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

TL;DR

This work investigates the pseudoeffective cones of $k$-cycles on the blow-up $X_r^n$ of $({ m P}^1)^n$ at $r$ very general points. It introduces fiber-generated cones ${ m CF}_k(X_r^n)$ and a nefness criterion that characterizes when $ ext{Eff}_k(X_r^n)$ equals their fiber-generated subcone, proving sharp bounds for $k=1$ and $k=n-1$ and a general bound $r eq n-k+1$ ensuring fiber-generation in many cases. The paper also analyzes when these cones fail to be fiber-generated for large $r$, discusses toric cases, gives explicit results in low dimensions (e.g., $X_4^4$), and connects to broader questions about Mori dream spaces and polyhedrality of the effective cones. Overall, it advances understanding of the structure of higher-codimension effective cones on blow-ups of products of projective lines and provides concrete criteria and bounds for fiber-generation.

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.

Theorems & Definitions (30)

• Theorem 1
• Theorem 2
• Proposition 1
• Remark 1.1
• Lemma 1.2
• proof
• Lemma 1.3
• proof
• Lemma 1.4
• proof