Universal cohomological expressions for singularities in families of genus 0 stable maps

We consider families of curve-to-curve maps that have no singularities except those of genus 0 stable maps and that satisfy a versality condition at each singularity. We provide a universal expression for the cohomology class Poincar\'e dual to the locus of any given singularity. Our expressions hold for any family of curve-to-curve maps satisfying the above properties.