Lower estimate of multiplicity of isolated complete intersection singularities with applications in weakly elliptic singularities (Q1882364)

Let \((V,0)\) be a two-dimensional complete intersection weakly elliptic singularity with muliplicity \(m\). It is proved that \((V,0)\) is a hypersurface singularity if and only if \(m=2\) or \(m=3\), \((V,0)\) is not a hypersurface if and only if \(m=4\). In this case \((V,0)\) is defined by two functions in \(\mathbb C^4\), each of which has multiplicity too. The proof is based on the following results: Let \((V,0)\) be an isolated complete intersection of dimension \(n\). Let \(n+k\) be the embedding dimension and \(f_1, \ldots, f_k\) defining equations of \((V,0)\). Then \(\text{mult}(V,0)\geq \text{mult} (f_1, 0)\cdot\ldots\cdot \text{mult} (f_k, 0)\). Let \((V,0)\) be the germ of a normal two-dimensional complete intersection weakly elliptic singularity. If \(\text{mult}(V,0)\geq 3\) then \(\text{mult}(V,0)= \text{embdim} (V,0)\).