Showing papers on "Intersection number published in 2013"

On the arithmetic fundamental lemma in the minuscule case

Abstract: The arithmetic fundamental lemma conjecture of the third author connects the derivative of an orbital integral on a symmetric space with an intersection number on a formal moduli space of -divisible groups of Picard type. It arises in the relative trace formula approach to the arithmetic Gan–Gross–Prasad conjecture. We prove this conjecture in the minuscule case.

Mirror Map as Generating Function of Intersection Numbers: Toric Manifolds with Two Kähler Forms

Abstract: In this paper, we extend our geometrical derivation of the expansion coefficients of mirror maps by localization computation to the case of toric manifolds with two Kahler forms. In particular, we consider Hirzebruch surfaces F 0, F 3 and Calabi-Yau hypersurface in weighted projective space P(1, 1, 2, 2, 2) as examples. We expect that our results can be easily generalized to arbitrary toric manifolds.

Geometric intersection of curves on punctured disks

Abstract: We give a recipe to compute the geometric intersection number of an integral lamination with a particular type of integral lamination on an $n$-times punctured disk. This provides a way to find the geometric intersection number of two arbitrary integral laminations when combined with an algorithm of Dynnikov and Wiest.

Subgroups generated by two Dehn twists on a nonorientable surface

Abstract: Let a and b be two simple closed curves on an orientable surface S such that their geometric intersection number is greater than 1. It is known that the group generated by corresponding Dehn twists t_a and t_b is isomorphic to the free group of rank 2. In this paper we extend this result to the case of a nonorientable surface.

Mirror Map as Generating Function of Intersection Numbers

Abstract: In this article, we review our recent results on geometrical reconstruction of the B-model data used in the mirror computation of projective hypersurfaces, which was presented in Jinzenji (Lett. Math. Phys. 86(2–3):99–114, 2008; Mirror map as generating function of intersection numbers: Toric manifolds with two Kahler forms. Preprint, arXiv:1006.0607).

Covering maps and hulls in the curve complex

Abstract: This thesis studies the coarse geometry of the curve complex using intersection number techniques. We show how weighted intersection numbers can be studied using appropriate singular Euclidean surfaces. We then introduce a coarse analogue of the convex hull of a finite set of vertices in the curve complex, called the short curve hull, and provide intersection number conditions to find nearest point projections to such hulls. We also obtain an upper bound for distances in the curve complex using a greedy algorithm due to Hempel. Covering maps between surfaces also play a significant part in this thesis. We give a new proof of a theorem of Rafi and Schleimer which states that a covering map between surfaces induces a natural quasi-isometric embedding between their corresponding curve complexes. Our proof employs a distance estimate via a suitable hyperbolic 3-manifold which arises from work on the proof of the Ending Lamination Theorem. We then define an operation using a given covering map and intersection number conditions and show that it approximates a nearest point projection to the image of Rafi-Schleimer's map. We also prove that this operation approximates a circumcentre of the orbit of a vertex in the curve complex under the deck transformation group of a regular cover

On Noether and strict stability, Hilbert exponent, and relative Nullstellensatz

Abstract: Conditions characterizing the membership of the ideal of a subvariety S arising from (effective) divisors in a product complex space Y × X are given. For the algebra OY [V ] of relative regular functions on an algebraic variety V , the strict stability is proved, in the case where Y is a normal space, and the Noether stability is established under a weakened condition. As a consequence (for both general and complete intersections) a global Nullstellensatz is derived for divisors in Y ×C , respectively, Y × P (C). Also obtained are a principal ideal theorem for relative divisors, a generalization of the Gauss decomposition rule, and characterizations of solid pseudospherical harmonics on a semi-Riemann domain. A result towards a more general case is as follows: Let Dj , 1 ≤ j ≤ p, be principal divisors in X associated to the components of a q-weakly normal map g = (g1, . . . , gp) : X → C, and S := ⋂ S|Dj |. Then for any proper slicing (φ, g,D) of {Dj}1≤j≤p (where D ⊂ X is a relatively compact open subset), there exists an explicitly determined Hilbert exponent hD1···Dp,D for the ideal of the subvariety S = Y × (S ∩D).

