| For projective 3-folds, M.N.O.P. raised a conjecture to identify partition functions of Donaldson-Thomas and Gromov-Witten theories after an exponential-type parameter change. In 2006, R.Pandharipande initiated the program to prove it by degenerations where the relative P 1 scroll plays a fundamental role. Let S be a projective surface, and L a line bundle on S. Consider the three-fold X to be the compactification of L. Assume further that S has a smooth canonical curve C. Then the author study the virtual cycle of moduli space of ideal sheaves by generalizing theta-localization methods in Gromov-Witten theory of pg>0 surfaces. In this work the author applied the program to the case where the base surface is K3 blown up at one point which corresponds to g c=0. For general gc a vanishing theorem is proved and asserts the only curve classes that contribute to the invariants are those whose components on S are multiples of [C]. |
