Laplace Transforms of a Class of Higher Dimensional Varieties in a Projective Space of n Dimensions

1989 ◽  
pp. 113-115
Author(s):  
Shiing-Shen Chern
Keyword(s):  
Projective Space ◽  
Laplace Transforms ◽  
Higher Dimensional

scholarly journals Stable Rationality of del Pezzo Fibrations of Low Degree Over Projective Spaces

2018 ◽  
Vol 2020 (23) ◽  
pp. 9075-9119 ◽  
Author(s):  
Igor Krylov ◽  
Takuzo Okada
Keyword(s):  
Projective Space ◽  
Three Dimensional ◽  
Projective Spaces ◽  
Canonical Divisor ◽  
Stable Rationality ◽  
Higher Dimensional ◽  
Low Degree ◽  
Del Pezzo

Abstract The main aim of this article is to show that a very general three-dimensional del Pezzo fibration of degrees 1, 2, and 3 is not stably rational except for a del Pezzo fibration of degree 3 belonging to explicitly described two families. Higher-dimensional generalizations are also discussed and we prove that a very general del Pezzo fibration of degrees 1, 2, and 3 defined over the projective space is not stably rational provided that the anti-canonical divisor is not ample.

scholarly journals THE WEINSTEIN CONJECTURE IN THE PRESENCE OF SUBMANIFOLDS HAVING A LEGENDRIAN FOLIATION

2011 ◽  
Vol 03 (04) ◽  
pp. 405-421 ◽  
Author(s):  
KLAUS NIEDERKRÜGER ◽  
ANA RECHTMAN
Keyword(s):  
Projective Space ◽  
Holomorphic Curves ◽  
Real Projective Space ◽  
Contact Manifolds ◽  
Connected Sum ◽  
Novel Technique ◽  
Weinstein Conjecture ◽  
Higher Dimensional

Helmut Hofer introduced in 1993 a novel technique based on holomorphic curves to prove the Weinstein conjecture. Among the cases where these methods apply are all contact 3-manifolds (M, ξ) with π2(M) ≠ 0. We modify Hofer's argument to prove the Weinstein conjecture for some examples of higher-dimensional contact manifolds. In particular, we are able to show that the connected sum with a real projective space always has a closed contractible Reeb orbit.

scholarly journals Singularities of Projective Embedding (Points of order n on an Elliptic Curve)

1972 ◽  
Vol 45 ◽  
pp. 97-107 ◽  
Author(s):  
Akikuni Kato
Keyword(s):  
Projective Space ◽  
Elliptic Curve ◽  
Vector Bundles ◽  
Neutral Element ◽  
Projective Embedding ◽  
Dual Formulation ◽  
Higher Dimensional ◽  
Dimensional Projective Space ◽  
Dimension One ◽  
Principal Parts

In the Plücker formula for a curve embedded in a higher dimensional projective space, one encounters the notion of stationary point (cf, [B], [W]). W. F. Pohl gave new view point about it in terms of vector bundles and he defined “the singularities of embedding” (cf. [P]). At first, we shall give dual formulation of Pohl’s one by means of the sheaf of principal parts of order n, and next we shall prove the following: If an elliptic curve is embedded in (n — l)-dimensional projective space as a curve of degree n, singularities of projective embedding of order n — 1 are exactly the points of order n with suitable choice of a neutral element on the curve which is an abelian variety of dimension one. The proof is given by making use of the relation between and Schwarzenberger’s secant bundle which we shall also give.

Thomas Gerald Room, 10 November 1902 - 2 April 1986

1987 ◽  
Vol 33 ◽  
pp. 573-601
Keyword(s):  
Projective Space ◽  
Mathematics Teachers ◽  
New South ◽  
New South Wales ◽  
South Wales ◽  
Higher Dimensional ◽  
Dimensional Projective Space ◽  
The University ◽  
Insight Into ◽  
Profound Insight

Thomas Gerald Room, Professor of Mathematics in the University of Sydney from 1935 until his retirement in 1968, died at his home near Sydney on 2 April 1986, at the age of 83. Born and educated in England, he came to Sydney in his early thirties and made it his adopted home. He was a classical geometer with unusual gifts of intuition and combinatorial skill, who will be particularly remembered for his profound insight into configurations in higher dimensional projective space. He and his slightly older contemporary, T. M. Cherry, were the elder statesmen of Australian mathematics in their time. He exerted enormous influence on the course of mathematics in New South Wales at both school and university levels and will be remembered by mathematicians, mathematics teachers and generations of students.

scholarly journals Desargues maps and the Hirota–Miwa equation

2009 ◽  
Vol 466 (2116) ◽  
pp. 1177-1200 ◽  
Author(s):  
Adam Doliwa
Keyword(s):  
Integral Equation ◽  
Fredholm Determinant ◽  
Laplace Transforms ◽  
Complex Field ◽  
Dressing Method ◽  
Commutative Case ◽  
Higher Dimensional ◽  
Non Local ◽  
Desargues Theorem

We study the Desargues maps , which generate lattices whose points are collinear with all their nearest (in positive directions) neighbours. The multi-dimensional compatibility of the map is equivalent to the Desargues theorem and its higher dimensional generalizations. The nonlinear counterpart of the map is the non-commutative (in general) Hirota–Miwa system. In the commutative case of the complex field we apply the non-local -dressing method to construct Desargues maps and the corresponding solutions of the system. In particular, we identify the Fredholm determinant of the integral equation inverting the non-local -dressing problem with the τ -function. Finally, we establish equivalence between the Desargues maps and quadrilateral lattices provided we take into consideration also their Laplace transforms.

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