On Degenerations of Moduli of Hitchin Pairs

The purpose of this note is to announce certain basic results on the construction of a degeneration of M H Xk(n, d) as the smooth curve Xk degenerates to an irreducible nodal curve with a single node. Let Xk be a smooth projective curve of genus g ≥ 2 over an algebraically closed field k of characteristic zero and let L be a line bundle on Xk. A Hitchin pair (E, θ) is comprised of a torsion-free OXk-module E together with a OXk-morphism θ : E → E⊗L called the Higgs structure. Let M Xk(n, d) denote the moduli space of semistable Hitchin pairs on Xk with Higgs structure given by the line bundle L. The geometry of Hitchin pairs or Higgs bundles has been extensively studied for over twenty-five years beginning with Hitchin ([4], [5]), Nitsure ([8]), and Simpson ([11], [12], [13]). More precisely, let R be a discrete valuation ring with quotient field K and residue field an algebraically closed field k, for instance R = k[[t]]. Let S = Spec R, and Spec K the generic point and let s be the closed point of S. Let X → S be a proper, flat family with generic fibre XK a smooth projective curve of genus g ≥ 2 and with closed fibre Xs a irreducible nodal curve C with a single node p ∈ C. Assume that X is regular as a scheme over k. Let L be a relative line bundle on X and assume that deg(L|C) > deg(ωC), where ωC is the dualizing sheaf on C. Let (n, d) be a pair of integers such that gcd(n, d) = 1. Received by the editors May 4, 2013 and, in revised form, September 20, 2013.