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Brill–Noether Theory
In algebraic geometry, Brill–Noether theory, introduced by , is the study of special divisors, certain divisors on a curve that determine more compatible functions than would be predicted. In classical language, special divisors move on the curve in a "larger than expected" linear system of divisors. Throughout, we consider a projective smooth curve over the complex numbers (or over some other algebraically closed field). The condition to be a special divisor can be formulated in sheaf cohomology terms, as the non-vanishing of the cohomology of the sheaf of sections of the invertible sheaf or line bundle associated to . This means that, by the Riemann–Roch theorem, the cohomology or space of holomorphic sections is larger than expected. Alternatively, by Serre duality, the condition is that there exist holomorphic differentials with divisor on the curve. Main theorems of Brill–Noether theory For a given genus , the moduli space for curves of genus should contain a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical problems about these sets of zeros. The fundamental objects of study in algebraic geometry are algebraic varieties, which are geometric manifestations of solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are: plane algebraic curves, which include lines, circles, parabolas, ellipses, hyperbolas, cubic curves like elliptic curves, and quartic curves like lemniscates and Cassini ovals. A point of the plane belongs to an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of the points of special interest like the singular points, the inflection points and the points at infinity. More advanced questions involve the topology of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Equivalence
In algebraic geometry, divisors are a generalization of codimension-1 subvarieties of algebraic varieties. Two different generalizations are in common use, Cartier divisors and Weil divisors (named for Pierre Cartier (mathematician), Pierre Cartier and André Weil by David Mumford). Both are derived from the notion of divisibility in the integers and algebraic number fields. Globally, every codimension-1 subvariety of projective space is defined by the vanishing of one homogeneous polynomial; by contrast, a codimension-''r'' subvariety need not be definable by only ''r'' equations when ''r'' is greater than 1. (That is, not every subvariety of projective space is a complete intersection.) Locally, every codimension-1 subvariety of a smooth scheme, smooth variety can be defined by one equation in a neighborhood of each point. Again, the analogous statement fails for higher-codimension subvarieties. As a result of this property, much of algebraic geometry studies an arbitrary variety ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Duke Mathematical Journal
''Duke Mathematical Journal'' is a peer-reviewed mathematics journal published by Duke University Press. It was established in 1935. The founding editors-in-chief were David Widder, Arthur Coble, and Joseph Miller Thomas Joseph Miller Thomas (16 January 1898 – 1979) was an American mathematician, known for the Thomas decomposition of algebraic and differential systems. Thomas received his Ph.D., supervised by Frederick Wahn Beal, from the University of Pennsylva .... The first issue included a paper by Solomon Lefschetz. Leonard Carlitz served on the editorial board for 35 years, from 1938 to 1973. The current managing editor is Richard Hain (Duke University). Impact According to the journal homepage, the journal has a 2018 impact factor of 2.194, ranking it in the top ten mathematics journals in the world. References External links * Mathematics journals Duke University, Mathematical Journal Publications established in 1935 Multilingual journals English-language jo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematische Annalen
''Mathematische Annalen'' (abbreviated as ''Math. Ann.'' or, formerly, ''Math. Annal.'') is a German mathematical research journal founded in 1868 by Alfred Clebsch and Carl Neumann. Subsequent managing editors were Felix Klein, David Hilbert, Otto Blumenthal, Erich Hecke, Heinrich Behnke, Hans Grauert, Heinz Bauer, Herbert Amann, Jean-Pierre Bourguignon, Wolfgang Lück, and Nigel Hitchin. Currently, the managing editor of Mathematische Annalen is Thomas Schick. Volumes 1–80 (1869–1919) were published by Teubner. Since 1920 (vol. 81), the journal has been published by Springer. In the late 1920s, under the editorship of Hilbert, the journal became embroiled in controversy over the participation of L. E. J. Brouwer on its editorial board, a spillover from the foundational Brouwer–Hilbert controversy. Between 1945 and 1947 the journal briefly ceased publication. References External links''Mathematische Annalen''homepage at Springer''Mathematische Annalen''archive (1869†... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
David Gieseker
David Arends Gieseker (born 23 November 1943 in Oakland, California) is an American mathematician, specializing in algebraic geometry. Gieseker received his bachelor's degree in 1965 from Reed College and his master's degree from Harvard University in 1967. In 1970 he received his Ph.D. under Robin Hartshorne with thesis ''Contributions to the Theory of Positive Embeddings in Algebraic Geometry''. Gieseker became a professor at the University of California, Los Angeles in 1975 and became professor emeritus in 2022. The topics of his research include geometric invariant theory and moduli of vector bundles over algebraic curves. Selected publications Articles *with Spencer Bloch: * * * * *with Jun Li (mathematician), Jun Li: *with Jun Li: BooksLectures on moduli of curves Tata Institute of Fundamental Research, Springer Verlag 1982; notes by D. R. Gokhale *with Eugene Trubowitz and Horst Knörrer''Geometry of algebraic Fermi curves'' Academic Press 1992 References External li ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Robert Lazarsfeld
