The classification problem for Higgs bundles of a fixed rank and degree on a compact Riemann surface is encoded in the moduli stack of such objects. Nitsure's GIT construction of the moduli space of semistable Higgs bundles produces a quasi-projective coarse moduli space (if the rank and degree are coprime) for the semistable stratum of this moduli stack. Nevertheless, GIT cannot be used to produce coarse moduli spaces for the complement of the semistable stratum. In this paper we use a recent generalisation of GIT, called Non-Reductive GIT, to construct two stratifications of the stack of Higgs bundles which satisfy the property that each stratum admits a quasi-projective coarse moduli space with an explicit projective completion. The first is a refinement of the Higgs Harder-Narasimhan stratification of the stack of Higgs bundles (defined by the instability type of the Higgs bundle), while the second is a refinement of the Harder-Narasimhan stratification (defined by the instability type of the underlying bundle). We provide a complete and explicit moduli-theoretic description of both refined stratifications in the rank 2 case.
We study three instability stratifications of the stack of twisted Higgs bundle of a fixed rank and degree on a smooth complex projective curve. The first is the Harder–Narasimhan (HN) stratification, defined by the instability type of the Higgs bundle. The second is the bundle Harder–Narasimhan (bHN) stratification, defined by the instability type of the underlying bundle. While an unstable HN stratum fibres over the stack parameterizing Higgs bundles which are isomorphic to their HN graded, this is not true for Higgs bundles of unstable bHN type. Obtaining such a fibration requires refining the bHN stratification; this is the third instability stratification. After introducing these three stratifications, we establish comparison results. In particular, we obtain explicit criteria for determining semistability of a Higgs bundle of low rank with unstable underlying bundle. Then we show how the HN and bHN stratifications can be used to filter the stack of Higgs bundles by global quotient stacks in two different ways. Finally, we use these filtrations to relate the HN and bHN stratifications to GIT instability stratifications and the refined bHN stratification to a Bialynicki–Birula stratification.
We address the problem of classifying complete C-subalgebras of C [[t]]. A discrete invariant for this classification problem is the semigroup of orders of the elements in a given C-subalgebra. Hence we can define the space RΓ of all C-subalgebras of C[[t]] with semigroup Γ. After relating this space to the Zariski moduli space of curve singularities and to a moduli space of global singular curves, we prove that RΓ is an affine variety by describing its defining equations in an ambient affine space in terms of an explicit algorithm. Moreover, we identify certain types of semigroups Γ for which RΓ is always an affine space, and for general Γ we describe the stratification of RΓ by embedding dimension. We also describe the natural map from RΓ to the Zariski moduli space in some special cases. Explicit examples are provided throughout.
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