Related papers: Multiplicity in difference geometry
We define a concept which we call multiplicity. First, multiplicity of a morphism is defined. Then the multiplicity of an object over another object is defined to be the minimum of the multiplicities of all morphisms from one to another.…
Let B be a finite collection of geometric (not necessarily convex) bodies in the plane. Clearly, this class of geometric objects naturally generalizes the class of disks, lines, ellipsoids, and even convex polygons. We consider geometric…
A uniform bound of intersection multiplicities of curves and divisors on abelian varieties is proved by algebraic geometric methods. It extends and improves a result obtained by A. Buium with a different method based on Kolchin's…
We generalize two classical formulas for complete intersection curves by introducing the the complete intersection discrepancy of a curve as a correction term. The first is a well-known multiplicity formula in singularity theory, due to…
The purpose of this paper is to provide a new account of multiplicity for finite morphisms between smooth projective varieties. Traditionally, this has been defined using commutative algebra in terms of the length of integral ring…
We provide a necessary and sufficient condition for the derived self-intersection of a smooth subscheme inside a smooth scheme to be a fibration over the subscheme. As a consequence we deduce a generalized HKR isomorphism. We also…
In this note, given a family of relative dimension one over a smooth curve, we determine the parity of the restriction of a relative theta characteristic to an arbitrary multiple of a fiber in terms of the parity of the restriction to a…
By a theorem of Chevalley the image of a morphism of varieties is a constructible set. The algebraic version of this fact is usually stated as a result on "extension of specializations" or "lifting of prime ideals". We present a difference…
A classical inequality which is due to Lickorish and Hempel says that the distance between two curves in the curve complex can be measured by their intersection number. In this paper, we show a converse version; the intersection number of…
The local Lipschitz property is shown for the graph avoiding multiple point intersection with lines directed in a given cone. The assumption is much stronger than those of Marstrand's well-known theorem, but the conclusion is much stronger…
We derive various inequalities involving the intersection number of the curves contained in geodesics and tight geodesics in the curve graph. While there already exist such inequalities on tight geodesics, our method applies in the setting…
We obtain a coarse relationship between geometric intersection numbers of curves and the sum of their subsurface projection distances with explicit quasi-constants. By using this relationship, we give applications in the studies of the…
The idea of slicing divergences has been proven to be successful when comparing two probability measures in various machine learning applications including generative modeling, and consists in computing the expected value of a `base…
For suitable subgroups of a finitely generated group, we define the intersection number of one subgroup with another subgroup and show that this number is symmetric. We also give an interpretation of this number.
We show that points in the intersection of the tropicalizations of subvarieties of a torus lift to algebraic intersection points with expected multiplicities, provided that the tropicalizations intersect in the expected dimension. We also…
In this paper we study geometric coincidence problems in the spirit of the following problems by B. Gr\"unbaum: How many affine diameters of a convex body in $\mathbb R^n$ must have a common point? How many centers (in some sense) of…
We consider ruled surfaces with finite multiplicity. We study behaviors of the striction curves and the singularities of the ruled surfaces. We also give geometric meanings of invariants related to the ruled surfaces.
On a smooth variety, Serre's intersection formula computes intersection multiplicities via an alternating sum of the lengths of Tor groups. When the variety is singular, the corresponding sum can be a divergent series. But there are…
For studying the local topology of maps, one uses deformations which split the singularities into simpler ones while preserving the general fibres. We give conditions under which such conservation holds.
We develop techniques for determining the fibers of a morphism of curves $\phi: C \to D$ over a nonarchimedean local field $K$. These results have applications to studying closed point on curves over global fields since closed points on $C$…