WDVV-Type Relations for Welschinger's Invariants: Applications

Abstract: We first recall Solomon's relations for Welschinger's invariants counting real curves in real symplectic fourfolds, announced in \cite{Jake2} and established in \cite{RealWDVV}, and the WDVV-style relations for Welschinger's invariants counting real curves in real symplectic sixfolds with some symmetry established in \cite{RealWDVV3}. We then explicitly demonstrate that in some important cases (projective spaces with standard conjugations, real blowups of the projective plane, and two- and three-fold products of the one-dimensional projective space with two involutions each), these relations provide complete recursions determining all Welschinger's invariants from basic input. We include extensive tables of Welschinger's invariants in low degrees obtained from these recursions with {\it Mathematica}. These invariants provide lower bounds for counts of real rational curves, including with curve insertions in smooth algebraic threefolds.