Cones of Cycles on blowups of $({\mathbb P}^1)^n$

TL;DR

This paper investigates the structure of pseudoeffective cycle cones on blowups of products of projective lines, revealing they are often generated by exceptional and fiber classes, with implications for algebraic geometry.

Contribution

It provides new insights into the structure of pseudoeffective cones on blowups of $(\mathbb{P}^1)^n$, highlighting cases where they are generated by specific classes.

Findings

Cones are generated by exceptional and fiber classes in certain cases.

Structural results depend on the position of points in the blowup.

The work advances understanding of cycle cones on blowups of product varieties.

Abstract

We study cones of pseudoeffective cycles on the blow up of $({\mathbb P}^1)^n$ at points in very general position, proving some results concerning their structure. In particular we show that in some cases they turn out to be generated by exceptional classes and fiber classes relatively to the projections onto a smaller number of copies of projective lines.