8 citations · 18 across the 10 of their papers we have counts for
10 papers
Cosmic String, Harvey-Moore Conjecture and Family Seiberg-Witten Theory
Ai-Ko Liu
In this paper, we study the enumeration of virtual numbers of immersed nodal curves along certain Calabi-Yau K3 fibrations. By using the concept of cosmic strings, we verify the mo…
The Residual Intersection Formula of Type II Exceptional Curves
Ai-Ko Liu
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).…
A Note About Universality Theorem as an Enumerative Riemann-Roch Theorem
Ai-Ko Liu
The paper is a short supplement of the longer paper "The Algebraic Proof of the Universality Theorem", preprint math.AG/0402045. In this short note, we outline the geometric meanin…
The Algebraic Proof of the Universality Theorem
Ai-Ko Liu
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 g…
Family Switching Formula and the Exceptional Rational Curves
Ai-Ko Liu
In this paper, we derive the family switching formula of -n two-sphere fiber bundle embedded in a smooth four-manifold fiber bundle. In the smooth category, it is a partial general…
A Note on Curve Counting Scheme in an Algebraic Family and The Admissible Decomposition Classes
Ai-Ko Liu
Mcduff had proposed in 1997 a way to modify the definition of Taubes' version of Gromov invariant when multiple coverings of -1 curves appear. In this paper we generalize Mcduff's…