Using the degeneration technique, we study the behavior of Welschinger invariants under the blow-up and obtain some blow-up formulae of Welschinger invariants. To analyze the variation of Welschinger invariants when replacing a pair of real points in the real configuration by a pair of conjugated points, Welschinger introduced the $\theta $-invariant. In this paper, we also verify that the $\theta $-invariant is the Welschinger invariant of the blow-up of the symplectic $4$-manifold.
"Welschinger invariants of blow-ups of symplectic 4-manifolds." Rocky Mountain J. Math. 48 (4) 1105 - 1144, 2018. https://doi.org/10.1216/RMJ-2018-48-4-1105