On pseudoeffective cones of projective bundles and volume function

TL;DR

This paper computes the pseudo-effective cones of projective bundles over higher-dimensional varieties and analyzes the volume function on fiber products of projective bundles, revealing polynomial behavior in specific cases.

Contribution

It provides explicit descriptions of pseudo-effective cones for certain projective bundles and characterizes the volume function on fiber products over curves, including polynomial cases.

Findings

Pseudo-effective cones computed for various projective bundles.

Volume function on fiber products over curves shown to be polynomial.

Results apply under specific assumptions on base varieties and bundles.

Abstract

In this article, we compute the pseudo-effective cones of various projective bundles $\mathbb{P}_X(E)$ over higher dimensional varieties $X$ under some assumptions on $X$ as well as on the vector bundle $E$. We also compute the volume function on fibre product $\mathbb{P}(E)\times_C\mathbb{P}(F)$ of two projective bundles over a smooth irreducible complex projective curve $C$. In particular, we show that the volume function on the fiber product of two ruled surfaces is of polynomial type.