MEGA 2007Effective Methods in Algebraic Geometry Strobl, Austria, June 25th - 29th |
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Electronic Proceedings
Abstract
| Title | Milnor algebras could be isomorphic to modular algebras |
| Keywords | deformation of singularities, modular stratum, milnor algebra |
| Abstract | We find and describe unexpected isomorphisms between two very
different objects associated to hypersurface singularities. One object is the Milnor algebra of a function, while the other object associated to a singularity is the local ring of the flatness stratum of the singular locus in a miniversal deformation, an invariant of the contact class of a defining function. Such isomorphisms exist for unimodal hypersurface singularities. However, for the moment it is badly understood, which principle causes these isomorphisms and how far this observation generalises. Let $X_0\subseteq \C^n$ be a germ of an isolated hypersurface singularity defined by an analytic function $f(x)=0$, $ f\in \C\{x\}$. An imported topological invariant of the germ is the Milnor number, which can be computed as the $\C$-dimension of the so-called Milnor algebra $Q(f)=\C\{x\}/(\partial f/\partial x)$. The Milnor algebra carries a canonical structure of a $\C[T]$-algebra defined by the multiplication with $f$. A special version of the Mather-Yau theorem states that the $\mathcal{R}$-class (right-equivalence class) of the function $f(x)$ with isolated critical point is fully determined by the isomorphism class of $Q(f)$ as $\C[T]$-algebra. However, there is richer structure on the Milnor algebra connected with the relative Milnor algebra associated to a universal unfolding $F(x,s)$ of the function $f(x)$ and the associated Frobenius manifold. A moduli space functions with respect to $\R$-equivalence can be constructed from it. This will be not discussed here. By computational experiments, we have found another occurrence of the Milnor algebra -- this time connected with the $\mathcal{K}$-class (the contact equivalence class) of $f(x)$, i.e. with the isomorphism-class of the germ $X_0$. Our observation concerns unimodal functions that are not quasihomogeneous. Here we consider a miniversal deformation $F:X\rightarrow S$ of the singularity $X_0$. It has a smooth base space of dimension $\tau$, with $\tau$ being the Tjurina number, i.e. the $\C$-dimension of the Tjurina algebra $T(f):=Q(f)/fQ(f)$. We consider the relative singular locus $Sing(X/S)$ of $X$ over $S$ and its flatness stratum $\F:=\F_S(Sing(X/S))\subset S$, which depends only on $X_0$, up to isomorphism. The flatness stratum is computable for sufficiently simple functions using a special algorithm. Surprisingly, the local ring of the flatness stratum of a unimodal singularity is isomorphic in all computed cases, either to the Milnor algebra of the defining function (in case $ dim(\F)=0$), or to the Milnor algebra of a 'nearby' function with non-isolated critical point, otherwise. The notion of a modular stratum was developed by Palamodov, in order to find a moduli space for singularities. It coincides with the flatness stratum $\F=\F_S(Sing(X/S))$. Only for some singularities from the T-series the modular stratum has expected dimension 1 with smooth curves and embedded fat points as primary components. The combinatorial pattern of its occurrence was found and the phenomenon of a splitting singular locus along a $\tau$-constant stratum was discovered. Here we extend our observation that the modular stratum is the spectrum of the Milnor algebra of an associated non-isolated limiting singularity. The modular stratum is a fat point of multiplicity $\mu$ isomorphic to ${\mathrm Spec}\, Q(f)$ in all other (computed) cases of $T$-series singularities. \begin{prop} We consider T-series singularities $f=x^p+y^q+z^r+xyz$ with Milnor number $\leq 45$. For their modular strata $M_{p,q,r}$ the following statements hold. \begin{eqnarray*} M_{4,4,4}\cong M_{6,3,3}&\cong&Spec\ Q(xyz)\\ M_{6,4,2}\cong M_{6,6,2}&\cong&Spec\ Q(x^2+xyz)\\ M_{6,6,3}&\cong& Spec\ Q(x^3+xyz) \end{eqnarray*} Except from the cases above, where several of the sub-series $T_{3,3,k}$, $T_{6,3,k}$, $T_{3,2,k}$, $T_{6,2,k}$, $T_{4,4,k}$, $T_{4,2,k}$ meet, we have \begin{eqnarray*} M_{3,3,k} &\cong& Spec\ Q(x^3+y^3 +xyz)\\ M_{3,2,k} &\cong& Spec\ Q(x^3+y^2 +xyz)\\ M_{6,3,k} &\cong& Spec\ Q(x^6+y^3 +xyz)\\ M_{6,2,k} &\cong& Spec\ Q(x^6+y^2+xyz)\\ M_{4,4,k} &\cong& Spec\ Q(x^4+y^4+xyz)\\ M_{4,2,k} &\cong& Spec\ Q(x^4+y^2+xyz). \end{eqnarray*} For all other cases we have $M_{p,q,r} \cong Spec\ Q(f)$. \end{prop} An analogue statement holds for all 14 exceptional and non-quasihomogeneous unimodal singularities. \begin{prop} All 14 exceptional semi-quasihomogeneous unimodal singularities fulfil: The local ring of their modular stratum is isomorphic to their Milnor algebra. In the case of a quasihomogenous exceptional singularity, the modular stratum is a smooth germ, hence corresponding to a trivial Milnor algebra. \end{prop} We give explicit examples for modular strata and non-trivial isomorphisms to the corresponding Milnor algebras. Finally an algorithmic approach is outlined, how to perform a check of algebra isomorphy using a computer algebra system, knowing that there is no practical algorithm in general. All computations were executed in the computer algebra system {\sc Singular}. |
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