  We develop new techniques to study regularity questions for moduli spaces of
pseudoholomorphic curves that are multiply covered. Among the main results, we
show that unbranched multiple covers of closed holomorphic curves are
generically regular, and simple index zero curves in dimensions greater than
four are generically super-rigid, implying e.g. that the Gromov-Witten
invariants of Calabi-Yau 3-folds reduce to sums of local invariants for finite
sets of embedded curves. We also establish partial results on super-rigidity in
dimension four and regularity of branched covers, and briefly discuss the
outlook for bifurcation analysis. The proofs are based on a general
stratification result for moduli spaces of multiple covers, framed in terms of
a representation-theoretic splitting of Cauchy-Riemann operators with
symmetries.
