The Residual Intersection Formula of Type II Exceptional Curves
  The paper is a part of our program to build up a theory of couting immersed nodal curve on algebraic surfaces, as an enumerative Riemann-Roch theory (outlined in math.AG/0405113). In this paper, we discuss the excess intersection theory of the so-called type two exceptional curves, which plays the analogous role as the "index of speciality" (h^2) in the classical surface Riemann-Roch formula.   We show that the algebraic family Seiberg-Witten theory of type I exceptional curves can be generalized to the theory of type II exceptional curves when the family moduli spaces are not regular of the expected dimension. 
Ai-Ko Liu