The general quadruple point formula (original) (raw)

Multiplicity as an Object of Study in Contemporary Differential Geometry

ANNALS OF THE ORADEA UNIVERSITY. Fascicle of Management and Technological Engineering., 2013

The study presents introductory concepts of multiplicity as one of the basic concepts of differential topology. The work gives the definition of Hausdorff space and some theorems concerning Hausdorff space.

Minicourse: Cohomology jump loci in geometry and topology. Part IV: Applications

The cohomology jumping loci of a space come in two basic flavors: the characteristic varieties, which are the jump loci for homology with coefficients in rank 1 local systems, and the resonance varieties, which are the jump loci for the homology of cochain complexes arising from multiplication by degree 1 classes in the cohomology ring. I will explain the algebraic notions underlying these constructions and will present some structural results, such as the influence of formality, the interplay between resonance and duality, connections to tropical geometry, and the relationship to the finiteness properties of spaces and groups. Along the way, I will illustrate the general theory with a number of examples, mostly drawn from complex algebraic geometry, singularity theory, and low-dimensional topology.

The double point formula with isolated singularities and canonical embeddings

Journal of the London Mathematical Society, 2020

Motivated by the embedding problem of canonical models in small codimension, we extend Severi's double point formula to the case of surfaces with rational double points, and we give more general double point formulae for varieties with isolated singularities. A concrete application is for surfaces with geometric genus pg = 5: the canonical model is embedded in P 4 if and only if we have a complete intersection of type (2, 4) or (3, 3). Contents 1. Introduction 2 2. The classical double point formula 6 3. Extension of Severi's Statement 9 4. Comparing normal sheaf and quotient bundle 12 5. Surfaces with rational double points 14 6. Double point formulae via symplectic approximations. 20 References 21

MULTIPLE-POINT FORMULAS—A NEW POINT OF VIEW

2002

On the basis of the Generalized Pontryagin-Thom construction (see Rimanyi & Szucs, 1998) and its application in computing Thom polynomials (see Rimanyi, 2001) here we introduce a new point of view in multiple-point theory. Using this approach we will first show how to reprove results of Kleiman and his followers (the corank 1 case) then we will prove some new multiple-point formulas which are not subject to the condition of corank≤ 1.

Minicourse: Cohomology jump loci in geometry and topology. Part II: Characteristic varieties

Quadratic Cohomology

Arnold Mathematical Journal, 2014

We study homological invariants of smooth families of real quadratic forms as a step towards a "Lagrange multipliers rule in the large" that intends to describe topology of smooth maps in terms of scalar Lagrange functions.

The link of {f(x, y) + z n = 0} and Zariski's conjecture

Compositio Mathematica, 2005

We consider suspension hypersurface singularities of type g = f (x, y) + z n , where f is an irreducible plane curve singularity. For such germs, we prove that the link of g determines completely the Newton pairs of f and the integer n except for two pathological cases, which can be completely described. Even in the pathological cases, the link and the Milnor number of g determine uniquely the Newton pairs of f and n. In particular, for such g, we verify Zariski's conjecture about the multiplicity. The result also supports the following conjecture formulated in the paper. If the link of an isolated hypersurface singularity is a rational homology 3-sphere then it determines the embedded topological type, the equivariant Hodge numbers and the multiplicity of the singularity. The conjecture is verified for weighted homogeneous singularities too.

Algebraic invariants in classification of 6-points in degenerations of surfaces

2010

In this paper, we find isomorphisms between certain invariant groups corresponding to different numerations on 6-points of surfaces. There is a combinatorial correspondence between four 6-point orderings obtained by exchanging two opposite labels. We derive isomorphisms between certain invariant quotient groups obtained from these 6-point nu-merations. This is a preliminary step towards an ultimate classification of 6-points invariants, and perhaps towards a proof that the invariant groups, or at least certain derived invariants, are independent of the arbitrary choice of the numeration.

Generalized Monodromy Conjecture in dimension two

Geometry & Topology, 2012

The aim of the article is an extension of the Monodromy Conjecture of Denef and Loeser in dimension two, incorporating zeta functions with differential forms and targeting all monodromy eigenvalues, and also considering singular ambient spaces. That is, we treat in a conceptual unity the poles of the (generalized) topological zeta function and the monodromy eigenvalues associated with an analytic germ f : (X, 0) → (C, 0) defined on a normal surface singularity (X, 0). The article targets the 'right' extension in the case when the link of (X, 0) is a homology sphere. As a first step, we prove a splice decomposition formula for the topological zeta function Z(f, ω; s) for any f and analytic differential form ω, which will play the key technical localization tool in the later definitions and proofs.

Thom polynomial computing strategies. A survey

Abstract. Thom polynomials compute the cohomology classes of degeneracy loci. In this paper we use a simple example to review the core ideas in different—mostly recently found—methods of computing Thom polynomials. Our goal is to show the underlying topology/geometry/algebra without involving combinatorics.