  On a space of stable maps, the psi classes are modified by subtracting
certain boundary divisors. The top products of modified psi classes, usual psi
classes, and classes pulled back along the evaluation maps are called twisted
descendants; it is shown that in genus 0, they admit a complete recursion and
are determined by the Gromov-Witten invariants. One motivation for this
construction is that all characteristic numbers (of rational curves) can be
interpreted as twisted descendants; this is explained in the second part, using
pointed tangency classes. As an example, some of Schubert's numbers of twisted
cubics are verified.
