2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/67522In 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.79 pages, supplement of the long paper: family blowup formula, admissible graphs and the enumeration of singular curves, IDifferential GeometryAlgebraic Geometry57R17; 14N99text