H-Instanton Bundles on Three-Dimensional Smooth Toric Varieties with Picard Number Two

TL;DR

This paper investigates $H$-instanton bundles on a family of smooth threefolds with Picard number two, providing monadic descriptions, characterizations, and existence results that extend prior work on special cases.

Contribution

It introduces two monadic descriptions of $H$-instanton bundles on $X_e$, generalizing classical monads on $P^3$, and explores their existence and moduli for various Chern classes.

Findings

Two monadic descriptions of $H$-instanton bundles on $X_e$

Characterization of bundles with specific second Chern class support

Existence of bundles for all admissible second Chern classes when $e \,\leq\, 3$

Abstract

We study $H$-instanton bundles on the infinite family of smooth three-dimensional varieties $X_e=\mathbb{P}(\mathcal{O}_{\mathbb{P}^2} \oplus \mathcal{O}_{\mathbb{P}^2}(e))$, for $e \geq 0$. We provide two distinct monadic descriptions of $H$-instanton bundles on $X_e$, generalizing the classical monads on $\mathbb P^3$. We then characterize $H$-instanton bundles with second Chern class supported in a single degree, and investigate their existence and moduli spaces. Finally, for $e\leq 3$, we prove the existence of $H$-instanton bundles for all admissible second Chern classes. These results extend previous constructions on specific cases and contribute to the study of instanton bundles on threefolds with higher Picard number.