Arcs on determinantal varieties (Q2838064)

Let \(\mathcal M = {\mathbb{A}}^{rs}\) denote the complex algebraic variety of \(r \times s\) complex matrices (\(r \leq s\)) and \(D^k\) the generic determinantal variety of rank \(k\), i.e., its points correspond to matrices in \(\mathcal M\) of rank \(\leq k\) (\(\leq r\)). In this paper the author studies the arc space and jet schemes of \(D^k\). An \(n\)-jet is a morphism \(\mathrm{Spec}({\mathbb{C}} [t]/(t^{n+1})) \to D^k\). These \(n\)-jets are represented by an algebraic \(\mathbb{C}\)-scheme \(D^k_n\). The inverse limit \(D^k_{\infty}\) of the schemes \(D^k_n\) (for \(n \to \infty\)) is a \({\mathbb{C}}\)-scheme (not necessarily of finite type) representing arcs on \(D^k\), i.e., morphisms \(\mathrm{Spec}({\mathbb{C}}[[t]]) \to D^k\).NEWLINENEWLINEThe author proves that for \(k = 0\) or \(k=r-1\) the jet schemes \(D^k_n\) are irreducible, and gives a formula (in terms of \(n\) and \(k\)) for the number of irreducible components of \(D^k_n\) for \(1 \leq k \leq r-2\).NEWLINENEWLINEThis analysis also yields a formula for the \textit{log canonical threshold} of the pair \((M,D^k)\), in terms of \(k\), \(r\) and \(s\). An expression for the topological zeta function of \((\mathcal M, D^k)\) (when \(r=s\)) is also obtained.NEWLINENEWLINEThe main idea of this research is to exploit an action of the group \(G=\mathrm{GL}_r \times \mathrm{GL}_s\) on \(\mathcal M\), which induces similar group actions on \(D_k\), \({\mathcal M}_{\infty}\) and \({\mathcal M}_n\), for all \(n\).NEWLINENEWLINEThe orbits under these actions are very important and, by a process of Gaussian elimination, each orbit in \({\mathcal M}_{\infty}\) is characterized by a sequence \(\lambda_1 \geq \dots \lambda _r \geq 0\) (where some \(\lambda _i\) could be infinite). This allows the author to use the Theory of Partitions (or a slight generalization thereof) to study, among other things, the question of incidence of closures of orbits. Other techniques that are used include the theory of \textit{contact loci} and \textit{motivic integration} (to compute motivic volumes of certain orbit closures). A number of auxiliary results, interesting in themselves, are obtained.NEWLINENEWLINEAlthough the material is technical the paper is well written and many necessary (known) results are briefly but clearly reviewed.