studies the geometric and algebraic structure of blow-ups of multiprojective spaces, specifically describing their Mori cones and Cox rings for certain cases, revealing new properties like being log Fano.

Contribution

It provides explicit descriptions of the Mori cone and Cox ring for blow-ups of multiprojective spaces, including new results on their log Fano property.

Findings

Mori cone described for r ≤ n+2 and r = n+3 when n ≤ 4

Cox ring presentation for the case r = n+1

Proof that certain blow-ups are log Fano

Abstract

Let $X^{1,n}_r$ be the blow-up of $\mathbb{P}^1\times\mathbb{P}^n$ in $r$ general points. We describe the Mori cone of $X^{1,n}_r$ for $r\leq n+2$ and for $r = n+3$ when $n\leq 4$. Furthermore, we prove that $X^{1,n}_{n+1}$ is log Fano and give an explicit presentation for its Cox ring.