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Nagata–Biran Conjecture
In mathematics, the Nagata–Biran conjecture, named after Masayoshi Nagata and Paul Biran, is a generalisation of Nagata's conjecture on curves to arbitrary polarised surfaces. Statement Let ''X'' be a smooth algebraic surface and ''L'' be an ample line bundle on ''X'' of degree ''d''. The Nagata–Biran conjecture states that for sufficiently large ''r'' the Seshadri constant In algebraic geometry, a Seshadri constant is an invariant of an ample line bundle ''L'' at a point ''P'' on an algebraic variety. It was introduced by Jean-Pierre Demailly, Demailly to measure a certain ''rate of growth'', of the tensor powers of ' ... satisfies : \varepsilon(p_1,\ldots,p_r;X,L) = . References *. *. See in particular page 3 of the pdf. Algebraic surfaces Conjectures {{algebraic-geometry-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Masayoshi Nagata
Masayoshi Nagata (Japanese: 永田 雅宜 ''Nagata Masayoshi''; February 9, 1927 – August 27, 2008) was a Japanese mathematician, known for his work in the field of commutative algebra. Work Nagata's compactification theorem shows that varieties can be embedded in complete varieties. The Chevalley–Iwahori–Nagata theorem describes the quotient of a variety by a group. In 1959 he introduced a counterexample to the general case of Hilbert's fourteenth problem on invariant theory. His 1962 book on local rings contains several other counterexamples he found, such as a commutative Noetherian ring that is not catenary, and a commutative Noetherian ring of infinite dimension. Nagata's conjecture on curves concerns the minimum degree of a plane curve specified to have given multiplicities at given points; see also Seshadri constant. Nagata's conjecture on automorphisms concerns the existence of wild automorphisms of polynomial algebra In mathematics, especially in the fi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paul Biran
Paul Ian Biran ( he, פאול בירן; born 25 February 1969) is an Israeli mathematician. He holds a chair at ETH Zurich. His research interests include symplectic geometry and algebraic geometry. Education Born in Romania in 1969, Biran's family moved to Israel in 1971. He attended Tel Aviv University, where he earned his Bachelor's degree in 1994 and Ph.D. in 1997 under supervision of Leonid Polterovich (thesis: ''Geometry of Symplectic Packing''). Career From 1997 to 1999, Biran was a "Szego Assistant Professor" at Stanford University. At Tel Aviv University, he was a lecturer from 1997 to 2001, a senior lecturer from 2001 to 2005, an associate professor in 2005, and a full professor in 2008. In 2009, Biran became a full professor of mathematics at ETH Zurich. Awards Biran was awarded the Oberwolfach Prize in 2003, the EMS Prize in 2004, and the Erdős Prize in 2006. In 2013 he became a member of the German Academy of Sciences Leopoldina The German National Acade ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nagata's Conjecture On Curves
In mathematics, the Nagata conjecture on curves, named after Masayoshi Nagata, governs the minimal degree required for a plane algebraic curve to pass through a collection of very general points with prescribed multiplicities. History Nagata arrived at the conjecture via work on the 14th problem of Hilbert, which asks whether the invariant ring of a linear group action on the polynomial ring over some field is finitely generated. Nagata published the conjecture in a 1959 paper in the American Journal of Mathematics, in which he presented a counterexample to Hilbert's 14th problem. Statement :Nagata Conjecture. Suppose are very general points in and that are given positive integers. Then for any curve in that passes through each of the points with multiplicity must satisfy ::\deg C > \frac\sum_^r m_i. The condition is necessary: The cases and are distinguished by whether or not the anti-canonical bundle on the blowup of at a collection of points is nef. In the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Surface
In mathematics, an algebraic surface is an algebraic variety of dimension two. In the case of geometry over the field of complex numbers, an algebraic surface has complex dimension two (as a complex manifold, when it is non-singular) and so of dimension four as a smooth manifold. The theory of algebraic surfaces is much more complicated than that of algebraic curves (including the compact Riemann surfaces, which are genuine surfaces of (real) dimension two). Many results were obtained, however, in the Italian school of algebraic geometry, and are up to 100 years old. Classification by the Kodaira dimension In the case of dimension one varieties are classified by only the topological genus, but dimension two, the difference between the arithmetic genus p_a and the geometric genus p_g turns to be important because we cannot distinguish birationally only the topological genus. Then we introduce the irregularity for the classification of them. A summary of the results (in det ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ample Line Bundle
In mathematics, a distinctive feature of algebraic geometry is that some line bundles on a projective variety can be considered "positive", while others are "negative" (or a mixture of the two). The most important notion of positivity is that of an ample line bundle, although there are several related classes of line bundles. Roughly speaking, positivity properties of a line bundle are related to having many global sections. Understanding the ample line bundles on a given variety ''X'' amounts to understanding the different ways of mapping ''X'' into projective space. In view of the correspondence between line bundles and divisors (built from codimension-1 subvarieties), there is an equivalent notion of an ample divisor. In more detail, a line bundle is called basepoint-free if it has enough sections to give a morphism to projective space. A line bundle is semi-ample if some positive power of it is basepoint-free; semi-ampleness is a kind of "nonnegativity". More strongly, a line bun ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Seshadri Constant
In algebraic geometry, a Seshadri constant is an invariant of an ample line bundle ''L'' at a point ''P'' on an algebraic variety. It was introduced by Jean-Pierre Demailly, Demailly to measure a certain ''rate of growth'', of the tensor powers of ''L'', in terms of the jet (mathematics), jets of the Section (category theory), sections of the ''L''''k''. The object was the study of the Fujita conjecture. The name is in honour of the Indian mathematician C. S. Seshadri. It is known that Nagata's conjecture on algebraic curves is equivalent to the assertion that for more than nine general points, the Seshadri constants of the projective plane are maximal. There is a general conjecture for algebraic surfaces, the Nagata–Biran conjecture. Definition Let be a smooth projective variety, an ample line bundle on it, a point of , = . . Here, denotes the intersection number of and , measures how many times passing through . Definition: One says that is the Seshadri constant of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Surfaces
In mathematics, an algebraic surface is an algebraic variety of dimension two. In the case of geometry over the field of complex numbers, an algebraic surface has complex dimension two (as a complex manifold, when it is non-singular) and so of dimension four as a smooth manifold. The theory of algebraic surfaces is much more complicated than that of algebraic curves (including the compact Riemann surfaces, which are genuine surfaces of (real) dimension two). Many results were obtained, however, in the Italian school of algebraic geometry, and are up to 100 years old. Classification by the Kodaira dimension In the case of dimension one varieties are classified by only the topological genus, but dimension two, the difference between the arithmetic genus p_a and the geometric genus p_g turns to be important because we cannot distinguish birationally only the topological genus. Then we introduce the irregularity for the classification of them. A summary of the results (in det ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
