Research Projects

A central problem in Algebraic Geometry is the classification of objects under a given equivalence relation, which involves constructing a Moduli space that effectively captures the structure and complexity of these objects. This is known as a Moduli problem. Our focus is on solving this problem for vector bundles on smooth, proper Deligne-Mumford stacks over an algebraically closed field of an arbitrary characteristic. In the special case of an orbifold curve, that is, a smooth proper one-dimensional Deligne-Mumford stack which is generically an algebraic curve (here, the structure of the stack is completely determined by an algebraic curve together with a finite data of certain Galois field extensions), this problem is open when the base field has prime characteristic. Recently, we have given a construction of the Moduli space of orbifold-semistable bundles on an orbifold curve. Our first goal is to obtain this Moduli space through invariant theory, to understand the geometric properties of this space (for example, the connected components, normality, dimensional estimate), and to understand the morphisms between such spaces. We also want to study the deformation of bundles in the corresponding Moduli stack. More precisely, we want to relate the notion of formal patching for covers using these deformations. The invariants of the Moduli problem are generally interpreted in terms of cohomological vanishing criteria. Such criteria are closely related to finding the Raynaud bundles, to understand the vanishing locus of the ’Theta-divisors’, Clifford’s Theorem, and the Brill-Noether Loci. Each of these problems lies at the core of the theory of bundles. We are also vastly interested in studying the interplay between covers of orbifold curves with their ramification-theoretic invariants and certain aspects of the moduli space of bundles on them. The study of covers is related to understanding the etale fundamental groups for curves, which remains a central difficult problem in Arithmetic Algebraic Geometry. This is our final goal: to construct Galois covers of curves and of orbifold curves from a bundle-theoretic perspective. The ongoing and proposed work relate different advanced fields in mathematics: algebraic stacks, bundles on them with their cohomologies, parabolic structures, and modular representation of finite groups.