Zhou, Shengtian (2011) Orbifold Riemann-Roch and Hilbert series. PhD thesis, University of Warwick.

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Abstract

A general Riemann-Roch formula for smooth Deligne-Mumford stacks was obtained by Toen [Toë99]. Using this formula, we obtain an explicit Riemann-Roch formula for quasismooth substacks of weighted projective space, following the ideas in [Nir]. The Riemann-Roch formula enables us to study polarized orbifolds in terms of the associated Hilbert series. Given a polarized projectively Gorenstein quasismooth pair (X, Od∈Z O(d)), we want to parse the Hilbert series P(t) = ∑d>=0 h0(X,OX (d))td according to the orbifold loci. For X with only isolated orbifold points, we give a parsing such that each orbifold point corresponds to a closed term, which only depends on the orbifold type of the point and has Goresntein symmetry property and integral coefficients. Similarly, for the case when X has dimension <= 1 orbifold loci, we can also parse the Hilbert series into closed terms corresponding to orbifold curves and dissident points as well as isolated orbifold points. Our parsing of Hilbert series reflects the global symmetry property of the Gorenstein ring Od>=0 H0(X,OX (d))td in terms of its local data.

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