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Castelnuovo Curve
In algebraic geometry, a Castelnuovo curve, studied by , is a curve in projective space P''n'' of maximal genus ''g'' among irreducible non-degenerate curves of given degree ''d''. Castelnuovo showed that the maximal genus is given by the Castelnuovo bound :g\le (n-1)m(m-1)/2+m\epsilon where ''m'' and ε are the quotient and remainder when dividing ''d''–1 by ''n''–1. Castelnuovo described the curves satisfying this bound, showing in particular that they lie on either a rational normal scroll or on the Veronese surface. References * *{{Citation , last1=Griffiths , first1=Phillip , author1-link=Phillip Griffiths , last2=Harris , first2=Joseph , author2-link=Joe Harris (mathematician) , title=Principles of algebraic geometry , publisher=John Wiley & Sons , location=New York , series=Wiley Classics Library , isbn=978-0-471-05059-9 , mr=1288523 , year=1994 Algebraic curves ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normal Scroll
In mathematics, a rational normal scroll is a ruled surface of degree ''n'' in projective space of dimension ''n'' + 1. Here "rational" means birational to projective space, "scroll" is an old term for ruled surface, and "normal" refers to projective normality (not normal schemes). A non-degenerate irreducible surface of degree ''m'' – 1 in P''m'' is either a rational normal scroll or the Veronese surface. Construction In projective space of dimension ''m'' + ''n'' + 1 choose two complementary linear subspaces of dimensions ''m'' > 0 and ''n'' > 0. Choose rational normal curves in these two linear subspaces, and choose an isomorphism φ between them. Then the rational normal surface consists of all lines joining the points ''x'' and ''φ''(''x''). In the degenerate case when one of ''m'' or ''n'' is 0, the rational normal scroll becomes a cone over a rational normal curve. If ''m'' < ''n'' then the ration ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Veronese Surface
In mathematics, the Veronese surface is an algebraic surface in five-dimensional projective space, and is realized by the Veronese embedding, the embedding of the projective plane given by the complete linear system of conics. It is named after Giuseppe Veronese (1854–1917). Its generalization to higher dimension is known as the Veronese variety. The surface admits an embedding in the four-dimensional projective space defined by the projection from a general point in the five-dimensional space. Its general projection to three-dimensional projective space is called a Steiner surface. Definition The Veronese surface is the image of the mapping :\nu:\mathbb^2\to \mathbb^5 given by :\nu: :y:z\mapsto ^2:y^2:z^2:yz:xz:xy/math> where :\cdots/math> denotes homogeneous coordinates. The map \nu is known as the Veronese embedding. Motivation The Veronese surface arises naturally in the study of conics. A conic is a degree 2 plane curve, thus defined by an equation: :Ax^2 + Bxy + Cy^2 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John Wiley & Sons
John Wiley & Sons, Inc., commonly known as Wiley (), is an American multinational publishing company founded in 1807 that focuses on academic publishing and instructional materials. The company produces books, journals, and encyclopedias, in print and electronically, as well as online products and services, training materials, and educational materials for undergraduate, graduate, and continuing education students. History The company was established in 1807 when Charles Wiley opened a print shop in Manhattan. The company was the publisher of 19th century American literary figures like James Fenimore Cooper, Washington Irving, Herman Melville, and Edgar Allan Poe, as well as of legal, religious, and other non-fiction titles. The firm took its current name in 1865. Wiley later shifted its focus to scientific, technical, and engineering subject areas, abandoning its literary interests. Wiley's son John (born in Flatbush, New York, October 4, 1808; died in East Orange, New Je ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