Robert Kendall Lazarsfeld (born April 15, 1953) is an American mathematician, currently a professor at Stony Brook University. He was previously the Raymond L. Wilder Collegiate Professor of Mathematics at the University of Michigan. He is the son of two sociologists, Paul Lazarsfeld and Patricia Kendall. His research focuses on algebraic geometry. During 2002–2009, Lazarsfeld was an editor at the '' Journal of the American Mathematical Society'' (Managing Editor, 2007–2009). In 2012–2013, he served as the Managing Editor of the ''Michigan Mathematical Journal''. Lazarsfeld went to Harvard for undergraduate studies and earned his doctorate from Brown University in 1980 under supervision of William Fulton. In 2006 Lazarsfeld was elected a Fellow of the American Academy of Arts and Sciences. In 2012 he became a fellow of the American Mathematical Society. In 2015 he was awarded the AMS Leroy P. Steele Prize The Leroy P. Steele Prizes are awarded every year by the A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
William Fulton (mathematician)
William Edgar Fulton (born August 29, 1939) is an American mathematician, specializing in algebraic geometry. Education and career He received his undergraduate degree from Brown University in 1961 and his doctorate from Princeton University in 1966. His Ph.D. thesis, written under the supervision of Gerard Washnitzer, was on ''The fundamental group of an algebraic curve''. Fulton worked at Princeton and Brandeis University from 1965 until 1970, when he began teaching at Brown. In 1987 he moved to the University of Chicago.Announcement of the 1996 Steele Prizes at the American Mathematical Society web site, accessed July 15, 2009. He is, as of 2011, a professor at the University of Michigan. Fulton is known as the author or coauthor of a number of popular texts, including ''Algebraic Curves'' and ''Representation Theory'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
George Kempf
George Rushing Kempf (Globe, Arizona, August 12, 1944 – Lawrence, Kansas, July 16, 2002) was a mathematician who worked on algebraic geometry, who proved the Riemann–Kempf singularity theorem, the Kempf–Ness theorem, the Kempf vanishing theorem, and who introduced Kempf varieties. Mumford on Kempf 'I met George in 1970 when he burst on the algebraic geometry scene with a spectacular PhD thesis. His thesis gave a wonderful analysis of the singularities of the subvarieties W_r of the Jacobian of a curve obtained by adding the curve to itself r times inside its Jacobian. This was one of the major themes that he pursued throughout his career: understanding the interaction of a curve with its Jacobian and especially to the map from the r-fold symmetric product of the curve to the Jacobian. In his thesis he gave a determinantal representation both of W_r and of its tangent cone at all its singular points, which gives you a complete understanding of the nature of these singulariti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subscheme
This is a glossary of algebraic geometry. See also glossary of commutative algebra, glossary of classical algebraic geometry, and glossary of ring theory. For the number-theoretic applications, see glossary of arithmetic and Diophantine geometry. For simplicity, a reference to the base scheme is often omitted; i.e., a scheme will be a scheme over some fixed base scheme ''S'' and a morphism an ''S''-morphism. !$@ A B C D E F G H I J K L M N O P ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Divisor Class
In algebraic geometry, divisors are a generalization of codimension-1 subvarieties of algebraic varieties. Two different generalizations are in common use, Cartier divisors and Weil divisors (named for Pierre Cartier and André Weil by David Mumford). Both are derived from the notion of divisibility in the integers and algebraic number fields. Globally, every codimension-1 subvariety of projective space is defined by the vanishing of one homogeneous polynomial; by contrast, a codimension-''r'' subvariety need not be definable by only ''r'' equations when ''r'' is greater than 1. (That is, not every subvariety of projective space is a complete intersection.) Locally, every codimension-1 subvariety of a smooth variety can be defined by one equation in a neighborhood of each point. Again, the analogous statement fails for higher-codimension subvarieties. As a result of this property, much of algebraic geometry studies an arbitrary variety by analysing its codimension-1 subvarieties an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Picard Variety
In mathematics, the Picard group of a ringed space ''X'', denoted by Pic(''X''), is the group of isomorphism classes of invertible sheaves (or line bundles) on ''X'', with the group operation being tensor product. This construction is a global version of the construction of the divisor class group, or ideal class group, and is much used in algebraic geometry and the theory of complex manifolds. Alternatively, the Picard group can be defined as the sheaf cohomology group :H^1 (X, \mathcal_X^).\, For integral schemes the Picard group is isomorphic to the class group of Cartier divisors. For complex manifolds the exponential sheaf sequence gives basic information on the Picard group. The name is in honour of Émile Picard's theories, in particular of divisors on algebraic surfaces. Examples * The Picard group of the spectrum of a Dedekind domain is its ''ideal class group''. * The invertible sheaves on projective space P''n''(''k'') for ''k'' a field, are the twisting sh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
