In this note, we continue the study of Seshadri constants on blow-ups of Hirzebruch surfaces initiated in (Hanumanthu, K. et al.: Math. Nachr. 298(2), 437–455 2025). Now we consider blow-ups of ruled surfaces more generally. We propose a conjecture for classifying all the negative self-intersection curves on the blow-up of a ruled surface at very general points, analogous to the (-1)-curves conjecture in P2. Assuming this conjecture is true, we exhibit an ample line bundle with an irrational Seshadri constant at a very general point on such a surface.

In this article, we study the smoothness of the moduli space of finite quiver vector bundles over the smooth complex projective curves.

Let C be a comb-like curve over ℂ and let E be a vector bundle of rank n on C. In this paper, we investigate the criteria for the semistability of the restriction of E onto the components of C when E is given to be semistable with respect to a polarization w. As an application, assuming that each irreducible component of C is general in its moduli space, we investigate the w-semistability of kernel bundles on such curves, extending the results (completely for rank two and partially for higher rank) known in the case of a reducible nodal curve with two smooth components, but here using different techniques.

Let (Formula presented.) be integers and let (Formula presented.) denote the Hirzebruch surface with invariant (Formula presented.). We compute the Seshadri constants of an ample line bundle at an arbitrary point of the (Formula presented.) -point blow-up of (Formula presented.) when (Formula presented.) and at a very general point when (Formula presented.) or (Formula presented.). We also discuss several conjectures on linear systems of curves on the blow-up of (Formula presented.) at (Formula presented.) very general points.

Let C be a chain-like curve over C. In this paper, we investigate the rationality of moduli spaces of w-semistable vector bundles on C of arbitrary rank and fixed determinant by putting some restrictions on the Euler characteristics.

Let C be a chain-like curve having n smooth components and n−1 nodes, where n≥2. Let E be a vector bundle on C and V⊆H0(E) be a linear subspace generating E. We investigate the (semi)stability of the kernel bundle ME,V associated to (E,V).

We study the splitting of the pullback of the tangent bundle Tℙ3 on an elliptic curve under the morphism given by a line bundle of degree less or equal to 5.

We study semi-stability of the pullback of the tangent bundle TP2 on an elliptic curve under the morphism given by a line bundle of degree less or equal to 4.