AI 中文总结
证明Mond猜想对从三维到四维的余秩一映射芽成立，通过拟齐次情形下的闭式公式及约化定理，将一般情形归结为齐次芽的不等式。
AI 中文摘要
我们证明了Mond猜想对$\mathcal A$-有限余秩一映射芽$(\mathbb C^3,0)\to(\mathbb C^4,0)$成立：$\mathcal A_e$-余维数至多等于像的Milnor数，且对拟齐次芽等式成立。对于拟齐次芽，我们通过双点曲面、其交叉帽曲线及其正规化，确定了其像的导子模去导子场后的结构。这给出了一个仅依赖于权重和次数的闭式公式，用于计算分次$\mathcal A_e$-正规空间的Hilbert级数；其在$t=1$处的值恰好与Ohmoto关于像的Milnor数的公式一致，该公式源自Segre--Schwartz--MacPherson Thom多项式。对于任意芽，我们证明互素次数的通用齐次余秩一芽在任意源维数下都是$\mathcal A$-有限的，并应用了Fernández de Bobadilla、Nuño-Ballesteros和Peñafort Sanchis的约化定理。在每一对良好维数中，这可将余秩一芽的Mond猜想约化为对通用齐次芽的一个不等式。
英文摘要
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 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.
