H-Instanton Bundles on Three-Dimensional Smooth Toric Varieties with Picard Number Two
Ozhan Genc, Francesco Malaspina
TL;DR
The paper extends the theory of mathematical instanton bundles to the threefolds $X_e=\mathbb{P}(\mathcal{O}_{\mathbb{P}^2}\oplus \mathcal{O}_{\mathbb{P}^2}(e))$ with Picard number two. It develops two monad descriptions of $H$-instanton bundles via Beilinson-type complexes, and characterizes those with second Chern class concentrated in a single degree, linking to pullbacks of linear monads on $\mathbb{P}^2$ and Ulrich pullbacks. Existence is established for $e\le 3$ using Hartshorne–Serre constructions, with detailed Ext-dimension computations and the notion of earnestness; the paper also constructs new instantons by elementary transformations along disjoint lines in $|f^2|$, showing that all admissible charges occur in this range. Overall, the work provides explicit monad descriptions, existence results, and moduli information for $H$-instantons on toric threefolds with higher Picard number, extending previous results on $e=0,1,2$.
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.