Families of jets of arc type and higher (co)dimensional du val singularities (Q6553455)

Suppose \(k\) is an algebraically closed field of characteristic 0, and \(X\) an algebraic variety over \(k\) with \(\pi: X_{\infty} \rightarrow X\) the projection of its space of arcs. Families of jets through the singular locus \(X_{\mathrm{sing}}\) have been investigated in relation with the components of \(\pi^{-1}(X_{\mathrm{sing}}) \subset X_{\infty}\), called Nash components. Each of them defines a divisorial valuation, here called Nash valuation. Characterization of these components is motivated by the understanding the essential divisors that appear birationally in every resolution of singularities of \(X\). Here in shown that there is natural correspondence between families of arcs and some families of sufficiently high order jets.\N\N{Theorem A.} Among all families of jets of high enough order centered at \(X_{\mathrm{sing}}\) there is a selection of them. Such selections are in natural one-to-one correspondence with the Nash families in \(X_{\infty}\), and these families of jets are called ``of arc type''.\N\NIf \(X\) is normal locally complete intersection variety, it is well known that \(\mathrm{ecodim}(\mathcal{O}_{X, x}) \leq \dim(\mathcal{O}_{X, x}) - \mathrm{mld}(X)\) for every \(x\). If the equality holds for every \(x\), we say that \(X\) has maximal embedding codimension singularities. These are isolated singularities. The varieties satisfying \(\mathrm{mld}_x(X) = \dim(\mathcal{O}_{X, x}) - \mathrm{ecodim}(\mathcal{O}_{X, x}) =1\) at all \(x\) are called higher Du Val singularities. In dimension two they coincide with the usual varieties with Du Val singularities. For \(X\) such a variety, \(x \in X\) is called higher compound Du Val singularity if the variety \(Y \subset X\) cut out by \(r \geq 0\) general hyperplane sections through \(x\) has higher Du Val singularities. For them holds the following result, which relates to and provides evidence in the direction of [\textit{H. Mourtada}, in: Valuation theory in interaction. Proceedings of the 2nd international conference and workshop on valuation theory, Segovia and El Escorial, Spain, July 18--29, 2011. Zürich: European Mathematical Society (EMS). 373--388 (2014; Zbl 1312.14047)].\N\N{Theorem B.} On an isolated higher compound Du Val singularity P, all families of jets of big enough order are of arc type.\N\NNext theorem gives a characterization of Nash valuations in terms of the terminal models, by the exceptional divisors on them. The corresponding valuations are called terminal.\N\N{Theorem C.} On a variety with higher Du Val singularities, for a divisorial valuation ord\(_E\) on it the following are equivalent:\N\begin{itemize}\N\item[i)] ord\(_E\) is a Nash valuation;\N\item[ii)] ord\(_E\) is a terminal valuation;\N\item[iii)] \(E\) is crepant exceptional divisor.\N\end{itemize}\N\NThe proof of the last theorem relies on inversion of adjunction and minimal model program. Based on the last theorem is obtained a positive solution of Nash problem for varieties with higher Du Val singularities.\N\NFinally, a directed graph is associated with arbitrary variety \(X\), whose vertices correspond to the irreducible components of the reduced subscheme of \(m\)-th jet spaces \(X_m\), \(m \geq 0\), where the subscheme consists of all jets centered at \(X_{\mathrm{sing}}\). For that graph is shown that the infinite branches are in bijection with Nash valuations on \(X\). In particular, the number of components of the subscheme for any \(m \geq 1\) is equal to the number of Nash components, supposed that \(X\) is with isolated higher compound Du Val singularities.