Enumeration of non-nodal real plane rational curves (Q6585669)

In his seminal paper [Invent. Math. 162, No. 1, 195--234 (2005; Zbl 1082.14052)], \textit{J.-Y. Welschinger} proved that the enumeration of real plane rational curves of a given degree \(d\), passing through a general conjugation-invariant configuration \(p\) of \(3d-1\) points, does not depend on the choice of \(p\), provided, the number of real points and the number of complex conjugate pairs in \(p\) are fixed, and each curve is provided with the so-called Welschinger sign \(\pm1\). More precisely, the sign of a real nodal curve is the product of contributions of all its nodes: a non-real node contributes factor~\(1\), a real hyperbolic node (the transverse intersection of two real smooth branches) contributes factor~\(1\), while a real elliptic node (the transverse intersection of two smooth complex conjugate branches) contributes factor~\(-1\). It is natural to ask whether Welschinger's idea provides invariants in other real enumerative problems. Welschinger showed that the invariance extends to enumeration of real pseudo-holomorphic curves in real rational symplectic four-folds [loc. cit.] and to enumeration of real rational curves in \(\mathbb{P}^3\) [\textit{J.-Y. Welschinger}, Duke Math. J. 127, No. 1, 89--121 (2005; Zbl 1084.14056)]. However, Welschinger's signed count fails to be invariant of the choice of constraints for the enumeration of real plane rational curves of a given degree, having one ordinary cusp and ordinary nodes as the rest of singularities [\textit{J.-Y. Welschinger}, Bull. Soc. Math. Fr. 134, No. 2, 287--325 (2006; Zbl 1118.53058)]. The reviewed paper aims to analyze in depth this situation and to answer the following\N\N{Question:} Does there exist a real enumerative invariant counting real plane rational curves of a given degree that have non-nodal singularities of prescribed types and nodes (if any) as the rest of singularities?\N\N{The author's answer} is: In a reasonable setting, there are no such real enumerative invariants except for a single case of three-cuspidal quartics.\N\NThe author shows that in the exceptional case the invariant enumerates real rational three-cuspidal quartics that pass through generically chosen four pairs of complex conjugate points. As a consequence, it is shown that through any generic configuration of four pairs of complex conjugate points, one can always trace at least two real rational three-cuspidal quartics.\N\NThe strategy of the proof is similar to the argument used in [\textit{J.-Y. Welschinger}, Bull. Soc. Math. Fr. 134, No. 2, 287--325 (2006; Zbl 1118.53058); Invent. Math. 162, No. 1, 195--234 (2005; Zbl 1082.14052); \textit{I. Itenberg} et al., in: Analysis meets geometry. The Mikael Passare memorial volume. Cham: Birkhäuser/Springer. 239--260 (2017; Zbl 1402.14074)] and consists in the analysis of wall-crossing events along a path joining two point constraints. As an example, the author considers real cuspidal cubics and describes two wall-crossing events breaking the invariance of enumeration (cf. [\textit{J.-Y. Welschinger}, Bull. Soc. Math. Fr. 134, No. 2, 287--325 (2006; Zbl 1118.53058)]). He proves three statements on the failure of enumeration invariance. At last, he demonstrates an example of a slightly modified real enumerative problem, in which the count of real curves appears to be invariant. A generalization of this observation will be considered in a forthcoming author's paper.