scholarly journals Nonproper intersection products and generalized cycles

2021 ◽  
Author(s):  
Mats Andersson ◽  
Dennis Eriksson ◽  
Håkan Samuelsson Kalm ◽  
Elizabeth Wulcan ◽  
Alain Yger
Keyword(s):  
Projective Space ◽  
Chow Group ◽  
Intersection Theory ◽  
Analytic Space ◽  
Intersection Numbers ◽  
Formula Group ◽  
Bezout Theorem ◽  
Join Construction ◽  
Chern Forms

AbstractWe develop intersection theory in terms of the $${{\mathscr {B}}}$$ B -group of a reduced analytic space. This group was introduced in a previous work as an analogue of the Chow group; it is generated by currents that are direct images of Chern forms and it contains all usual cycles. However, contrary to Chow classes, the $${{\mathscr {B}}}$$ B -classes have well-defined multiplicities at each point. We focus on a $${{\mathscr {B}}}$$ B -analogue of the intersection theory based on the Stückrad–Vogel procedure and the join construction in projective space. Our approach provides global $${{\mathscr {B}}}$$ B -classes which satisfy a Bézout theorem and have the expected local intersection numbers. We also introduce $${{\mathscr {B}}}$$ B -analogues of more classical constructions of intersections using the Gysin map of the diagonal. These constructions are connected via a $${{\mathscr {B}}}$$ B -variant of van Gastel’s formulas. Furthermore, we prove that our intersections coincide with the classical ones on cohomology level.

scholarly journals On the arithmetic of a family of degree - two K3 surfaces

2018 ◽  
Vol 166 (3) ◽  
pp. 523-542 ◽  
Author(s):  
FLORIAN BOUYER ◽  
EDGAR COSTA ◽  
DINO FESTI ◽  
CHRISTOPHER NICHOLLS ◽  
MCKENZIE WEST
Keyword(s):  
Projective Space ◽  
Function Field ◽  
Weighted Projective Space ◽  
Module Structure ◽  
K3 Surface ◽  
Galois Module ◽  
Generic Element ◽  
The Family ◽  
Formula Group ◽  
Image Position

AbstractLet ℙ denote the weighted projective space with weights (1, 1, 1, 3) over the rationals, with coordinates x, y, z and w; let $\mathcal{X}$ be the generic element of the family of surfaces in ℙ given by \begin{equation*} X\colon w^2=x^6+y^6+z^6+tx^2y^2z^2. \end{equation*} The surface $\mathcal{X}$ is a K3 surface over the function field ℚ(t). In this paper, we explicitly compute the geometric Picard lattice of $\mathcal{X}$, together with its Galois module structure, as well as derive more results on the arithmetic of $\mathcal{X}$ and other elements of the family X.

INTERSECTION THEORY AND TWO-DIMENSIONAL GRAVITY AT GENUS 3 AND 4

1990 ◽  
Vol 05 (26) ◽  
pp. 2127-2134 ◽  
Author(s):  
JAMES H. HORNE
Keyword(s):  
Moduli Space ◽  
Intersection Theory ◽  
Gravity Theory ◽  
Two Dimensional ◽  
Intersection Numbers ◽  
Dimensional Gravity

We show that the k = 1 two-dimensional gravity amplitudes at genus 3 agree precisely with the results from intersection theory on moduli space. Predictions for the genus 4 intersection numbers follow easily from the two-dimensional gravity theory.

Note on the Grothendieck Group of Subspaces of Rational Functions and Shokurov's Cartier b-divisors

2014 ◽  
Vol 57 (3) ◽  
pp. 562-572
Author(s):  
Kiumars Kaveh ◽  
A. G. Khovanskii
Keyword(s):  
Algebraic Variety ◽  
Short Note ◽  
Grothendieck Group ◽  
Rational Functions ◽  
Intersection Theory ◽  
Point Of View ◽  
Intersection Numbers ◽  
Ground Field

Abstract.In a previous paper the authors developed an intersection theory for subspaces of rational functions on an algebraic variety X over k = ℂ. In this short note, we first extend this intersection theory to an arbitrary algebraically closed ground field k. Secondly we give an isomorphism between the group of Cartier b-divisors on the birational class of X and the Grothendieck group of the semigroup of subspaces of rational functions on X. The constructed isomorphism moreover preserves the intersection numbers. This provides an alternative point of view on Cartier b-divisors and their intersection theory.

scholarly journals The Af condition and relative conormal spaces for functions with nonvanishing derivative

2019 ◽  
Vol 30 (10) ◽  
pp. 1950050
Author(s):  
Terence Gaffney ◽  
Antoni Rangachev
Keyword(s):  
Analytic Function ◽  
Analytic Space ◽  
Numerical Criterion ◽  
Join Construction ◽  
Complex Analytic Space ◽  
Text Condition

We introduce a join construction, as a way of completing the description of the relative conormal space of an analytic function on a complex analytic space that has a non-vanishing derivative at the origin. Then we show how to obtain a numerical criterion for Thom’s [Formula: see text] condition.

scholarly journals Even-Relative-Dimensional Vanishing Cycles in Bivariant Intersection Theory

2007 ◽  
Vol 187 ◽  
pp. 49-73 ◽  
Author(s):  
Hiroshi Saito
Keyword(s):  
Cohomology Ring ◽  
Ring Homomorphism ◽  
Chow Groups ◽  
Double Cover ◽  
Chow Group ◽  
Base Change ◽  
Intersection Theory ◽  
Generic Fibre ◽  
Vanishing Cycles ◽  
Special Fibre

