Mond's conjecture for corank-one maps from ℂ³ to ℂ⁴

@misc{rimanyi2026monds,
  title = {{Mond's conjecture for corank-one maps from $\mathbb C^3$ to $\mathbb C^4$}},
  author = {Richard Rimanyi},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.03331},
  eprint = {2610.03331},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.03331},
}

TY  - UNPB
TI  - Mond's conjecture for corank-one maps from ℂ^3 to ℂ^4
AU  - Richard Rimanyi
PY  - 2026
DA  - 2026/10/02
PB  - arXiv
DO  - 10.48550/arxiv.2610.03331
UR  - https://arxiv.org/abs/2610.03331
AB  - We prove Mond's conjecture for 𝒜-finite corank-one map germs (ℂ^3,0)→(ℂ^4,0): the 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 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 𝒜-finite in every source dimension, and apply the reduction theorem of Fernández de Bobadilla, Nuño-Ballesteros and Peñafort Sanchis. In every pair of nice dimensions this reduces Mond's conjecture for corank-one germs to an inequality for generic homogeneous germs.
ER  -