Title:

On the Hilbert series of polarised orbifolds

We are interested in calculating the Hilbert series of a polarised orbifold (X;D) (that is D is an ample divisor on an orbifold X). Indeed, its numerical data is encoded in its Hilbert series, so that calculating this sometimes gives us information about the ring, notably possible generators and relations, using the Hilbert syzygies theorem. Vaguely, we have PX(t) = Num/Denom where Num is given by the relations and syzygies of R and Denom is given by the generators. Thus in particular we hope that we can use the numerical data of the ring to deduce possible explicit constructions. A reasonable goal is therefore to calculate the Hilbert series of a polarised (X;D); we write it in closed form, where each term corresponds to an orbifold stratum, is Gorenstein symmetric and with integral numerator of "short support". The study of the Hilbert series where the singular locus has dimension at most 1 leads to questions about more general rational functions of the form __N___ II(1tai) with N integral and symmetric. We prove various parsings in terms of the poles at the Uai ; each individual term is Gorenstein symmetric, with integral numerator of "short support" and geometrically corresponds to some orbifold locus. Chapters 1 and 2 are expository material: Chapter 1 is basic introductory material whilst in Chapter 2 we explain the Hilbert series parsing in the isolated singularity case, as solved in Buckley et al. [2013] and Zhou [2011] and go over worked examples for practice. Chapter 3 uses the structure of the parsing in the isolated case and the expected structure in the nonisolated case to discuss generalisations to arbitrary rational functions with symmetry and poles only at certain roots of unity. We prove some special cases. Chapter 4 discusses the Hilbert series parsing in the curve orbifold locus case in a more geometrical setting. Chapter 5 discusses further generalisations and issues. In particular we discuss how the strategies used in Chapter 3 could work in a more general section, and the non symmetric case.