AbstractFor a smooth variety proper over a curve having a fibre with isolated ordinary quadratic singularities, it is well-known that we have the vanishing cycles associated to the singularities in the étale cohomology of the geometric generic fibre. The base-change by a double cover of the base curve ramified at the image of the singular fibre has singularities corresponding to the singularities in the fibre. In this note, we show that in the even relative-dimensional case, there exist elements of the bivariant Chow group of the base-change with supports in the singularities and hence their images in the bivariant Chow group with supports in the special fibre and that the usual cohomological vanishing cycles are obtained as their images by a natural map, a kind of “cycle map” so that the elements in the bivariant Chow groups can be regarded as the vanishing cycles. The bivariant Chow group with supports in the special fibre has a ring structure and the natural map is a ring homomorphism to the cohomology ring of the geometric generic fibre. Also discussed is the relation of the bivariant Chow group with supports in the special fibre to the specialization map of Chow groups.

scholarly journals Decomposition of Feynman integrals by multivariate intersection numbers

2021 ◽  
Vol 2021 (3) ◽  
Author(s):  
Hjalte Frellesvig ◽  
Federico Gasparotto ◽  
Stefano Laporta ◽  
Manoj K. Mandal ◽  
Pierpaolo Mastrolia ◽  
...  
Keyword(s):  
Differential Equations ◽  
Recursive Algorithm ◽  
Intersection Theory ◽  
Direct Decomposition ◽  
Intersection Numbers ◽  
Top Down ◽  
Feynman Integrals ◽  
Direct Derivation ◽  
Potential Applications ◽  
Recent Idea

AbstractWe present a detailed description of the recent idea for a direct decomposition of Feynman integrals onto a basis of master integrals by projections, as well as a direct derivation of the differential equations satisfied by the master integrals, employing multivariate intersection numbers. We discuss a recursive algorithm for the computation of multivariate intersection numbers, and provide three different approaches for a direct decomposition of Feynman integrals, which we dub thestraight decomposition, thebottom-up decomposition, and thetop-down decomposition. These algorithms exploit the unitarity structure of Feynman integrals by computing intersection numbers supported on cuts, in various orders, thus showing the synthesis of the intersection-theory concepts with unitarity-based methods and integrand decomposition. We perform explicit computations to exemplify all of these approaches applied to Feynman integrals, paving a way towards potential applications to generic multi-loop integrals.

scholarly journals C2 building and projective space

2004 ◽  
Vol 76 (3) ◽  
pp. 383-402
Author(s):  
K. F. Lai
Keyword(s):  
Projective Space ◽  
Local Field ◽  
Convex Sets ◽  
Analytic Space ◽  
Rigid Analytic Space ◽  
The Stability ◽  
Stable Points

AbstractWe study the stability map from the rigid analytic space of semistable points in P3 to convex sets in the building of Sp2 over a local field and construct a pure affinoid covering of the space of stable points.

scholarly journals A METHOD TO COMPUTE SEGRE CLASSES OF SUBSCHEMES OF PROJECTIVE SPACE

2012 ◽  
Vol 12 (02) ◽  
pp. 1250142 ◽  
Author(s):  
DAVID EKLUND ◽  
CHRISTINE JOST ◽  
CHRIS PETERSON
Keyword(s):  
Projective Space ◽  
Software Package ◽  
Complex Projective Space ◽  
Intersection Theory ◽  
Software System ◽  
Numerical Implementation ◽  
Complex Projective

We present a method to compute the degrees of the Segre classes of a subscheme of complex projective space. The method is based on generic residuation and intersection theory. We provide a symbolic implementation using the software system Macaulay2 and a numerical implementation using the software package Bertini.

scholarly journals A FAMILY OF MARKED CUBIC SURFACES AND THE ROOT SYSTEM D5

2007 ◽  
Vol 18 (05) ◽  
pp. 505-525 ◽  
Author(s):  
ELISABETTA COLOMBO ◽  
BERT VAN GEEMEN
Keyword(s):  
Projective Space ◽  
Root System ◽  
Moduli Space ◽  
Tangent Bundle ◽  
Chow Group ◽  
Cubic Surfaces ◽  
Dimensional Projective Space

We define and study a family of cubic surfaces in the projectivized tangent bundle over a four-dimensional projective space associated to the root system D5. The 27 lines are rational over the base and we determine the classifying map to the moduli space of marked cubic surfaces. This map has degree two and we use it to get short proofs for some results on the Chow group of the moduli space of marked cubic surfaces.

scholarly journals A Characterization of the Hermitian Variety in Finite 3-Dimensional Projective Spaces

10.37236/3416 ◽  
2015 ◽  
Vol 22 (1) ◽  
Author(s):  
Vito Napolitano
Keyword(s):  
Projective Space ◽  
Projective Spaces ◽  
Intersection Numbers ◽  
Combinatorial Characterization ◽  
Hermitian Variety ◽  
3 Dimensional ◽  
Dimensional Projective Space

A combinatorial characterization of a non-singular Hermitian variety of the finite 3-dimensional projective space via its intersection numbers with respect to lines and planes is given.   A corrigendum was added on March 29, 2019.

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