Electronic Thesis and Dissertation Repository

Thesis Format

Monograph

Degree

Doctor of Philosophy

Program

Mathematics

Supervisor

Dhillon, Ajneet

2nd Supervisor

Lemire, Nicole

Co-Supervisor

Abstract

Essential dimension of a geometric object is roughly the number of algebraically independent parameters needed to define the object. In this thesis we give upper bounds for the essential dimension of parabolic bundles over a non-singular curve X of genus g greater than or equal to 2 using Borne's correspondence between parabolic bundles on a curve and vector bundles on a root stack. This is a generalization of the work of Biswas, Dhillon and Hoffmann on the essential dimension of vector bundles, by following their method for curves and adapting it to root stacks. In this process, we invoke the Riemann-Roch theorem of Toen for Deligne-Mumford stacks and use it to estimate the dimension of stack of parabolic bundles on X.

Summary for Lay Audience

Algebraic geometry is the study of solution sets of polynomials called `Varieties'. For example, by solving the equations $ax + by + c = 0$ and $x^{2} + y^{2} = r^{2}$, we find that a line and a circle intersect at a maximum of two points in the Cartesian plane. One can ask solutions of higher degree polynomials in several variables. One of the less understood, yet seemingly tractable problem is the study of algebraic curves, these are roughly the varieties of dimension $1$. One of the key methods of studying varieties is the study of functions on them. A vector bundle is roughly an assignment of a vector space to each point in the curve. The concept of vector bundles, in some sense, encode `generalized functions' on varieties. Parabolic vector bundles are vector bundles along with the extra data of a sequence of subspaces at finitely many points on the curve.

It is useful to study the `moduli space of parabolic vector bundles' on a curve. In a naive sense this is just the collection of all parabolic vector bundles on the curve. But we want more than that: It would be useful to have the collection of parabolic vector bundles on a curve as a honest 'variety'. This is not possible unless we restrict the class of vector bundles to a special type. Simply put, the reason for the failure is the `symmetries' of the vector bundles. Taking these symmetries into consideration, we can construct a `moduli stack of vector bundles'. This is not exactly a variety but it enjoys several geometric properties of a variety. Hence one can ask, what are the `degrees of freedom' in the moduli stack of parabolic bundles. This number turns out to be the `essential dimension' of the stack.

The essential dimension of an object, in essence, is the number of parameters required to determine the object. In this thesis we use several techniques of modern algebraic geometry, especially of algebraic stacks to find the maximum number of parameters required to determine a parabolic bundle over a large class of nice algebraic curves.

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