Yongqiang Liu, Guillermo Peñafort Sanchis, Matthias Zach

• Let \(f: (\mathbb {C}^n,S)\rightarrow (\mathbb {C}^p,0)\) be a finite map-germ with \(n < p\) and \(Y_\delta\) the image of a small perturbation \(f_\delta\) . We show that the reduced cohomology of $Y_\delta$ is concentrated in a range of degrees determined by the dimension of the instability locus of \(f\) . In the case \(n\ge p\) , we obtain an analogous result, replacing finiteness by \(\mathcal {K}\) -finiteness and \(Y_\delta\) by the discriminant \(\Delta (f_\delta)\) . We also study the monodromy associated to the perturbation \(f_\delta\) .