## Lech inequalities for deformations of singularities defined by power products of degree 2.(English)Zbl 1052.14008

Summary: Using a result from B. Herzog [Kodaira-Spencer maps in local algebra. Lect. Notes Math. 1597 (1994; Zbl 0809.13011)] we prove the following. Let $$(B_0,{\mathfrak n}_0)$$ be an artinian local algebra of embedding dimension $$v$$ over some field $$L$$ with tangent cone $$\text{gr}(B_0)\cong L[X_1,\ldots ,X_v]/I_0$$. Suppose the ideal $$I_0$$ is generated by power products of degree 2. Then for every residually rational flat local homomorphism $$(A,{\mathfrak m})\to (B,{\mathfrak n})$$ of local $$L$$-algebras that has a special fiber isomorphic to $$B_0$$ the $$(v+1)$$th sum transforms of the local Hilbert series of $$A$$ and $$B$$ satisfy the Lech inequality $$H_A^{v+1}\leq H_B^{v+1}$$.

### MSC:

 14B12 Local deformation theory, Artin approximation, etc. 13H15 Multiplicity theory and related topics

### Keywords:

Lech problem; $$L$$-algebras; local rings

Zbl 0809.13011
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