Examples of vanishing Gromov-Witten-Welschinger invariants (Q2788592)

It is known that the Gromov-Witten invariants of the plane, projective spaces, and many other convex varieties almost never vanish. In turn, Welschinger invariants (which are a kind of open Gromov-Witten invariants) counting real rational curves in the plane or in the projective three-space vanish in a series of cases, in particular, all Welschinger invariants of \({\mathbb P}^3\) of any even degree vanish. This primarily due to the fact that Welschinger invariant counts real rational curves with signs, which cancel out in summation. The author addresses the question in which cases the vanishing of the Weslchinger invariant is caused by the lack of real rational curves matching certain constraints.NEWLINENEWLINEThe main author's result states that, for any \(d\geq2\) there exists a configuration of \(2d\) real points in \({\mathbb P}^3\) in general position such that no real rational curve of degree \(d\) hits the entire given configuration (while for \(d\geq3\) there many complex rational curves passing through such a configuration). The core of the proof is a construction of a specific configuration of \(2d\) points on a smooth real elliptic quartic curve in \({\mathbb P}^3\) represented as an intersection of two real quadric surfaces. The constructed configuration possesses the property that it is not contained in the conjugation-invariant union of rational curves of the total degree \(\leq2d\). Pushing this configuration into a general position one obtains a configuration not lying on any real rational curve of degree \(2d\). The author considers some other similar examples of small degree like plane rational cubics passing through four pairs of complex conjugate points, or conics in \({\mathbb P}^3\) hitting conjugation-invariant configurations of points and lines.