Identification of Relations in Region Connection Calculus: 9-Intersection Reduced to 3 + -Intersection Predicates

Abstract: The intersection between objects relates to a class of problems where either precise intersection is required or imprecise intersection is acceptable. The calculation of intersection between two 2D/3D objects is a computation-intensive process. For qualitative spatio-temporal reasoning, it is sufficient to know the existence of intersection instead of the precise intersection. In order to identify RCC8 relations, the 9-Intersection model considers the pairwise intersection of interiors, boundaries, and exteriors of objects. It was determined that the 9-Intersection is sufficient for identifying spatial relations. Later, it was shown that a 4-Intersection model is sufficient to achieve the same results making the definition (and implementation) of the RCC8 relations worth studying in greater detail. Herein we prove that the 9-Intersection model can be further reduced to almost three intersection predicates, producing a 3 + -Intersection model. This results in improved algorithmic and computational efficiency as a consequence of fewer predicates and faster intersection operations.

Intersection Number of Plane Curves

Abstract: One of the most familiar objects in algebraic geometry is the plane curve. A plane curve is the vanishing set of a polynomial in two variables. One of the goals in algebraic geometry is to describe properties of geometric objects such as these curves in algebraic terms. Intersection theory is a branch of algebraic geometry motivated by the following geometric and topological question: Given a space X and a collection of subspaces X1, . . . , Xn ⊆ X, how many points lie in the intersection ∩k=1Xk? In this paper we highlight the special case where X has dimension 2, and the subspaces in question are two plane curves F and G. We are also concerned with counting how many times F and G intersect at a given point P , which is called the intersection number of F and G at P . Intuitively, the intersection number should be the product of the multiplicities of the curves at P , but this is only true in the simplest of cases. Examining the more complicated situations, we can produce a list of additional geometricallymotivated properties the intersection number should satisfy. Given the rich and delicate geometry at play here, it is perhaps unexpected that we can state the intersection number of two curves as an explicit, simple algebraic quantity. In order to answer the questions above, we must specify our ambient space X. We have stated that it should have dimension 2, but that is all. Different spaces can yield significantly different answers. The distinction between affine and projective spaces is particularly important. Projective n-space over a field k is a completion of affine n-space to include a set of points at infinity. The inclusion of these points gives projective space a nicer geometry than affine space, and in some cases includes intersection points that may have been “missed” in affine space. The choice of k is also important. While it is often convenient to visualize a curve in R, many important results only hold over an algebraically closed field such as C. For this reason, throughout this paper we will always be working over an algebraically closed field. Nevertheless, many interesting and important results are true over fields which are not algebraically closed. For instance, working over the rationals has many important applications to algebraic number theory.

The dimension of the leafwise reduced cohomology

Abstract: Geometric conditions are given so that the leafwise reduced cohomology is of infinite dimension, specially for foliations with dense leaves on closed manifolds. The main new definition involved is the intersection number of subfoliations with "appropriate coefficients". The leafwise reduced cohomology is also described for homogeneous foliations with dense leaves on closed nilmanifolds.

Jacobian Conjecture in two dimension

Abstract: Let $(P, Q)$ be a pair of Jacobian polynomials. We can show that $ +l+2g(P)-2= 0= $, where $ $ is the intersection number of $f, g\in \CC[x, y]$ in the affine plane, $l$ is the number of branch at point at infinity and $g(P)$ is the geometric genus of affine curve defined by $P$. Hence we can show that every Jacobian polynomial defines a smooth rational curve with one point at infinity. It is sufficient to fix the Jacobian conjecture in two dimension by the Abhyankar theorem or the Abhyankar-Moh-Suzuki theorem.

Intersection numbers of families of ideals

Abstract: We study the intersection number of families of tall ideals. We show that the intersection number of the class of analytic P-ideals is equal to the bounding number \({\mathfrak{b}}\), the intersection number of the class of all meager ideals is equal to \({\mathfrak{h}}\) and the intersection number of the class of all Fσ ideals is between \({\mathfrak{h}}\) and \({\mathfrak{b}}\), consistently different from both.

An Efficient Method for Calculating the Intersection Curve between a Ruled Surface and a Plane

Abstract: Based on the unique geometric features of the ruled surface, a new efficient algorithm for ruled surface/plane intersection is proposed. The ruled surface is firstly dispersed into a set of line segments, and the ruled surface/plane intersection is transformed to the intersection of a group line segments with a plane. Then a set of ordered intersection points can be obtained by the proposed line/plane intersection algorithm. According to the serial number of every intersection point, all the intersection points are grouped and reorganized. Each group point corresponds to an intersection curve. All the intersection curves can be reconstructed by curve interpolation. Compared with the traditional tracing method, the proposed algorithm can avoid complex calculation including initial points searching and intersection points sorting, which is more efficient and stable.