Rendiconti dell’Istituto di Matematica dell’Università di Trieste: an International Journal of Mathematics vol.57 (2025)

We investigate the existence of some sporadic rank-r ⩾ 1 Ulrich vector bundles on suitable 3-fold scrolls X over the Hirzebruch surface F0, which arise as tautological embeddings of projectivization of very-ample vector bundles on F0 that are uniform in the sense of Brosius and Aprodu–Brinzanescu, cf. [11] and [4] respectively. Such Ulrich bundles arise as deformations of “iterative” extensions by means of sporadic Ulrich line bundles which have been contructed in our former paper [30] (where instead higher-rank sporadic bundles were not investigated therein). We explicitely describe irreducible components of the corresponding sporadic moduli spaces of rank r ⩾ 1 vector bundles which are Ulrich with respect to the tautological polarization on X. In some cases, such irreducible components turn out to be a singleton, in some other such components are generically smooth, whose positive dimension has been computed and whose general point turns out to be a slope-stable, indecomposable vector bundle.

Hecke modifications of vector bundles have played a significant role in several areas of mathematics. They appear in subjects ranging from number theory to complex geometry. This article intends to be a friendly introduction to the subject. We give an overview of how Hecke modifications appear in the literature, explain their origin and their importance in number theory and classical algebraic geometry. Moreover, we report the progress made in describing Hecke modifications explicitly and why these explicit descriptions are important. We describe all the Hecke modifications of the trivial rank 2 vector bundle over a fixed point of degree 5 on the projective line, as well as all the vector bundles over a certain elliptic curve, which admit a rank 2 and degree 0 trace bundle as a Hecke modification. This result is not present in existing literature.

These notes aim to develop a tool for constructing polynomial differential p-forms vanishing on prescribed loci through syzygies of homogeneous ideals. Examples are provided through implementing this method in Macaulay2, particularly examples of instanton bundles of charges 4 and 5 on P3 that arise in this construction.

This paper aims to investigate the existence of distributional solutions in ˚W 1,−→p (・)(Ω) (i.e. the anisotropic Sobolev space with variable exponents and zero boundary) for a class of nonlinear anisotropic elliptic equations with variable exponents and a lower-order term that has natural growth with respect to |∂iu|, i = 1, . . . ,N. The datum f on the right-hand side belongs to the space L(p∗)′(・)(Ω), where Ω ⊂ RN (N ≥ 2) is a bounded open Lipschitz domain and (p∗)′(・) represents the H¨older conjugate of the Sobolev conjugate p(・).

We classify involutions acting on spherical 3-manifolds up to conjugacy. Our geometric approach provides insights into numerous topological properties of these involutions.