Instanton Sheaves on Ruled Fano 3-folds of Picard Rank 2 and Index 1

We study rank 2 $h$-instanton sheaves on projective threefolds. We demonstrate that any orientable rank 2, non-locally free $h$-instanton sheaf with defect 0 on a threefold can be obtained as an elementary transformation of a locally free $h$-instanton sheaf. Our focus then shifts to ruled Fano threefolds of Picard rank 2 and index 1, of which there are five deformation classes. We establish the existence of orientable rank 2 $h$-instanton bundles on such varieties. Additionally, we prove the existence of Ulrich bundles on such varieties, which correspond to $h$-instanton sheaves of minimum charge.

💡 Research Summary

The paper develops a systematic theory of (h)-instanton sheaves on three‑dimensional projective varieties, with a focus on ruled Fano threefolds of Picard rank 2 and index 1. An (h)-instanton sheaf (defect 0) is defined as a torsion‑free sheaf (E) on a polarized threefold ((X,h)) satisfying the cohomology vanishings
(H^0(E(-h))=H^3(E(-3h))=0) and (H^1(E(-2h))=H^2(E(-2h))=0).
The integer (k(E)=h^1(E(-h))=h^2(E(-3h))) is called the charge; (k(E)=0) precisely characterises Ulrich bundles. The authors also introduce the notion of orientability, requiring (c_1(E)=(4h+K_X)\operatorname{rk}(E)/2), and define rank‑0 (h)-instantons as pure one‑dimensional sheaves (T) with (H^p(T(-2h))=0) for all (p). The degree (d_T=\chi(T(-h))) plays the role of a “charge” for rank‑0 objects.

A key structural result (Theorem 3.2) shows that any non‑locally‑free, orientable rank‑2 (h)-instanton sheaf (E) on a smooth irreducible threefold can be recovered as an elementary transformation of a locally free (h)-instanton sheaf. Concretely, the double dual (E^{\vee\vee}) is locally free, and the quotient (T_E:=E^{\vee\vee}/E) is a rank‑0 (h)-instanton sheaf. Conversely, given a locally free instanton sheaf and a rank‑0 instanton (T), the kernel of an epimorphism (E\to T) yields a new instanton sheaf whose charge is increased by (d_T). This establishes a bridge between locally free and non‑locally free instantons and provides a method to construct higher‑charge examples.

The paper then specializes to the five deformation families of ruled Fano threefolds (X_c=\mathbb P(\mathcal F_c)) where (\mathcal F_c) is a rank‑2 Fano bundle on (\mathbb P^2). The Picard group is generated by (\xi=c_1(\mathcal O_{X_c}(1))) and (f=\pi^*c_1(\mathcal O_{\mathbb P^2}(1))); the polarization is taken to be (h=\xi+f), which is ample. For each (c) the authors prove the existence of orientable, (\mu)-stable rank‑2 instanton bundles with prescribed Chern classes \