Log Fano blowups of mixed products of projective spaces and their effective cones

TL;DR

This paper computes the effective cones of certain blowups of product spaces and demonstrates their log Fano property, providing a conceptual understanding of their rational polyhedral cones.

Contribution

It explicitly determines the effective cones of blowups of b1 b7 b2 and b1 b7 b3 in up to 6 points and proves these varieties are log Fano.

Findings

Effective cones are rational polyhedral.

All studied blowups are log Fano.

Provides a conceptual explanation for the polyhedrality.

Abstract

We compute the cones of effective divisors on blowups of $\mathbb{P}^1 \times \mathbb{P}^2$ and $\mathbb{P}^1 \times \mathbb{P}^3$ in up to 6 points. We also show that all these varieties are log Fano, giving a conceptual explanation for the fact that all the cones we compute are rational polyhedral.