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In two very detailed, technical, and fundamental works, Jun Li constructed a theory of Gromov-Witten invariants for a singular scheme of the gluing form Y_1\cup_D Y_2 that arises from a degeneration Y_1\cup_D Y_2 and a theory of relative Gromov-Witten invariants for a codimension-1 relative pair Y_1\cup_D Y_2 . As a summit, he derived a degeneration formula that relates a finite summation of the usual Gromov-Witten invariants of a general smooth fiber Y_1\cup_D Y_2 of Y_1\cup_D Y_2 to the Gromov-Witten invariants of the singular fiber Y_1\cup_D Y_2 via gluing the relative pairs Y_1\cup_D Y_2 and Y_1\cup_D Y_2 . The finite sum mentioned above depends on a relative ample line bundle Y_1\cup_D Y_2 on Y_1\cup_D Y_2 . His theory has already applications to string theory and mathematics alike. For other new applications of Jun Li's theory, one needs a refined degeneration formula that depends on a curve class Y_1\cup_D Y_2 in Y_1\cup_D Y_2 or Y_1\cup_D Y_2 , rather than on the line bundle Y_1\cup_D Y_2 . Some monodromy effect has to be taken care of to deal with this. For the simple but useful case of a degeneration Y_1\cup_D Y_2 that arises from blowing up a trivial family Y_1\cup_D Y_2 , we explain how the details of Jun Li's work can be employed to reach such a desired degeneration formula. The related set Y_1\cup_D Y_2 of admissible triples adapted to Y_1\cup_D Y_2 that appears in the formula can be obtained via an analysis on the intersection numbers of relevant cycles and a study of Mori cones that appear in the problem. This set is intrinsically determined by Y_1\cup_D Y_2 and the normal bundle Y_1\cup_D Y_2 of the smooth subscheme Y_1\cup_D Y_2 in Y_1\cup_D Y_2 to be blown up.

We obtain a growth estimate for the number of lattice points inside any Q-Gorenstein cone. Our proof uses the result of Futaki-Ono-Wang on Sasaki-Einstein metric for the toric Sasakian manifold associated to the cone, a Yau’s inequality, and the Kawasaki-Riemann-Roch formula for orbifolds.

For any nonsimply laced Lie group G and elliptic curve , we show that the moduli space of flat G bundles over  can be identified with the moduli space of rational surfaces with G-configurations which contain  as an anticanonical curve. We also construct Lie(G)-bundles over these surfaces. The corresponding results for simply laced groups were obtained by the authors in another paper. Thus, we have established a natural identification for these two kinds of moduli spaces for any Lie group G.

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