Enumerative geometry for real varieties

We survey the problem of whether a given problem in enumerative geometry can have all of its solutions be real. After a survey of known results, we show how real effective rational equivalence can be used to show some enumerative problems involving the Schubert calculus on Grassmannians may have all of their solutions be real. This is illustrated with a new example; the problem of 42 planes in P⁵ meeting 9 general planes is fully real. We conclude with a description of the work of Ronga-Tognoli-Vust showing that there are 5 general real conics in the plane so that all of the 3264 conics tangent to the 5 are real. A supplement contains diagrams illustrating some configurations of real conics. In the last section, we describe an application of these ideas to algorithms to find exact solutions to certain enumerative problems, and strong evidence for a conjecture of Shapiro and Shapiro.