From Hitchin Systems to Rational Elliptic Surfaces with C*-actions via Orbifold Hilbert Schemes

Using orbifold Hilbert schemes, we compactify all two-dimensional Hitchin systems corresponding to types A0-tilde, D4-tilde, E6-tilde, E7-tilde, and E8-tilde, thereby obtaining four rational elliptic surfaces with C*-actions. Their singular fibers and relative minimal models are listed in the main table. A particularly interesting point is that we found they can all be obtained by performing a finite number of blow-ups on the second Hirzebruch surface. To this end, we prove that Hilbert schemes of orbifold surfaces are connected smooth projective schemes under suitable conditions, and we use the Hilbert-Chow morphism to construct the minimal resolutions of the coarse moduli spaces.