The Family Blowup Formula of the Family Seiberg-Witten Invariants

TL;DR

This paper formulates and derives the family blowup formula for family Seiberg-Witten invariants, crucial for enumerative geometry and counting singular curves on algebraic surfaces.

Contribution

It provides both topological and algebraic derivations of the family blowup formula for Seiberg-Witten invariants, expanding their applicability.

Findings

Derived the family blowup formula using family index theorem.

Defined algebraic family Seiberg-Witten invariants for algebraic surfaces.

Established the algebraic derivation of the blowup formula.

Abstract

In the paper we formulate and derive the family blowup formula of family Seiberg-Witten invariants. The formula has been used in the enumerative application of counting singular curves on algebraic surfaces. We first give a topological derivation of the formula by using family index theorem. Then we define the algebraic (family) Seiberg-Witten invariants for algebraic surfaces and then give an algebraic derivation of the family blowup formula for the algebraic family Seiberg-Witten invariants.