Related papers: Arcs on Determinantal Varieties
This paper is a survey on arc spaces, a recent topic in algebraic geometry and singularity theory. The geometry of the arc space of an algebraic variety yields several new geometric invariants and brings new light to some classical…
We construct algorithms and topological invariants that allow us to distinguish the topological type of a surface, as well as functions and vector fields for their topological equivalence. In the first part we discus the main structures…
For $m\in \IN, m\geq 1,$ we determine the irreducible components of the $m-th$ jet scheme of a toric surface $S.$ For $m$ big enough, we connect the number of a class of these irreducible components to the number of exceptional divisors on…
We use the theory of motivic integration for singular spaces to give a characterization of minimal log discrepencies in terms of the codimension of certain subsets of spaces of arcs. This is done for arbitrary pairs $(X,Y)$, with $X$ normal…
We construct algorithms and topological invariants that allow us to distinguish the topological type of a surface, as well as functions and vector fields for their topological equivalence. In the first part (arXiv:2501.15657), we discused…
Basis sets of atomic orbitals are very efficient for density functional calculations but lack a systematic variational convergence. We present a variational method to optimize numerical atomic orbitals using a single parameter to control…
We introduce jet topics: a framework to identify underlying classes of jets from collider data. Because of a close mathematical relationship between distributions of observables in jets and emergent themes in sets of documents, we can apply…
From a group action on a space, define a variant of the configuration space by insisting that no two points inhabit the same orbit. When the action is almost free, this "orbit configuration space" is the complement of an arrangement of…
Over a smooth complex projective curve, we study an algebraic versal deformation space with fixed determinant of a coherent sheaf. The algebraic versal deformation space decomposes into a disjoint union of Shatz strata, namely locally…
We consider the scheme $X_{r,d,n}$ parametrizing $n$ ordered points in projective space $\mathbb{P}^r$ that lie on a common hypersurface of degree $d$. We show that this scheme has a determinantal structure and we prove that it is…
We study geometric properties of varieties associated with invariant subspaces of nilpotent operators. There are reductive algebraic groups acting on these varieties. We give dimensions of orbits of these actions. Moreover, a combinatorial…
We study logarithmic jet schemes of a log scheme and generalize a theorem of M. Mustata from the case of ordinary jet schemes to the logarithmic case. If X is a normal local complete intersection log variety, then X has canonical…
We seek to determine a real algebraic variety from a fixed finite subset of points. Existing methods are studied and new methods are developed. Our focus lies on aspects of topology and algebraic geometry, such as dimension and defining…
We study the arc space of the Grassmannian from the point of view of the singularities of Schubert varieties. Our main tool is a decomposition of the arc space of the Grassmannian that resembles the Schubert cell decomposition of the…
We present a deterministic polynomial time algorithm for computing the zeta function of an arbitrary variety of fixed dimension over a finite field of small characteristic. One consequence of this result is an efficient method for computing…
We determine the number of ${\mathbb{F}}_q$-rational points of hyperplane sections of classical determinantal varieties defined by the vanishing of minors of a fixed size of a generic matrix, and identify sections giving the maximum number…
We study dynamical systems arising from word maps on simple groups. We develop a geometric method based on the classical trace map for investigating periodic points of such systems. These results lead to a new approach to the search of…
We introduce new notions of log jet spaces. Mildly singular spaces are ``smooth'' in log geometry, so their log jet spaces behave like the jet spaces of smooth varieties. Myriad examples contrast log jet spaces with the usual jet spaces of…
We study varieties defined over nonstandard fields using techniques of nonstandard mathematics.
Algorithmicists are well-aware that fast dynamic programming algorithms are very often the correct choice when computing on compositional (or even recursive) graphs. Here we initiate the study of how to generalize this folklore intuition to…