Mond's conjecture for corank-one maps from $\mathbb C^3$ to $\mathbb C^4$

Abstract

We prove Mond's conjecture for $\mathcal A$-finite corank-one map germs $(\mathbb C^3,0)\to(\mathbb C^4,0)$: the $\mathcal A_e$-codimension is at most the image Milnor number, with equality for quasihomogeneous germs. For a quasihomogeneous germ we determine the derivations of its image modulo the conductor fields, in terms of the double-point surface, its cross-cap curve and its normalisation. This yields a closed formula, depending only on the weights and degrees, for the Hilbert series of the graded $\mathcal A_e$-normal space; its value at $t=1$ coincides with Ohmoto's formula for the image Milnor number, which comes from Segre--Schwartz--MacPherson Thom polynomials. For arbitrary germs we show that generic homogeneous corank-one germs of coprime degrees are $\mathcal A$-finite in every source dimension, and apply the reduction theorem of Fern\'andez de Bobadilla, Nu\~no-Ballesteros and Pe\~nafort Sanchis. In every pair of nice dimensions this reduces Mond's conjecture for corank-one germs to an inequality for generic homogeneous germs.