Enumeration of real conics and maximal configurations (Q388770)

The Gromov-Witten invariant of the complex projective space is the number of rational curves of a given degree, passing through a generic collection of subspaces of given dimensions. An important problem of real algebraic geometry is the real version of this question: is it possible that, for certain generic real subspaces, none of the corresponding rational curves are real, or, on the contrary, all of them are real?NEWLINENEWLINERegarding the first part of the question, a major breakthrough was the definition of the Welschinger invariant, which gave a negative answer for every degree in 2 and 3 dimensions (see [\textit{J.~Y.~Welschinger}, Invent. Math. 162, No. 1, 195--234 (2005; Zbl 1082.14052)]; [Duke Math. J. 127, No. 1, 89--121 (2005; Zbl 1084.14056)] and [\textit{I. V. Itenberg} et al., Russ. Math. Surv. 59, No. 6, 1093--1116 (2004); translation from Usp. Mat. Nauk 59, No. 6, 85--110 (2004; Zbl 1086.14047)]). Regarding the second part, the only general result was the positive answer for degree 1 in any dimension, see [\textit{F.~Sottile}, Duke Math. J. 87, No. 1, 59--85 (1997; Zbl 0986.14033)] and [\textit{R.~Vakil}, Ann. Math. (2) 164, No. 2, 489--512 (2006; Zbl 1115.14043)] (even this linear algebraic special case is very complicated).NEWLINENEWLINEIn the present paper, a positive answer is given for degree 2 in any dimension: any Gromov-Witten enumerative problem for conics admits a generic real collection of subspaces such that all conics passing through these subspaces are real. The proof is based on Mikhalkin's tropical correspondence theorem (see [\textit{G. Mikhalkin}, J. Am. Math. Soc. 18, 313--377 (2005; Zbl 1092.14068)] and [\textit{B. Siebert} and \textit{T. Nishinou}, Duke Math. J. 135, No. 1, 1--51 (2006; Zbl 1105.14073)]) and the technique of floor diagrams (which was invented to combinatorially simplify the tropical correspondence and was already used earlier in [\textit{E. Brugallé} and \textit{G. Mikhalkin}, C. R., Math., Acad. Sci. Paris 345, No. 6, 329--334 (2007; Zbl 1124.14047)] to prove some special cases of the result of this paper).