  In the long paper "Family Blowup formula, Admissible Graphs and the
Enumeration of Singular Curves (I)" (appearing in JDG), the author solved the
enumeration problem of nodal (or general singular) curve counting on algebraic
surfaces by using techniques from differential topology/symplectic geometry
(including family Seiberg-Witten theory) and some ideas derived from Taubes'
"SW=Gr". In the current paper, we offer an algebraic proof of the "universality
theorem", showing that the counting of nodal curves for 5$\delta$-1 very ample
complete linear systems are controled by universal polynomials of the
characteristic classes. The theorem (implicitly) was the backbone of the
earlier long paper(cited above). In the current paper, we derive the result by
using intersection theory and the concept of localized contributions of top
Chern classes, and therefore relaxing the dependence on the symplectic
techniques.
