The local Nash problem on arc families of singularities. (English) Zbl 1116.14030

Let \(X\) be a \(k\)-scheme. The arc space \(X_\infty\) is characterized by the property \[ \text{Hom}_k(Y, X_\infty)\simeq \text{Hom}_k(Y\widehat{\times}_{\text{Spec}(k)}\text{Spec}(k[[t]]),X) \] for an arbitrary \(k\)-scheme \(Y\). Let \(\pi_X:X_\infty\to X\) be the canoncial projection sending an arc \(\alpha:\text{Spec}\;k[[t]]\to X\) to \(\alpha(0)\). A morphism \(\varphi: Z\to X\) induces a canonical morphism \(\varphi_\infty: Z_\infty\to X_\infty\), \(\alpha\mapsto \varphi\circ \alpha\).
An irreducible component \(C\) of \(\overline{\pi_X^{-1}(x)}\) is called local Nash component of \((X,x)\) if \(C\) is not contained in \((\text{Sing }(X))_\infty\). Let \(\varphi: Y\to X\) be a resolution of the singularities such that \(\overline{\varphi^{-1}(x)}\) is a union of nonsingular divisors. Let \(\overline{\varphi^{-1}(x)}=\bigcup E_j\) be the decomposition into irreducible components. Let \(\{C_i\}\) be the local Nash components of \((X,x)\). Then the morphism \(\varphi_\infty:\bigcup_j\pi_Y^{-1}(E_j)\to\bigcup_iC_i\) is dominant and injective outside \((\text{Sing }(X))_\infty\). For each \(C_i\) there is a unique \(E_{j_i}\) such that \(\pi^{-1}_Y(E_{j_i})\) is dominant to \(C_i\). \(E_{j_i}\) is an essential divisor over \((X, x)\). The map \[ l\mathcal N:\{\text{local Nash components of }(X,x)\}\to\{\text{essential divisors over }(X,x)\}, \] \(l\mathcal N(C_i)=E_{j_i}\), is called the local Nash map. This map is injective. The local Nash Problem is the question whether this map is bijective. An affirmative answer to the local Nash problem is given for toric singularities and so-called analytically pretoric singularities. This includes the class of quasi-ordinary singularities.


14J17 Singularities of surfaces or higher-dimensional varieties
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
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